This huge crank belongs to a Wartsila-Sulzer RTA96-C turbocharged two-stroke diesel engine. In this chapter you will learn to perform the kinematic

Transcription

1 This huge crank belongs to a Wartsila-Sulzer RT96- turbocharged two-stroke diesel engine. In this chapter ou will learn to perform the kinematic analsis of rigid bodies that undergo translation, fied ais rotation, and general plane motion. 914

2 15 H P T E R Kinematics of Rigid odies 915

3 hapter 15 Kinematics of Rigid odies 15.1 Introduction 15.2 Translation 15.3 Rotation about a Fied is 15.4 Equations efining the Rotation of a Rigid od about a Fied is 15.5 General Plane Motion 15.6 bsolute and Relative Velocit in Plane Motion 15.7 Instantaneous enter of Rotation in Plane Motion 15.8 bsolute and Relative cceleration in Plane Motion 15.9 nalsis of Plane Motion in Terms of a Parameter Rate of hange of a Vector with Respect to a Rotating Frame Plane Motion of a Particle Relative to a Rotating Frame. oriolis cceleration Motion about a Fied Point General Motion Three-imensional Motion of a Particle Relative to a Rotating Frame. oriolis cceleration Frame of Reference in General Motion INTRUTIN In this chapter, the kinematics of rigid bodies will be considered. You will investigate the relations eisting between the time, the positions, the velocities, and the accelerations of the various particles forming a rigid bod. s ou will see, the various tpes of rigid-bod motion can be convenientl grouped as follows: 1 Fig Translation. motion is said to be a translation if an straight line inside the bod keeps the same direction during the motion. It can also be observed that in a translation all the particles forming the bod move along parallel paths. If these paths are straight lines, the motion is said to be a rectilinear translation (Fig. 15.1); if the paths are curved lines, the motion is a curvilinear translation (Fig. 15.2). 2. Rotation about a Fied is. In this motion, the particles forming the rigid bod move in parallel planes along circles centered on the same fied ais (Fig. 15.3). If this ais, called the ais of rotation, intersects the rigid bod, the particles located on the ais have zero velocit and zero acceleration. Rotation should not be confused with certain tpes of curvilinear translation. For eample, the plate shown in Fig. 15.4a is in curvilinear translation, with all its particles moving along parallel circles, while the plate shown in Fig. 15.4b is in rotation, with all its particles moving along concentric circles Fig (b) Rotation 1 (a) urvilinear translation Fig Fig. 15.4

4 In the first case, an given straight line drawn on the plate will maintain the same direction, whereas in the second case, point remains fied. ecause each particle moves in a given plane, the rotation of a bod about a fied ais is said to be a plane motion. 3. General Plane Motion. There are man other tpes of plane motion, i.e., motions in which all the particles of the bod move in parallel planes. n plane motion which is neither a rotation nor a translation is referred to as a general plane motion. Two eamples of general plane motion are given in Fig Introduction 917 (a) Rolling wheel Fig (b) Sliding rod 4. Motion about a Fied Point. The three-dimensional motion of a rigid bod attached at a fied point, e.g., the motion of a top on a rough floor (Fig. 15.6), is known as motion about a fied point. 5. General Motion. n motion of a rigid bod which does not fall in an of the categories above is referred to as a general motion. fter a brief discussion in Sec of the motion of translation, the rotation of a rigid bod about a fied ais is considered in Sec The angular velocit and the angular acceleration of a rigid bod about a fied ais will be defined, and ou will learn to epress the velocit and the acceleration of a given point of the bod in terms of its position vector and the angular velocit and angular acceleration of the bod. The following sections are devoted to the stud of the general plane motion of a rigid bod and to its application to the analsis of mechanisms such as gears, connecting rods, and pin-connected linkages. Resolving the plane motion of a slab into a translation and a rotation (Secs and 15.6), we will then epress the velocit of a point of the slab as the sum of the velocit of a reference point and of the velocit of relative to a frame of reference translating with (i.e., moving with but not rotating). The same approach is used later in Sec to epress the acceleration of in terms of the acceleration of and of the acceleration of relative to a frame translating with. Fig. 15.6

5 918 Kinematics of Rigid odies n alternative method for the analsis of velocities in plane motion, based on the concept of instantaneous center of rotation, is given in Sec. 15.7; and still another method of analsis, based on the use of parametric epressions for the coordinates of a given point, is presented in Sec The motion of a particle relative to a rotating frame of reference and the concept of oriolis acceleration are discussed in Secs and 15.11, and the results obtained are applied to the analsis of the plane motion of mechanisms containing parts which slide on each other. The remaining part of the chapter is devoted to the analsis of the three-dimensional motion of a rigid bod, namel, the motion of a rigid bod with a fied point and the general motion of a rigid bod. In Secs and 15.13, a fied frame of reference or a frame of reference in translation will be used to carr out this analsis; in Secs and 15.15, the motion of the bod relative to a rotating frame or to a frame in general motion will be considered, and the concept of oriolis acceleration will again be used TRNSLTIN onsider a rigid bod in translation (either rectilinear or curvilinear translation), and let and be an two of its particles (Fig. 15.7a). enoting, respectivel, b r and r the position vectors of and with respect to a fied frame of reference and b r / the vector joining and, we write r 5 r 1 r / (15.1) Photo 15.1 This replica of a battering ram at hâteau des au, France undergoes curvilinear translation. Let us differentiate this relation with respect to t. We note that from the ver definition of a translation, the vector r / must maintain a constant direction; its magnitude must also be constant, since and a v a r / r v r z (a) z (b) z (c) Fig. 15.7

6 belong to the same rigid bod. Thus, the derivative of r / is zero and we have 15.3 Rotation about a Fied is 919 v 5 v (15.2) ifferentiating once more, we write a 5 a (15.3) Thus, when a rigid bod is in translation, all the points of the bod have the same velocit and the same acceleration at an given instant (Fig. 15.7b and c). In the case of curvilinear translation, the velocit and acceleration change in direction as well as in magnitude at ever instant. In the case of rectilinear translation, all particles of the bod move along parallel straight lines, and their velocit and acceleration keep the same direction during the entire motion RTTIN UT FIXE XIS onsider a rigid bod which rotates about a fied ais 9. Let P be a point of the bod and r its position vector with respect to a fied frame of reference. For convenience, let us assume that the frame is centered at point on 9 and that the z ais coincides with 9 (Fig. 15.8). Let be the projection of P on 9; since P must remain at a constant distance from, it will describe a circle of center and of radius r sin f, where f denotes the angle formed b r and 9. The position of P and of the entire bod is completel defined b the angle u the line P forms with the z plane. The angle u is known as the angular coordinate of the bod and is defined as positive when viewed as counterclockwise from 9. The angular coordinate will be epressed in radians (rad) or, occasionall, in degrees ( ) or revolutions (rev). We recall that 1 rev 5 2p rad We recall from Sec that the velocit v 5 dr/dt of a particle P is a vector tangent to the path of P and of magnitude v 5 ds/dt. bserving that the length s of the arc described b P when the bod rotates through u is s 5 (P) u 5 (r sin f) u and dividing both members b t, we obtain at the limit, as t approaches zero, v 5 ds dt 5 ru. sin f (15.4) where u denotes the time derivative of u. (Note that the angle u depends on the position of P within the bod, but the rate of change u is itself independent of P.) We conclude that the velocit v of P is a vector perpendicular to the plane containing 9 and r, and of Fig q f r Photo 15.2 For the central gear rotating about a fied ais, the angular velocit and angular acceleration of that gear are vectors directed along the vertical ais of rotation. P ' z

7 920 Kinematics of Rigid odies magnitude v defined b (15.4). ut this is precisel the result we would obtain if we drew along 9 a vector V 5 u k and formed the ' vector product V 3 r (Fig. 15.9). We thus write k i Fig j f w = qk r P v v 5 dr dt 5 V 3 r (15.5) The vector V 5 vk 5 u k (15.6) which is directed along the ais of rotation, is called the angular velocit of the bod and is equal in magnitude to the rate of change u of the angular coordinate; its sense ma be obtained b the righthand rule (Sec. 3.6) from the sense of rotation of the bod. The acceleration a of the particle P will now be determined. ifferentiating (15.5) and recalling the rule for the differentiation of a vector product (Sec ), we write a 5 dv dt 5 d (V 3 r) dt 5 dv dt 3 r 1 V 3 dr dt 5 dv dt 3 r 1 V 3 v (15.7) The vector dv/dt is denoted b and is called the angular acceleration of the bod. Substituting also for v from (15.5), we have a 5 3 r 1 V 3 (V 3 r) (15.8) ifferentiating (15.6) and recalling that k is constant in magnitude and direction, we have 5 ak 5 v k 5 ük (15.9) Thus, the angular acceleration of a bod rotating about a fied ais is a vector directed along the ais of rotation, and is equal in magnitude to the rate of change v of the angular velocit. Returning to (15.8), we note that the acceleration of P is the sum of two vectors. The first vector is equal to the vector product 3 r; it is tangent to the circle described b P and therefore represents the tangential component of the acceleration. The second vector is equal to the vector triple product V 3 (V 3 r) obtained b forming the vector product of V and V 3 r; since V 3 r is tangent to the circle described b P, the vector triple product is directed toward the center of the circle and therefore represents the normal component of the acceleration. It will be shown in Sec in the more general case of a rigid bod rotating simultaneousl about aes having different directions that angular velocities obe the parallelogram law of addition and thus are actuall vector quantities.

8 Rotation of a Representative Slab. The rotation of a rigid bod about a fied ais can be defined b the motion of a representative slab in a reference plane perpendicular to the ais of rotation. Let us choose the plane as the reference plane and assume that it coincides with the plane of the figure, with the z ais pointing out of the paper (Fig ). Recalling from (15.6) that V 5 vk, we 15.3 Rotation about a Fied is 921 v = wk r r P w = wk Fig note that a positive value of the scalar v corresponds to a counterclockwise rotation of the representative slab, and a negative value to a clockwise rotation. Substituting vk for V into Eq. (15.5), we epress the velocit of an given point P of the slab as v 5 vk 3 r (15.10) Since the vectors k and r are mutuall perpendicular, the magnitude of the velocit v is v 5 rv (15.109) and its direction can be obtained b rotating r through 90 in the sense of rotation of the slab. Substituting V 5 vk and 5 ak into Eq. (15.8), and observing that cross-multipling r twice b k results in a 180 rotation of the vector r, we epress the acceleration of point P as a 5 ak 3 r 2 v 2 r (15.11) Resolving a into tangential and normal components (Fig ), we write a t 5 ak 3 r a t 5 ra (15.119) a n 5 2v 2 r a n 5 rv 2 The tangential component a t points in the counterclockwise direction if the scalar a is positive, and in the clockwise direction if a is negative. The normal component a n alwas points in the direction opposite to that of r, that is, toward. Fig a t = a k r P a n = w 2 r w = wk a = ak

9 922 Kinematics of Rigid odies 15.4 EQUTINS EFINING THE RTTIN F RIGI Y UT FIXE XIS The motion of a rigid bod rotating about a fied ais 9 is said to be known when its angular coordinate u can be epressed as a known function of t. In practice, however, the rotation of a rigid bod is seldom defined b a relation between u and t. More often, the conditions of motion will be specified b the tpe of angular acceleration that the bod possesses. For eample, a ma be given as a function of t, as a function of u, or as a function of v. Recalling the relations (15.6) and (15.9), we write v 5 du dt (15.12) a 5 dv dt 5 d2 u dt 2 (15.13) or, solving (15.12) for dt and substituting into (15.13), Photo 15.3 If the lower roll has a constant angular velocit, the speed of the paper being wound onto it increases as the radius of the roll increases. dv a 5 v du (15.14) Since these equations are similar to those obtained in hap. 11 for the rectilinear motion of a particle, their integration can be performed b following the procedure outlined in Sec Two particular cases of rotation are frequentl encountered: 1. Uniform Rotation. This case is characterized b the fact that the angular acceleration is zero. The angular velocit is thus constant, and the angular coordinate is given b the formula u 5 u 0 1 vt (15.15) 2. Uniforml ccelerated Rotation. In this case, the angular acceleration is constant. The following formulas relating angular velocit, angular coordinate, and time can then be derived in a manner similar to that described in Sec The similarit between the formulas derived here and those obtained for the rectilinear uniforml accelerated motion of a particle is apparent. v 5 v 0 1 at u 5 u 0 1 v 0 t 1 1 2at 2 (15.16) v 2 5 v a(u 2 u 0 ) It should be emphasized that formula (15.15) can be used onl when a 5 0, and formulas (15.16) can be used onl when a 5 constant. In an other case, the general formulas (15.12) to (15.14) should be used.

10 SMPLE PRLEM in. 5 in. Load is connected to a double pulle b one of the two inetensible cables shown. The motion of the pulle is controlled b cable, which has a constant acceleration of 9 in./s 2 and an initial velocit of 12 in./s, both directed to the right. etermine (a) the number of revolutions eecuted b the pulle in 2 s, (b) the velocit and change in position of the load after 2 s, and (c) the acceleration of point on the rim of the inner pulle at t 5 0. SLUTIN v w v a. Motion of Pulle. Since the cable is inetensible, the velocit of point is equal to the velocit of point and the tangential component of the acceleration of is equal to the acceleration of. (v ) 0 5 (v ) in./s (a ) t 5 a 5 9 in./s 2 Noting that the distance from to the center of the pulle is 3 in., we write (v ) 0 5 rv 0 12 in./s 5 (3 in.)v 0 V rad/s i (a ) t 5 ra 9 in./s 2 5 (3 in.)a 5 3 rad/s 2 i (a ) n v (a ) t w a a Using the equations of uniforml accelerated motion, we obtain, for t 5 2 s, v 5 v 0 1 at 5 4 rad/s 1 (3 rad/s 2 )(2 s) 5 10 rad/s V 5 10 rad/s i u 5 v 0 t 1 1 2at 2 5 (4 rad/s)(2 s) 1 1 2(3 rad/s 2 )(2 s) rad u 5 14 rad i Number of revolutions 5 (14 rad)a 1 rev b rev 2p rad b. Motion of Load. Using the following relations between linear and angular motion, with r 5 5 in., we write v 5 rv 5 (5 in.)(10 rad/s) 5 50 in./s v 5 50 in./s 5 ru 5 (5 in.)(14 rad) 5 70 in in. upward a (a ) n = 48 in./s 2 (a ) t = 9 in./s 2 f a c. cceleration of Point at t 5 0. The tangential component of the acceleration is (a ) t 5 a 5 9 in./s 2 Since, at t 5 0, v rad/s, the normal component of the acceleration is (a ) n 5 r v (3 in.)(4 rad/s) in./s 2 (a ) n 5 48 in./s 2 w The magnitude and direction of the total acceleration can be obtained b writing tan f 5 (48 in./s 2 )/(9 in./s 2 ) f a sin in./s 2 a in./s 2 a in./s 2 c

11 SLVING PRLEMS N YUR WN In this lesson we began the stud of the motion of rigid bodies b considering two particular tpes of motion of rigid bodies: translation and rotation about a fied ais. 1. Rigid bod in translation. t an given instant, all the points of a rigid bod in translation have the same velocit and the same acceleration (Fig. 15.7). 2. Rigid bod rotating about a fied ais. The position of a rigid bod rotating about a fied ais was defined at an given instant b the angular coordinate u, which is usuall measured in radians. Selecting the unit vector k along the fied ais and in such a wa that the rotation of the bod appears counterclockwise as seen from the tip of k, we defined the angular velocit V and the angular acceleration of the bod: V 5 u k 5 ük (15.6, 15.9) In solving problems, keep in mind that the vectors V and are both directed along the fied ais of rotation and that their sense can be obtained b the righthand rule. a. The velocit of a point P of a bod rotating about a fied ais was found to be v 5 V 3 r (15.5) where V is the angular velocit of the bod and r is the position vector drawn from an point on the ais of rotation to point P (Fig. 15.9). b. The acceleration of point P was found to be a 5 3 r 1 V 3 (V 3 r) (15.8) Since vector products are not commutative, be sure to write the vectors in the order shown when using either of the above two equations. 3. Rotation of a representative slab. In man problems, ou will be able to reduce the analsis of the rotation of a three-dimensional bod about a fied ais to the stud of the rotation of a representative slab in a plane perpendicular to the fied ais. The z ais should be directed along the ais of rotation and point out of the paper. Thus, the representative slab will be rotating in the plane about the origin of the coordinate sstem (Fig ). To solve problems of this tpe ou should do the following: a. raw a diagram of the representative slab, showing its dimensions, its angular velocit and angular acceleration, as well as the vectors representing the velocities and accelerations of the points of the slab for which ou have or seek information. 924

12 b. Relate the rotation of the slab and the motion of points of the slab b writing the equations v 5 rv (15.109) a t 5 ra a n 5 rv 2 (15.119) Remember that the velocit v and the component a t of the acceleration of a point P of the slab are tangent to the circular path described b P. The directions of v and a t are found b rotating the position vector r through 90 in the sense indicated b V and, respectivel. The normal component a n of the acceleration of P is alwas directed toward the ais of rotation. 4. Equations defining the rotation of a rigid bod. You must have been pleased to note the similarit eisting between the equations defining the rotation of a rigid bod about a fied ais [Eqs. (15.12) through (15.16)] and those in hap. 11 defining the rectilinear motion of a particle [Eqs. (11.1) through (11.8)]. ll ou have to do to obtain the new set of equations is to substitute u, v, and a for, v, and a in the equations of hap

13 PRLEMS 15.1 The motion of a cam is defined b the relation u 5 t 3 2 9t t, where u is epressed in radians and t in seconds. etermine the angular coordinate, the angular velocit, and the angular acceleration of the cam when (a) t 5 0, (b) t 5 3 s For the cam of Prob. 15.1, determine the time, angular coordinate, and angular acceleration when the angular velocit is zero The motion of an oscillating crank is defined b the relation u 5 u 0 sin (pt/t) 2 (0.5u 0 ) sin (2 pt/t) where u is epressed in radians and t in seconds. Knowing that u rad and T 5 4 s, determine the angular coordinate, the angular velocit, and the angular acceleration of the crank when (a) t 5 0, (b) t 5 2 s Solve Prob. 15.4, when t 5 1 s The motion of a disk rotating in an oil bath is defined b the relation u 5 u 0 (1 2 e 2t/4 ), where u is epressed in radians and t in seconds. Knowing that u rad, determine the angular coordinate, velocit, and acceleration of the disk when (a) t 5 0, (b) t 5 3 s, (c) t 5 ` The angular acceleration of an oscillating disk is defined b the relation a 5 2ku. etermine (a) the value of k for which v 5 8 rad/s when u 5 0 and u 5 4 rad when v 5 0, (b) the angular velocit of the disk when u 5 3 rad. Fig. P When the power to an electric motor is turned on the motor reaches its rated speed of 3300 rpm in 6 s, and when the power is turned off the motor coasts to rest in 80 s. ssuming uniforml accelerated motion, determine the number of revolutions that the motor eecutes (a) in reaching its rated speed, (b) in coasting to rest The rotor of a gas turbine is rotating at a speed of 6900 rpm when the turbine is shut down. It is observed that 4 min is required for the rotor to coast to rest. ssuming uniforml accelerated motion, determine (a) the angular acceleration, (b) the number of revolutions that the rotor eecutes before coming to rest The angular acceleration of a shaft is defined b the relation a v, where a is epressed in rad/s 2 and v in rad/s. Knowing that at t 5 0 the angular velocit of the shaft is 20 rad/s, determine (a) the number of revolutions the shaft will eecute before coming to rest, (b) the time required for the shaft to come to rest, (c) the time required for the angular velocit of the shaft to be reduced to 1 percent of its initial value.

14 15.10 The assembl shown consists of the straight rod which passes through and is welded to the rectangular plate EFH. The assembl rotates about the ais with a constant angular velocit of 9 rad/s. Knowing that the motion when viewed from is counterclockwise, determine the velocit and acceleration of corner F. Problems mm 175 mm z F E 100 mm 100 mm 100 mm H 100 mm Fig. P In Prob , determine the acceleration of corner H, assuming that the angular velocit is 9 rad/s and decreases at a rate of 18 rad/s The bent rod E rotates about a line joining points and E with a constant angular velocit of 9 rad/s. Knowing that the rotation is clockwise as viewed from E, determine the velocit and acceleration of corner. 200 mm 250 mm 150 mm z 400 mm E 150 mm Fig. P In Prob , determine the velocit and acceleration of corner, assuming that the angular velocit is 9 rad/s and increases at the rate of 45 rad/s 2.

15 928 Kinematics of Rigid odies triangular plate and two rectangular plates are welded to each other and to the straight rod. The entire welded unit rotates about ais with a constant angular velocit of 5 rad/s. Knowing that at the instant considered the velocit of corner E is directed downward, determine the velocit and acceleration of corner. z E 350 mm 300 mm 400 mm Fig. P In Prob , determine the acceleration of corner, assuming that the angular velocit is 5 rad/s and decreases at the rate of 20 rad/s The earth makes one complete revolution on its ais in 23 h 56 min. Knowing that the mean radius of the earth is 3960 mi, determine the linear velocit and acceleration of a point on the surface of the earth (a) at the equator, (b) at Philadelphia, latitude 40 north, (c) at the North Pole The earth makes one complete revolution around the sun in das. ssuming that the orbit of the earth is circular and has a radius of 93,000,000 mi, determine the velocit and acceleration of the earth. q r The circular plate shown is initiall at rest. Knowing that r mm and that the plate has a constant angular acceleration of 0.3 rad/s 2, determine the magnitude of the total acceleration of point when (a) t 5 0, (b) t 5 2 s, (c) t 5 4 s The angular acceleration of the 600-mm-radius circular plate shown is defined b the relation a 5 a 0 e 2t. Knowing that the plate is at rest when t 5 0 and that a rad/s 2, determine the magnitude of the total acceleration of point when (a) t 5 0, (b) t s, (c) t 5 `. Fig. P15.18, P15.19, and P15.20 a w The 250-mm-radius circular plate shown is initiall at rest and has an angular acceleration defined b the relation a 5 a 0 cos (pt/t). Knowing that T s and a rad/s 2, determine the magnitude of the total acceleration of point when (a) t 5 0, (b) t s, (c) t s.

16 15.21 series of small machine components being moved b a conveor belt pass over a 6-in.-radius idler pulle. t the instant shown, the velocit of point is 15 in./s to the left and its acceleration is 9 in./s 2 to the right. etermine (a) the angular velocit and angular acceleration of the idler pulle, (b) the total acceleration of the machine component at. Problems in. Fig. P15.21 and P series of small machine components being moved b a conveor belt pass over a 6-in.-radius idler pulle. t the instant shown, the angular velocit of the idler pulle is 4 rad/s clockwise. etermine the angular acceleration of the pulle for which the magnitude of the total acceleration of the machine component at is 120 in./s The belt sander shown is initiall at rest. If the driving drum has a constant angular acceleration of 120 rad/s 2 counterclockwise, determine the magnitude of the acceleration of the belt at point when (a) t s, (b) t 5 2 s. 25 mm Fig. P15.23 and P mm The rated speed of drum of the belt sander shown is 2400 rpm. When the power is turned off, it is observed that the sander coasts from its rated speed to rest in 10 s. ssuming uniforml decelerated motion, determine the velocit and acceleration of point of the belt, (a) immediatel before the power is turned off, (b) 9 s later.

17 930 Kinematics of Rigid odies Ring has an inside radius of 55 mm and an outside radius of 60 mm and is positioned between two wheels and, each of 24 mm 24-mm outside radius. Knowing that wheel rotates with a constant angular velocit of 300 rpm and that no slipping occurs, determine (a) the angular velocit of the ring and of wheel, (b) the acceleration of the points and which are in contact with. Fig. P mm Ring has an inside radius r 2 and hangs from the horizontal shaft as shown. Knowing that shaft rotates with a constant angular velocit v and that no slipping occurs, derive a relation in terms of r 1, r 2, r 3, and v for (a) the angular velocit of ring, (b) the accelerations of the points of shaft and ring which are in contact. r 1 r 2 z r 3 Fig. P15.26 and P Ring has an inside radius r 2 and hangs from the horizontal shaft as shown. Shaft rotates with a constant angular velocit of 25 rad/s and no slipping occurs. Knowing that r mm, r mm, and r mm, determine (a) the angular velocit of ring, (b) the accelerations of the points of shaft and ring which are in contact, (c) the magnitude of the acceleration of a point on the outside surface of ring ft linder is moving downward with a velocit of 9 ft/s when the brake is suddenl applied to the drum. Knowing that the clinder moves 18 ft downward before coming to rest and assuming uniforml accelerated motion, determine (a) the angular acceleration of the drum, (b) the time required for the clinder to come to rest. Fig. P15.28 and P The sstem shown is held at rest b the brake-and-drum sstem. fter the brake is partiall released at t 5 0, it is observed that the clinder moves 16 ft in 5 s. ssuming uniforml accelerated motion, determine (a) the angular acceleration of the drum, (b) the angular velocit of the drum at t 5 4 s.

18 15.30 pulle and two loads are connected b inetensible cords as shown. Load has a constant acceleration of 300 mm/s 2 and an initial velocit of 240 mm/s, both directed upward. etermine (a) the number of revolutions eecuted b the pulle in 3 s, (b) the velocit and position of load after 3 s, (c) the acceleration of point on the rim of the pulle at t mm 120 mm Problems pulle and two loads are connected b inetensible cords as shown. The pulle starts from rest at t 5 0 and is accelerated at the uniform rate of 2.4 rad/s 2 clockwise. t t 5 4 s, determine the velocit and position (a) of load, (b) of load isk is at rest when it is brought into contact with disk which is rotating freel at 450 rpm clockwise. fter 6 s of slippage, during which each disk has a constant angular acceleration, disk reaches a final angular velocit of 140 rpm clockwise. etermine the angular acceleration of each disk during the period of slippage. Fig. P15.30 and P in. 5 in. Fig. P15.32 and P and simple friction drive consists of two disks and. Initiall, disk has a clockwise angular velocit of 500 rpm and disk is at rest. It is known that disk will coast to rest in 60 s. However, rather than waiting until both disks are at rest to bring them together, disk is given a constant angular acceleration of 2.5 rad/s 2 counterclockwise. etermine (a) at what time the disks can be brought together if the are not to slip, (b) the angular velocit of each disk as contact is made Two friction disks and are both rotating freel at 240 rpm counterclockwise when the are brought into contact. fter 8 s of slippage, during which each disk has a constant angular acceleration, disk reaches a final angular velocit of 60 rpm counterclockwise. etermine (a) the angular acceleration of each disk during the period of slippage, (b) the time at which the angular velocit of disk is equal to zero. 80 mm 60 mm Fig. P15.34 and P15.35

19 932 Kinematics of Rigid odies *15.36 In a continuous printing process, paper is drawn into the presses at a constant speed v. enoting b r the radius of the paper roll at an given time and b b the thickness of the paper, derive an epression for the angular acceleration of the paper roll. a b v w r a b Fig. P15.36 Fig. P15.37 w *15.37 Television recording tape is being rewound on a VR reel which rotates with a constant angular velocit v 0. enoting b r the radius of the reel and tape at an given time and b b the thickness of the tape, derive an epression for the acceleration of the tape as it approaches the reel GENERL PLNE MTIN s indicated in Sec. 15.1, we understand b general plane motion a plane motion which is neither a translation nor a rotation. s ou will presentl see, however, a general plane motion can alwas be considered as the sum of a translation and a rotation. onsider, for eample, a wheel rolling on a straight track (Fig ). ver a certain interval of time, two given points and will have moved, respectivel, from 1 to 2 and from 1 to 2. The same result could be obtained through a translation which would bring and into 2 and 9 1 (the line remaining vertical), followed b a rotation about bringing into 2. lthough the original rolling motion differs from the combination of translation and rotation when these motions are taken in succession, the original motion can be eactl duplicated b a combination of simultaneous translation and rotation. 1 1 ' 1 ' 1 2 = Fig Plane motion = Translation with + Rotation about

20 1 1 ' 1 ' General Plane Motion = Plane motion = Translation with + Rotation about (a) = ' 1 ' 1 Plane motion = Translation with + Rotation about (b) Fig nother eample of plane motion is given in Fig , which represents a rod whose etremities slide along a horizontal and a vertical track, respectivel. This motion can be replaced b a translation in a horizontal direction and a rotation about (Fig a) or b a translation in a vertical direction and a rotation about (Fig b). In the general case of plane motion, we will consider a small displacement which brings two particles and of a representative slab, respectivel, from 1 and 1 into 2 and 2 (Fig ). This displacement can be divided into two parts: in one, the particles move into 2 and 9 1 while the line maintains the same direction; in the other, moves into 2 while remains fied. The first part of the motion is clearl a translation and the second part a rotation about. Recalling from Sec the definition of the relative motion of a particle with respect to a moving frame of reference as opposed to its absolute motion with respect to a fied frame of reference we can restate as follows the result obtained above: Given two particles and of a rigid slab in plane motion, the relative motion of with respect to a frame attached to and of fied orientation is a rotation. To an observer moving with but not rotating, particle will appear to describe an arc of circle centered at Fig ' 1 2

21 934 Kinematics of Rigid odies 15.6 SLUTE N RELTIVE VELITY IN PLNE MTIN We saw in the preceding section that an plane motion of a slab can be replaced b a translation defined b the motion of an arbitrar reference point and a simultaneous rotation about. The absolute velocit v of a particle of the slab is obtained from the relativevelocit formula derived in Sec , v 5 v 1 v / (15.17) Photo 15.4 Planetar gear sstems are used to high reduction ratios with minimum space and weight. The small gears undergo general plane motion. v where the right-hand member represents a vector sum. The velocit v corresponds to the translation of the slab with, while the relative velocit v / is associated with the rotation of the slab about and is measured with respect to aes centered at and of fied orientation (Fig ). enoting b r / the position vector of relative to, and b vk the angular velocit of the slab with respect to aes of fied orientation, we have from (15.10) and (15.109) v / 5 vk 3 r / v / 5 rv (15.18) v ' v v = + (fied) wk r / ' v/ v v / v Plane motion = Translation with + Rotation about Fig v = v + v / where r is the distance from to. Substituting for v / from (15.18) into (15.17), we can also write v 5 v 1 vk 3 r / (15.179) s an eample, let us again consider the rod of Fig ssuming that the velocit v of end is known, we propose to find the velocit v of end and the angular velocit V of the rod, in terms of the velocit v, the length l, and the angle u. hoosing as a reference point, we epress that the given motion is equivalent to a translation with and a simultaneous rotation about (Fig ). The absolute velocit of must therefore be equal to the vector sum v 5 v 1 v / (15.17) We note that while the direction of v / is known, its magnitude lv is unknown. However, this is compensated for b the fact that the direction of v is known. We can therefore complete the diagram of Fig Solving for the magnitudes v and v, we write v 5 v tan u v 5 v / l 5 v l cos u (15.19)

22 v v / 15.6 bsolute and Relative Velocit in Plane Motion 935 v q l q l = + q l w v v (fied) Plane motion = Translation with + Rotation about Fig v q v v / v = v + v / The same result can be obtained b using as a point of reference. Resolving the given motion into a translation with and a simultaneous rotation about (Fig ), we write the equation v 5 v 1 v / (15.20) which is represented graphicall in Fig We note that v / and v / have the same magnitude lv but opposite sense. The sense of the relative velocit depends, therefore, upon the point of reference which has been selected and should be carefull ascertained from the appropriate diagram (Fig or 15.17). (fied) w v q l = v q l + l v v q v / v v / v Plane motion = Translation with + Rotation about v = v + v / Fig Finall, we observe that the angular velocit V of the rod in its rotation about is the same as in its rotation about. It is measured in both cases b the rate of change of the angle u. This result is quite general; we should therefore bear in mind that the angular velocit V of a rigid bod in plane motion is independent of the reference point. Most mechanisms consist not of one but of several moving parts. When the various parts of a mechanism are pin-connected, the analsis of the mechanism can be carried out b considering each part as a rigid bod, keeping in mind that the points where two parts are connected must have the same absolute velocit (see Sample Prob. 15.3). similar analsis can be used when gears are involved, since the teeth in contact must also have the same absolute velocit. However, when a mechanism contains parts which slide on each other, the relative velocit of the parts in contact must be taken into account (see Secs and 15.11).

23 R SMPLE PRLEM 15.2 r 1 = 150 mm v = 1.2 m/s r 2 = 100 mm The double gear shown rolls on the stationar lower rack; the velocit of its center is 1.2 m/s directed to the right. etermine (a) the angular velocit of the gear, (b) the velocities of the upper rack R and of point of the gear. SLUTIN a. ngular Velocit of the Gear. Since the gear rolls on the lower rack, its center moves through a distance equal to the outer circumference 2pr 1 for each full revolution of the gear. Noting that 1 rev 5 2p rad, and that when moves to the right (. 0) the gear rotates clockwise (u, 0), we write 2pr 1 52 u 2p 52r 1 u ifferentiating with respect to the time t and substituting the known values v m/s and r mm m, we obtain v 5 2r 1 v 1.2 m/s 5 2(0.150 m)v v 5 28 rad/s V 5 vk 5 2(8 rad/s)k where k is a unit vector pointing out of the paper. b. Velocities. The rolling motion is resolved into two component motions: a translation with the center and a rotation about the center. In the translation, all points of the gear move with the same velocit v. In the rotation, each point P of the gear moves about with a relative velocit v P/ 5 vk 3 r P/, where r P/ is the position vector of P relative to. v / v / v v v v w = 8k v + = (fied) v v v / v = 0 v / v Translation + Rotation = Rolling Motion Velocit of Upper Rack. The velocit of the upper rack is equal to the velocit of point ; we write v R 5 v 5 v 1 v / 5 v 1 vk 3 r / 5 (1.2 m/s)i 2 (8 rad/s)k 3 (0.100 m)j 5 (1.2 m/s)i 1 (0.8 m/s)i 5 (2 m/s)i v R 5 2 m/s Velocit of Point v v 5 v 1 v / 5 v 1 vk 3 r / 5 (1.2 m/s)i 2 (8 rad/s)k 3 ( m)i 5 (1.2 m/s)i 1 (1.2 m/s)j v m/s a

24 l = 8 in. r = 3 in. G 40 b P SMPLE PRLEM 15.3 In the engine sstem shown, the crank has a constant clockwise angular velocit of 2000 rpm. For the crank position indicated, determine (a) the angular velocit of the connecting rod, (b) the velocit of the piston P. SLUTIN w 3 in v Motion of rank. The crank rotates about point. Epressing v in rad/s and writing v 5 rv, we obtain v 5 a2000 rev min b a 1 min 2p rad b a b rad/s 60 s 1 rev v 5 ()v 5 (3 in.)(209.4 rad/s) in./s v in./s c 50 Motion of onnecting Rod. We consider this motion as a general plane motion. Using the law of sines, we compute the angle b between the connecting rod and the horizontal: sin 40 8 in. 5 sin b 3 in. b The velocit v of the point where the rod is attached to the piston must be horizontal, while the velocit of point is equal to the velocit v obtained above. Resolving the motion of into a translation with and a rotation about, we obtain v 50 v b = 50 v v 50 (fied) w b = l Plane motion = Translation + Rotation + v / v v = in./s v / b = Epressing the relation between the velocities v, v, and v /, we write v 5 v 1 v / We draw the vector diagram corresponding to this equation. Recalling that b , we determine the angles of the triangle and write v sin v / in./s 5 sin 50 sin v / in./s v / in./s a v in./s ft/s v ft/s v P 5 v ft/s Since v / 5 lv, we have in./s 5 (8 in.)v V rad/s l 937

25 SLVING PRLEMS N YUR WN In this lesson ou learned to analze the velocit of bodies in general plane motion. You found that a general plane motion can alwas be considered as the sum of the two motions ou studied in the last lesson, namel, a translation and a rotation. To solve a problem involving the velocit of a bod in plane motion ou should take the following steps. 1. Whenever possible determine the velocit of the points of the bod where the bod is connected to another bod whose motion is known. That other bod ma be an arm or crank rotating with a given angular velocit [Sample Prob. 15.3]. 2. Net start drawing a diagram equation to use in our solution (Figs and 15.16). This equation will consist of the following diagrams. a. Plane motion diagram: raw a diagram of the bod including all dimensions and showing those points for which ou know or seek the velocit. b. Translation diagram: Select a reference point for which ou know the direction and/or the magnitude of the velocit v, and draw a second diagram showing the bod in translation with all of its points having the same velocit v. c. Rotation diagram: onsider point as a fied point and draw a diagram showing the bod in rotation about. Show the angular velocit V 5 vk of the bod and the relative velocities with respect to of the other points, such as the velocit v / of relative to. 3. Write the relative-velocit formula v 5 v 1 v / While ou can solve this vector equation analticall b writing the corresponding scalar equations, ou will usuall find it easier to solve it b using a vector triangle (Fig ). 4. different reference point can be used to obtain an equivalent solution. For eample, if point is selected as the reference point, the velocit of point is epressed as v 5 v 1 v / Note that the relative velocities v / and v / have the same magnitude but opposite sense. Relative velocities, therefore, depend upon the reference point that has been selected. The angular velocit, however, is independent of the choice of reference point. 938

26 PRLEMS The motion of rod is guided b pins attached at and which slide in the slots shown. t the instant shown, u 5 40 and the pin at moves upward to the left with a constant velocit of 6 in./s. etermine (a) the angular velocit of the rod, (b) the velocit of the pin at end The motion of rod is guided b pins attached at and which slide in the slots shown. t the instant shown, u 5 30 and the pin at moves downward with a constant velocit of 9 in./s. etermine (a) the angular velocit of the rod, (b) the velocit of the pin at end. q 20 in Small wheels have been attached to the ends of rod and roll freel along the surfaces shown. Knowing that wheel moves to the left with a constant velocit of 1.5 m/s, determine (a) the angular velocit of the rod, (b) the velocit of end of the rod ollar moves upward with a constant velocit of 1.2 m/s. t the instant shown when u 5 25, determine (a) the angular velocit of rod, (b) the velocit of collar. 15 Fig. P15.38 and P mm Fig. P15.40 q 500 mm Fig. P15.41 and P ollar moves downward to the left with a constant velocit of 1.6 m/s. t the instant shown when u 5 40, determine (a) the angular velocit of rod, (b) the velocit of collar. 939

27 940 Kinematics of Rigid odies Rod moves over a small wheel at while end moves to the right with a constant velocit of 500 mm/s. t the instant shown, determine (a) the angular velocit of the rod, (b) the velocit of end of the rod. 400 mm 140 mm The plate shown moves in the plane. Knowing that (v ) 5 12 in./s, (v ) 5 24 in./s, and (v ) in./s, determine (a) the angular velocit of the plate, (b) the velocit of point. v = (v ) i + (v ) j Fig. P mm 4 in. 2 in. v = (v ) i + (v ) j v = (v ) i + (v ) j 6 in. 2 in. Fig. P In Prob , determine (a) the velocit of point, (b) the point on the plate with zero velocit The plate shown moves in the plane. Knowing that (v ) mm/s, (v ) mm/s, and (v ) mm/s, determine (a) the angular velocit of the plate, (b) the velocit of point. v = (v ) i + (v ) j 180 mm v = (v ) i + (v ) j 180 mm v = (v ) i + (v ) j 180 mm 180 mm Fig. P In Prob , determine (a) the velocit of point, (b) the point of the plate with zero velocit.

28 15.48 In the planetar gear sstem shown, the radius of gears,,, and is 3 in. and the radius of the outer gear E is 9 in. Knowing that gear E has an angular velocit of 120 rpm clockwise and that the central gear has an angular velocit of 150 rpm clockwise, determine (a) the angular velocit of each planetar gear, (b) the angular velocit of the spider connecting the planetar gears. Problems 941 E Fig. P15.48 and P In the planetar gear sstem shown the radius of the central gear is a, the radius of each of the planetar gears is b, and the radius of the outer gear E is a 1 2b. The angular velocit of gear is V clockwise, and the outer gear is stationar. If the angular velocit of the spider is to be V /5 clockwise, determine (a) the required value of the ratio b/a, (b) the corresponding angular velocit of each planetar gear Gear rotates with an angular velocit of 120 rpm clockwise. Knowing that the angular velocit of arm is 90 rpm clockwise, determine the corresponding angular velocit of gear. 90 mm 60 mm 120 mm Fig. P15.50 and P rm rotates with an angular velocit of 42 rpm clockwise. etermine the required angular velocit of gear for which (a) the angular velocit of gear is 20 rpm counterclockwise, (b) the motion of gear is a curvilinear translation. 50 mm rm rotates with an angular velocit of 20 rad/s counterclockwise. Knowing that the outer gear is stationar, determine (a) the angular velocit of gear, (b) the velocit of the gear tooth located at point. Fig. P15.52

29 942 Kinematics of Rigid odies and rm rotates about point with an angular velocit of 40 rad/s counterclockwise. Two friction disks and are pinned at their centers to arm as shown. Knowing that the disks roll without slipping at surfaces of contact, determine the angular velocit of (a) disk, (b) disk. 2.4 in. 1.2 in. 0.9 in. 0.3 in. 2.4 in. 1.8 in. 1.5 in. 0.6 in. 1.5 in. 0.6 in. Fig. P15.53 Fig. P Knowing that crank has a constant angular velocit of 160 rpm counterclockwise, determine the angular velocit of rod and the velocit of collar when (a) u 5 0, (b) u in. 3 in. q 6 in. Fig. P15.55 and P Knowing that crank has a constant angular velocit of 160 rpm counterclockwise, determine the angular velocit of rod and the velocit of collar when u 5 60.

30 15.57 In the engine sstem shown, l mm and b 5 60 mm. Knowing that the crank rotates with a constant angular velocit of 1000 rpm clockwise, determine the velocit of the piston P and the angular velocit of the connecting rod when (a) u 5 0, (b) u P Problems In the engine sstem shown in Fig. P15.57 and P15.58, l mm and b 5 60 mm. Knowing that crank rotates with a constant angular velocit of 1000 rpm clockwise, determine the velocit of the piston F and the angular velocit of the connecting rod when u l straight rack rests on a gear of radius r and is attached to a block as shown. enoting b v the clockwise angular velocit of gear and b u the angle formed b the rack and the horizontal, derive epressions for the velocit of block and the angular velocit of the rack in terms of r, u, and v. q b Fig. P15.57 and P15.58 r q Fig. P15.59, P15.60, and P straight rack rests on a gear of radius r 5 75 mm and is attached to a block as shown. Knowing that at the instant shown the angular velocit of gear is 15 rpm counterclockwise and u 5 20, determine (a) the velocit of block, (b) the angular velocit of the rack straight rack rests on a gear of radius r 5 60 mm and is attached to a block as shown. Knowing that at the instant shown the velocit of block is 200 mm/s to the right and u 5 25, determine (a) the angular velocit of gear, (b) the angular velocit of the rack In the eccentric shown, a disk of 2-in.-radius revolves about shaft that is located 0.5 in. from the center of the disk. The distance between the center of the disk and the pin at is 8 in. Knowing that the angular velocit of the disk is 900 rpm clockwise, determine the velocit of the block when u q 2 in. 1 in. 2 Fig. P in.

31 944 Kinematics of Rigid odies through In the position shown, bar has an angular velocit of 4 rad/s clockwise. etermine the angular velocit of 3 in. 8 in. 7 in. 4 in. bars and E. 250 mm 150 mm 60 mm 100 mm Fig. P15.63 E Fig. P15.64 E 300 mm E 500 mm 400 mm Fig. P mm In the position shown, bar E has a constant angular velocit of 10 rad/s clockwise. Knowing that h mm, determine (a) the angular velocit of bar F, (b) the velocit of point F. F 100 mm 200 mm E 120 mm h Fig. P15.66 and P mm 100 mm

32 15.67 In the position shown, bar E has a constant angular velocit of 10 rad/s clockwise. etermine (a) the distance h for which the velocit of point F is vertical, (b) the corresponding velocit of point F. Problems In the position shown, bar has zero angular acceleration and an angular velocit of 20 rad/s counterclockwise. etermine (a) the angular velocit of member H, (b) the velocit of point G. 3 in. 5 in. 5 in. 3 in. E 4 in. G 10 in. H Fig. P15.68 and P In the position shown, bar has zero angular acceleration and an angular velocit of 20 rad/s counterclockwise. etermine (a) the angular velocit of member H, (b) the velocit of point H n automobile travels to the right at a constant speed of 48 mi/h. If the diameter of a wheel is 22 in., determine the velocities of points,,, and E on the rim of the wheel E 22 in. Fig. P15.70

33 946 Kinematics of Rigid odies The 80-mm-radius wheel shown rolls to the left with a velocit of 900 mm/s. Knowing that the distance is 50 mm, determine the velocit of the collar and the angular velocit of rod when (a) b 5 0, (b) b mm 250 mm 160 mm b Fig. P15.71 *15.72 For the gearing shown, derive an epression for the angular velocit v of gear and show that v is independent of the radius of gear. ssume that point is fied and denote the angular velocities of rod and gear b v and v respectivel. r r r Fig. P15.72 Photo 15.5 If the tires of this car are rolling without sliding the instantaneous center of rotation of a tire is the point of contact between the road and the tire INSTNTNEUS ENTER F RTTIN IN PLNE MTIN onsider the general plane motion of a slab. We propose to show that at an given instant the velocities of the various particles of the slab are the same as if the slab were rotating about a certain ais perpendicular to the plane of the slab, called the instantaneous ais of rotation. This ais intersects the plane of the slab at a point, called the instantaneous center of rotation of the slab. We first recall that the plane motion of a slab can alwas be replaced b a translation defined b the motion of an arbitrar reference point and b a rotation about. s far as the velocities are concerned, the translation is characterized b the velocit v of the reference point and the rotation is characterized b the angular velocit V of the slab (which is independent of the choice of ). Thus, the velocit v of point and the angular velocit V of the slab define

34 w 15.7 Instantaneous enter of Rotation in Plane Motion 947 r = v /w w v v Fig (a) (b) completel the velocities of all the other particles of the slab (Fig a). Now let us assume that v and V are known and that the are both different from zero. (If v 5 0, point is itself the instantaneous center of rotation, and if V 5 0, all the particles have the same velocit v.) These velocities could be obtained b letting the slab rotate with the angular velocit V about a point located on the perpendicular to v at a distance r 5 v /v from as shown in Fig b. We check that the velocit of would be perpendicular to and that its magnitude would be rv 5 (v /v)v 5 v. Thus the velocities of all the other particles of the slab would be the same as originall defined. Therefore, as far as the velocities are concerned, the slab seems to rotate about the instantaneous center at the instant considered. The position of the instantaneous center can be defined in two other was. If the directions of the velocities of two particles and of the slab are known and if the are different, the instantaneous center is obtained b drawing the perpendicular to v through and the perpendicular to v through and determining the point in which these two lines intersect (Fig a). If the velocities v and v of two particles and are perpendicular to the line and if their magnitudes are known, the instantaneous center can be found b intersecting the line with the line joining the etremities of the vectors v and v (Fig b). Note that if v and v were parallel v v v v Fig (a) (b)

35 948 Kinematics of Rigid odies in Fig a or if v and v had the same magnitude in Fig b, the instantaneous center would be at an infinite distance and V would be zero; all points of the slab would have the same velocit. To see how the concept of instantaneous center of rotation can be put to use, let us consider again the rod of Sec rawing the perpendicular to v through and the perpendicular to v through (Fig ), we obtain the instantaneous center. t the w v l q v Fig od centrode Space centrode Fig instant considered, the velocities of all the particles of the rod are thus the same as if the rod rotated about. Now, if the magnitude v of the velocit of is known, the magnitude v of the angular velocit of the rod can be obtained b writing v 5 v 5 v l cos u The magnitude of the velocit of can then be obtained b writing v v 5 ()v 5 l sin u l cos u 5 v tan u Note that onl absolute velocities are involved in the computation. The instantaneous center of a slab in plane motion can be located either on the slab or outside the slab. If it is located on the slab, the particle coinciding with the instantaneous center at a given instant t must have zero velocit at that instant. However, it should be noted that the instantaneous center of rotation is valid onl at a given instant. Thus, the particle of the slab which coincides with the instantaneous center at time t will generall not coincide with the instantaneous center at time t 1 t; while its velocit is zero at time t, it will probabl be different from zero at time t 1 t. This means that, in general, the particle does not have zero acceleration and, therefore, that the accelerations of the various particles of the slab cannot be determined as if the slab were rotating about. s the motion of the slab proceeds, the instantaneous center moves in space. ut it was just pointed out that the position of the instantaneous center on the slab keeps changing. Thus, the instantaneous center describes one curve in space, called the space centrode, and another curve on the slab, called the bod centrode (Fig ). It can be shown that at an instant, these two curves are tangent at and that as the slab moves, the bod centrode appears to roll on the space centrode.

36 SMPLE PRLEM 15.4 Solve Sample Prob. 15.2, using the method of the instantaneous center of rotation. SLUTIN v r v v r = 150 mm r = 250 mm a. ngular Velocit of the Gear. Since the gear rolls on the stationar lower rack, the point of contact of the gear with the rack has no velocit; point is therefore the instantaneous center of rotation. We write v 5 r v 1.2 m/s 5 (0.150 m)v V 5 8 rad/s i b. Velocities. s far as velocities are concerned, all points of the gear seem to rotate about the instantaneous center. Velocit of Upper Rack. Recalling that v R 5 v, we write v R 5 v 5 r v v R 5 (0.250 m)(8 rad/s) 5 2 m/s v R 5 2 m/s Velocit of Point. Since r 5 (0.150 m) m, we write v 5 r v v 5 ( m)(8 rad/s) m/s v m/s a 45 SMPLE PRLEM 15.5 Solve Sample Prob. 15.3, using the method of the instantaneous center of rotation. SLUTIN Motion of rank. Referring to Sample Prob. 15.3, we obtain the velocit of point ; v in./s c 50. Motion of the onnecting Rod. We first locate the instantaneous center b drawing lines perpendicular to the absolute velocities v and v. Recalling from Sample Prob that b and that 5 8 in., we solve the triangle v 40 b b 90 v g b g b sin sin in. sin in in. Since the connecting rod seems to rotate about point, we write v 5 ()v in./s 5 (10.14 in.)v V rad/s l v 5 ()v 5 (8.44 in.)(62.0 rad/s) in./s ft/s v P 5 v ft/s 949

37 SLVING PRLEMS N YUR WN In this lesson we introduced the instantaneous center of rotation in plane motion. This provides us with an alternative wa for solving problems involving the velocities of the various points of a bod in plane motion. s its name suggests, the instantaneous center of rotation is the point about which ou can assume a bod is rotating at a given instant, as ou determine the velocities of the points of the bod at that instant.. To determine the instantaneous center of rotation of a bod in plane motion, ou should use one of the following procedures. 1. If the velocit v of a point and the angular velocit V of the bod are both known (Fig ): a. raw a sketch of the bod, showing point, its velocit v, and the angular velocit V of the bod. b. From draw a line perpendicular to v on the side of v from which this velocit is viewed as having the same sense as V. c. Locate the instantaneous center on this line, at a distance r 5 v /v from point. 2. If the directions of the velocities of two points and are known and are different (Fig a): a. raw a sketch of the bod, showing points and and their velocities v and v. b. From and draw lines perpendicular to v and v, respectivel. The instantaneous center is located at the point where the two lines intersect. c. If the velocit of one of the two points is known, ou can determine the angular velocit of the bod. For eample, if ou know v, ou can write v 5 v /, where is the distance from point to the instantaneous center. 3. If the velocities of two points and are known and are both perpendicular to the line (Fig b): a. raw a sketch of the bod, showing points and with their velocities v and v drawn to scale. b. raw a line through points and, and another line through the tips of the vectors v and v. The instantaneous center is located at the point where the two lines intersect. 950

38 c. The angular velocit of the bod is obtained b either dividing v b or v b. d. If the velocities v and v have the same magnitude, the two lines drawn in part b do not intersect; the instantaneous center is at an infinite distance. The angular velocit V is zero and the bod is in translation.. nce ou have determined the instantaneous center and the angular velocit of a bod, ou can determine the velocit v P of an point P of the bod in the following wa. 1. raw a sketch of the bod, showing point P, the instantaneous center of rotation, and the angular velocit V. 2. raw a line from P to the instantaneous center and measure or calculate the distance from P to. 3. The velocit v P is a vector perpendicular to the line P, of the same sense as V, and of magnitude v P 5 (P)v. Finall, keep in mind that the instantaneous center of rotation can be used onl to determine velocities. It cannot be used to determine accelerations. 951

39 PRLEMS ft beam E is being lowered b means of two overhead cranes. t the instant shown it is known that the velocit of point is 24 in./s downward and the velocit of point E is 36 in./s downward. etermine (a) the instantaneous center of rotation of the beam, (b) the velocit of point. w E 3 ft 4 ft 3 ft Fig. P15.73 z Fig. P helicopter moves horizontall in the direction at a speed of 120 mi/h. Knowing that the main blades rotate clockwise with an angular velocit of 180 rpm, determine the instantaneous ais of rotation of the main blades The spool of tape shown and its frame assembl are pulled upward at a speed v mm/s. Knowing that the 80-mm-radius spool has an angular velocit of 15 rad/s clockwise and that at the instant shown the total thickness of the tape on the spool is 20 mm, determine (a) the instantaneous center of rotation of the spool, (b) the velocities of points and. v 80 mm v Fig. P15.75 and P The spool of tape shown and its frame assembl are pulled upward at a speed v mm/s. Knowing that end of the tape is pulled downward with a velocit of 300 mm/s and that at the instant shown the total thickness of the tape on the spool is 20 mm, determine (a) the instantaneous center of rotation of the spool, (b) the velocit of point of the spool.

40 15.77 Solve Sample Prob. 15.2, assuming that the lower rack is not stationar but moves to the left with a velocit of 0.6 m/s. Problems double pulle is attached to a slider block b a pin at. The 30-mm-radius inner pulle is rigidl attached to the 60-mm-radius outer pulle. Knowing that each of the two cords is pulled at a constant speed as shown, determine (a) the instantaneous center of rotation of the double pulle, (b) the velocit of the slider block, (c) the number of millimeters of cord wrapped or unwrapped on each pulle per second. 200 mm/s E Solve Prob , assuming that cord E is pulled upward at a speed of 160 mm/s and cord F is pulled downward at a speed of 200 mm/s and in.-radius drum is rigidl attached to a 5-in.- radius drum as shown. ne of the drums rolls without sliding on the surface shown, and a cord is wound around the other drum. Knowing that end E of the cord is pulled to the left with a velocit of 6 in./s, determine (a) the angular velocit of the drums, (b) the velocit of the center of the drums, (c) the length of cord wound or unwound per second. F 160 mm/s Fig. P in. 5 in. 3 in. 5 in. E E Fig. P15.80 Fig. P Knowing that at the instant shown the angular velocit of rod is 15 rad/s clockwise, determine (a) the angular velocit of rod, (b) the velocit of the midpoint of rod. 0.2 m 0.25 m E 0.2 m 0.6 m Fig. P15.82 and P Knowing that at the instant shown the velocit of point is 2.4 m/s upward, determine (a) the angular velocit of rod, (b) the velocit of the midpoint of rod.

41 954 Kinematics of Rigid odies Rod is guided b wheels at and that roll in horizontal and vertical tracks. Knowing that at the instant b 5 60 and the velocit of wheel is 40 in./s downward, determine (a) the angular velocit of the rod, (b) the velocit of point. b 15 in. 15 in n overhead door is guided b wheels at and that roll in horizontal and vertical tracks. Knowing that when u 5 40 the velocit of wheel is 1.5 ft/s upward, determine (a) the angular velocit of the door, (b) the velocit of end of the door. Fig. P15.84 q 5 ft E 192 mm 5 ft 240 mm mm Fig. P Knowing that at the instant shown the angular velocit of rod E is 4 rad/s counterclockwise, determine (a) the angular velocit of rod, (b) the velocit of collar, (c) the velocit of point. Fig. P15.86 and P Knowing that at the instant shown the velocit of collar is 1.6 m/s upward, determine (a) the angular velocit of rod, (b) the velocit of point, (c) the velocit of point Rod can slide freel along the floor and the inclined plane. enoting b v the velocit of point, derive an epression for (a) the angular velocit of the rod, (b) the velocit of end. l v b q Fig. P15.88 and P Rod can slide freel along the floor and the inclined plane. Knowing that u 5 20, b 5 50, l m, and v 5 3 m/s, determine (a) the angular velocit of the rod, (b) the velocit of end.

42 15.90 rm is connected b pins to a collar at and to crank E. Knowing that the velocit of collar is 400 mm/s upward, determine (a) the angular velocit of arm, (b) the velocit of point. Problems mm E 90 mm 160 mm 300 mm 4 in. 6 in. 18 in. 8 in. G 7 in. 180 mm Fig. P15.90 and P mm 6 in. F rm is connected b pins to a collar at and to crank E. Knowing that the angular velocit of crank E is 1.2 rad/s counterclockwise, determine (a) the angular velocit of arm, (b) the velocit of point. 8 in. E Two slots have been cut in plate FG and the plate has been placed so that the slots fit two fied pins and. Knowing that at the instant shown the angular velocit of crank E is 6 rad/s clockwise, determine (a) the velocit of point F, (b) the velocit of point G Two identical rods F and E are connected b a pin at. Knowing that at the instant shown the velocit of point is 10 in./s upward, determine the velocit of (a) point E, (b) point F Rod is attached to a collar at and is fitted with a small wheel at. Knowing that when u 5 60 the velocit of the collar is 250 mm/s upward, determine (a) the angular velocit of rod, (b) the velocit of point. 3.6 in. Fig. P15.92 Fig. P in. 6 in. E F 300 mm 200 mm q Fig. P15.94

43 956 Kinematics of Rigid odies Two collars and move along the vertical rod shown. Knowing that the velocit of the collar is 660 mm/s downward, determine (a) the velocit of collar, (b) the angular velocit of member Two 500-mm rods are pin-connected at as shown. Knowing that moves to the left with a constant velocit of 360 mm/s, determine at the instant shown (a) the angular velocit of each rod, (b) the velocit of E. 320 mm 100 mm 100 mm Fig. P mm E 500 imensions in mm Fig. P15.96 E 8 in. 8 in Two rods and E are connected as shown. Knowing that point moves to the left with a velocit of 40 in./s, determine (a) the angular velocit of each rod, (b) the velocit of point Two rods and E are connected as shown. Knowing that point moves downward with a velocit of 60 in./s, determine (a) the angular velocit of each rod, (b) the velocit of point E. Fig. P in. 8 in. 8 in. 8 in. E 8 in. 15 in. 6 in. 9 in. Fig. P escribe the space centrode and the bod centrode of rod of Prob (Hint: The bod centrode need not lie on a phsical portion of the rod.) escribe the space centrode and the bod centrode of the gear of Sample Prob as the gear rolls on the stationar horizontal rack Using the method of Sec. 15.7, solve Prob Using the method of Sec. 15.7, solve Prob Using the method of Sec. 15.7, solve Prob Using the method of Sec. 15.7, solve Prob

44 15.8 SLUTE N RELTIVE ELERTIN IN PLNE MTIN We saw in Sec that an plane motion can be replaced b a translation defined b the motion of an arbitrar reference point and a simultaneous rotation about. This propert was used in Sec to determine the velocit of the various points of a moving slab. The same propert will now be used to determine the acceleration of the points of the slab. We first recall that the absolute acceleration a of a particle of the slab can be obtained from the relative-acceleration formula derived in Sec , 15.8 bsolute and Relative cceleration in Plane Motion 957 a 5 a 1 a / (15.21) Photo 15.6 The central gear rotates about a fied ais and is pin-connected to three bars which are in general plane motion. where the right-hand member represents a vector sum. The acceleration a corresponds to the translation of the slab with, while the relative acceleration a / is associated with the rotation of the slab about and is measured with respect to aes centered at and of fied orientation. We recall from Sec that the relative acceleration a / can be resolved into two components, a tangential component (a / ) t perpendicular to the line, and a normal component (a / ) n directed toward (Fig ). enoting b r / the position vector of relative to and, respectivel, b vk and ak the angular velocit and angular acceleration of the slab with respect to aes of fied orientation, we have (a / ) t 5 ak 3 r / (a / ) t 5 ra (a / ) n 5 2v 2 r / (a / ) n 5 rv 2 (15.22) where r is the distance from to. Substituting into (15.21) the epressions obtained for the tangential and normal components of a /, we can also write a 5 a 1 ak 3 r / 2 v 2 r / (15.219) ' a a = + wk (fied) ak r / a a (a / ) n a ' / a (a / ) n a / (a / ) t a (a / ) t Fig Plane motion = Translation with + Rotation about

45 958 Kinematics of Rigid odies a a (a / ) t θ l = + l a w (a / ) n a a (fied) Plane motion = Translation with + Rotation about Fig (a / ) n a a q (a / ) t (a) (a / ) n s an eample, let us again consider the rod whose etremities slide, respectivel, along a horizontal and a vertical track (Fig ). ssuming that the velocit v and the acceleration a of are known, we propose to determine the acceleration a of and the angular acceleration of the rod. hoosing as a reference point, we epress that the given motion is equivalent to a translation with and a rotation about. The absolute acceleration of must be equal to the sum a (a / ) t q (b) a q (a / ) n (a / ) t (c) a q (a / ) n (d) Fig a a a (a / ) t a 5 a 1 a / (15.23) 5 a 1 (a / ) n 1 (a / ) t where (a / ) n has the magnitude lv 2 and is directed toward, while (a / ) t has the magnitude la and is perpendicular to. Students should note that there is no wa to tell whether the tangential component (a / ) t is directed to the left or to the right, and therefore both possible directions for this component are indicated in Fig Similarl, both possible senses for a are indicated, since it is not known whether point is accelerated upward or downward. Equation (15.23) has been epressed geometricall in Fig Four different vector polgons can be obtained, depending upon the sense of a and the relative magnitude of a and (a / ) n. If we are to determine a and a from one of these diagrams, we must know not onl a and u but also v. The angular velocit of the rod should therefore be separatel determined b one of the methods indicated in Secs and The values of a and a can then be obtained b considering successivel the and components of the vectors shown in Fig In the case of polgon a, for eample, we write 1 components: 0 5 a 1 lv 2 sin u 2 la cos u 1 components: 2a 5 2lv 2 cos u 2 la sin u and solve for a and a. The two unknowns can also be obtained b direct measurement on the vector polgon. In that case, care should be taken to draw first the known vectors a and (a / ) n. It is quite evident that the determination of accelerations is considerabl more involved than the determination of velocities. Yet

46 in the eample considered here, the etremities and of the rod were moving along straight tracks, and the diagrams drawn were relativel simple. If and had moved along curved tracks, it would have been necessar to resolve the accelerations a and a into normal and tangential components and the solution of the problem would have involved si different vectors. When a mechanism consists of several moving parts which are pin-connected, the analsis of the mechanism can be carried out b considering each part as a rigid bod, keeping in mind that the points at which two parts are connected must have the same absolute acceleration (see Sample Prob. 15.7). In the case of meshed gears, the tangential components of the accelerations of the teeth in contact are equal, but their normal components are different nalsis of Plane Motion in Terms of a Parameter 959 *15.9 NLYSIS F PLNE MTIN IN TERMS F PRMETER In the case of certain mechanisms, it is possible to epress the coordinates and of all the significant points of the mechanism b means of simple analtic epressions containing a single parameter. It is sometimes advantageous in such a case to determine the absolute velocit and the absolute acceleration of the various points of the mechanism directl, since the components of the velocit and of the acceleration of a given point can be obtained b differentiating the coordinates and of that point. Let us consider again the rod whose etremities slide, respectivel, in a horizontal and a vertical track (Fig ). The coordinates and of the etremities of the rod can be epressed in terms of the angle u the rod forms with the vertical: 5 l sin u 5 l cos u (15.24) ifferentiating Eqs. (15.24) twice with respect to t, we write v 5 ẋ 5 lu cos u a 5 ẍ 5 2lu 2 sin u 1 lü cos u v 5 ẏ 5 2lu sin u a 5 ÿ 5 2lu 2 cos u 2 lü sin u Recalling that u 5 v and ü 5 a, we obtain v 5 lv cos u v 5 2lv sin u (15.25) a 5 2lv 2 sin u 1 la cos u a 5 2lv 2 cos u 2 la sin u (15.26) We note that a positive sign for v or a indicates that the velocit v or the acceleration a is directed to the right; a positive sign for v or a indicates that v or a is directed upward. Equations (15.25) can be used, for eample to determine v and v when v and u are known. Substituting for v in (15.26), we can then determine a and a if a is known. q l Fig

47 R SMPLE PRLEM 15.6 r 1 = 150 mm v = 1.2 m/s a = 3 m/s 2 r 2 = 100 mm The center of the double gear of Sample Prob has a velocit of 1.2 m/s to the right and an acceleration of 3 m/s 2 to the right. Recalling that the lower rack is stationar, determine (a) the angular acceleration of the gear, (b) the acceleration of points,, and of the gear. SLUTIN a. ngular cceleration of the Gear. In Sample Prob. 15.2, we found that 5 2r 1 u and v 5 2r 1 v. ifferentiating the latter with respect to time, we obtain a 5 2r 1 a. v 5 2r 1 v 1.2 m/s 5 2(0.150 m)v v 5 28 rad/s a 5 2r 1 a 3 m/s 2 5 2(0.150 m)a a rad/s 2 5 ak 5 2(20 rad/s 2 )k b. ccelerations. The rolling motion of the gear is resolved into a translation with and a rotation about. (a/ ) t (a / ) t a a (a / ) n a + a w = (a / ) n (fied) a a a a (a / ) t (a/ ) n a Translation + Rotation = Rolling motion a a (a / ) t (a / ) n cceleration of Point. dding vectoriall the accelerations corresponding to the translation and to the rotation, we obtain a 5 a 1 a / 5 a 1 (a / ) t 1 (a / ) n 5 a 1 ak 3 r / 2 v 2 r / 5 (3 m/s 2 )i 2 (20 rad/s 2 )k 3 (0.100 m)j 2 (8 rad/s) 2 (0.100 m)j 5 (3 m/s 2 )i 1 (2 m/s 2 )i 2 (6.40 m/s 2 )j a m/s 2 c 52.0 (a / ) n cceleration of Point a a (a / ) t a a 5 a 1 a / 5 a 1 ak 3 r / 2 v 2 r / 5 (3 m/s 2 )i 2 (20 rad/s 2 )k 3 ( m)j 2 (8 rad/s) 2 ( m)j 5 (3 m/s 2 )i 2 (3 m/s 2 )i 1 (9.60 m/s 2 )j a m/s 2 cceleration of Point a (a / ) n (a /) t a 5 a 1 a / 5 a 1 ak 3 r / 2 v 2 r / 5 (3 m/s 2 )i 2 (20 rad/s 2 )k 3 ( m)i 2 (8 rad/s) 2 ( m)i 5 (3 m/s 2 )i 1 (3 m/s 2 )j 1 (9.60 m/s 2 )i a m/s 2 a

48 r = 3 in. 40 b l = 8 in. G P SMPLE PRLEM 15.7 rank of the engine sstem of Sample Prob has a constant clockwise angular velocit of 2000 rpm. For the crank position shown, determine the angular acceleration of the connecting rod and the acceleration of point. SLUTIN r = 3 in. 40 a Motion of rank. Since the crank rotates about with constant v rpm rad/s, we have a 5 0. The acceleration of is therefore directed toward and has a magnitude a 5 rv 2 5 ( 3 12 ft)(209.4 rad/s) ,962 ft/s 2 a 5 10,962 ft/s 2 d 40 Motion of the onnecting Rod. The angular velocit V and the value of b were obtained in Sample Prob. 15.3: V rad/s l b The motion of is resolved into a translation with and a rotation about. The relative acceleration a / is resolved into normal and tangential components: (a / ) n 5 ()v 2 5 ( 8 12 ft)(62.0 rad/s) ft/s 2 (a / ) n ft/s 2 b (a / ) t 5 ()a 5 ( 12)a a (a / ) t a za While (a / ) t must be perpendicular to, its sense is not known. a a a G = a + a w G (a / ) n a (a / ) t Plane motion = Translation + Rotation Noting that the acceleration a must be horizontal, we write (a / ) t a / (a / ) n a a 40 a 5 a 1 a / 5 a 1 (a / ) n 1 (a / ) t [a G ] 5 [10,962 d 40 ] 1 [2563 b ] 1 [0.6667a za ] Equating and components, we obtain the following scalar equations: 1 components: 2a 5 210,962 cos cos a sin components: ,962 sin sin a cos Solving the equations simultaneousl, we obtain a rad/s 2 and a ft/s 2. The positive signs indicate that the senses shown on the vector polgon are correct; we write a rad/s 2 l a ft/s 2 z 961

49 3 in. 14 in. w 1 17 in. E SMPLE PRLEM 15.8 The linkage E moves in the vertical plane. Knowing that in the position shown crank has a constant angular velocit V 1 of 20 rad/s counterclockwise, determine the angular velocities and angular accelerations of the connecting rod and of the crank E. 8 in. 12 in. 17 in. SLUTIN r / r r r = 8i + 14j r = 17i + 17j r / = 12i + 3j E This problem could be solved b the method used in Sample Prob In this case, however, the vector approach will be used. The position vectors r, r, and r / are chosen as shown in the sketch. Velocities. Since the motion of each element of the linkage is contained in the plane of the figure, we have V 5 v k 5 (20 rad/s)k V 5 v k V E 5 v E k where k is a unit vector pointing out of the paper. We now write v 5 v 1 v / v E k 3 r 5 v k 3 r 1 v k 3 r / v E k 3 (217i 1 17j) 5 20k 3 (8i 1 14j) 1 v k 3 (12i 1 3j) 217v E j 2 17v E i 5 160j 2 280i 1 12v j 2 3v i Equating the coefficients of the unit vectors i and j, we obtain the following two scalar equations: 217v E v 217v E v V 5 2(29.33 rad/s)k V E 5 (11.29 rad/s)k ccelerations. Noting that at the instant considered crank has a constant angular velocit, we write a k E 5 a E k a 5 a 1 a / (1) Each term of Eq. (1) is evaluated separatel: a 5 a E k 3 r 2 ver 2 5 a E k 3 (217i 1 17j) 2 (11.29) 2 (217i 1 17j) 5 217a E j 2 17a E i i j a 5 a k 3 r 2 v 2 r (20) 2 (8i 1 14j) i j a / 5 a k 3 r / 2 v 2 r / 5 a k 3 (12i 1 3j) 2 (29.33) 2 (12i 1 3j) 5 12a j 2 3a i 2 10,320i j Substituting into Eq. (1) and equating the coefficients of i and j, we obtain 217a E 1 3a 5 215, a E 2 12a (645 rad/s 2 )k E 5 (809 rad/s 2 )k 962

50 SLVING PRLEMS N YUR WN This lesson was devoted to the determination of the accelerations of the points of a rigid bod in plane motion. s ou did previousl for velocities, ou will again consider the plane motion of a rigid bod as the sum of two motions, namel, a translation and a rotation. To solve a problem involving accelerations in plane motion ou should use the following steps: 1. etermine the angular velocit of the bod. To find V ou can either a. onsider the motion of the bod as the sum of a translation and a rotation as ou did in Sec. 15.6, or b. Use the instantaneous center of rotation of the bod as ou did in Sec However, keep in mind that ou cannot use the instantaneous center to determine accelerations. 2. Start drawing a diagram equation to use in our solution. This equation will involve the following diagrams (Fig ). a. Plane motion diagram. raw a sketch of the bod, including all dimensions, as well as the angular velocit V. Show the angular acceleration with its magnitude and sense if ou know them. lso show those points for which ou know or seek the accelerations, indicating all that ou know about these accelerations. b. Translation diagram. Select a reference point for which ou know the direction, the magnitude, or a component of the acceleration a. raw a second diagram showing the bod in translation with each point having the same acceleration as point. c. Rotation diagram. onsidering point as a fied reference point, draw a third diagram showing the bod in rotation about. Indicate the normal and tangential components of the relative accelerations of other points, such as the components (a / ) n and (a / ) t of the acceleration of point with respect to point. 3. Write the relative-acceleration formula a 5 a 1 a / or a 5 a 1 (a / ) n 1 (a / ) t The sample problems illustrate three different was to use this vector equation: a. If is given or can easil be determined, ou can use this equation to determine the accelerations of various points of the bod [Sample Prob. 15.6]. (continued) 963

51 b. If cannot easil be determined, select for point a point for which ou know the direction, the magnitude, or a component of the acceleration a and draw a vector diagram of the equation. Starting at the same point, draw all known acceleration components in tip-to-tail fashion for each member of the equation. omplete the diagram b drawing the two remaining vectors in appropriate directions and in such a wa that the two sums of vectors end at a common point. The magnitudes of the two remaining vectors can be found either graphicall or analticall. Usuall an analtic solution will require the solution of two simultaneous equations [Sample Prob. 15.7]. However, b first considering the components of the various vectors in a direction perpendicular to one of the unknown vectors, ou ma be able to obtain an equation in a single unknown. ne of the two vectors obtained b the method just described will be (a / ) t, from which ou can compute a. nce a has been found, the vector equation can be used to determine the acceleration of an other point of the bod. c. full vector approach can also be used to solve the vector equation. This is illustrated in Sample Prob The analsis of plane motion in terms of a parameter completed this lesson. This method should be used onl if it is possible to epress the coordinates and of all significant points of the bod in terms of a single parameter (Sec. 15.9). differentiating twice with respect to t the coordinates and of a given point, ou can determine the rectangular components of the absolute velocit and absolute acceleration of that point. 964

52 PRLEMS mm rod rests on a horizontal table. force P applied as shown produces the following accelerations: a m/s 2 to the right, a 5 6 rad/s 2 counterclockwise as viewed from above. etermine the acceleration (a) of point G, (b) of point. P G 0.45 m 0.45 m Fig. P and P In Prob , determine the point of the rod that (a) has no acceleration, (b) has an acceleration of 2.4 m/s 2 to the right ft steel beam is lowered b means of two cables unwinding at the same speed from overhead cranes. s the beam approaches the ground, the crane operators appl brakes to slow down the unwinding motion. t the instant considered the deceleration of the cable attached at is 12 ft/s 2, while that of the cable at is 5 ft/s 2. etermine (a) the angular acceleration of the beam, (b) the acceleration of point The acceleration of point is 1 ft/s 2 downward and the angular acceleration of the beam is 0.8 rad/s 2 clockwise. Knowing that the angular velocit of the beam is zero at the instant considered, determine the acceleration of each cable. 9 ft 1 ft Fig. P and P and ar E is attached to two links and. Knowing that at the instant shown link has zero angular acceleration and an angular velocit of 3 rad/s clockwise, determine the acceleration (a) of point, (b) of point E. 240 mm 240 mm 150 mm 150 mm 150 mm 150 mm E E 180 mm 180 mm Fig. P Fig. P

53 966 Kinematics of Rigid odies n automobile travels to the left at a constant speed of 48 mi/h. Knowing that the diameter of the wheel is 22 in., determine the acceleration (a) of point, (b) of point, (c) of point in. Fig. P Fig. P carriage is supported b a caster and a clinder, each of 50-mm diameter. Knowing that at the instant shown the carriage has an acceleration of 2.4 m/s 2 and a velocit of 1.5 m/s, both directed to the left, determine (a) the angular accelerations of the caster and of the clinder, (b) the accelerations of the centers of the caster and of the clinder. G 75 mm Fig. P and P E The motion of the 75-mm-radius clinder is controlled b the cord shown. Knowing that end E of the cord has a velocit of 300 mm/s and an acceleration of 480 mm/s 2, both directed upward, determine the acceleration (a) of point, (b) of point The motion of the 75-mm-radius clinder is controlled b the cord shown. Knowing that end E of the cord has a velocit of 300 mm/s and an acceleration of 480 mm/s 2, both directed upward, determine the accelerations of points and of the clinder and in.-radius drum is rigidl attached to a 5-in.- radius drum as shown. ne of the drums rolls without sliding on the surface shown, and a cord is wound around the other drum. Knowing that at the instant shown end of the cord has a velocit of 8 in./s and an acceleration of 30 in./s 2, both directed to the left, determine the accelerations of points,, and of the drums. 3 in. 5 in. G 3 in. 5 in. G Fig. P Fig. P15.116

54 The 150-mm-radius drum rolls without slipping on a belt that moves to the left with a constant velocit of 300 mm/s. t an instant when the velocit and acceleration of the center of the drum are as shown, determine the accelerations of points,, and of the drum. Problems mm/s 150 mm 900 mm/s mm/s Fig. P The 18-in.-radius flwheel is rigidl attached to a 1.5-in.-radius shaft that can roll along parallel rails. Knowing that at the instant shown the center of the shaft has a velocit of 1.2 in./s and an acceleration of 0.5 in./s 2, both directed down to the left, determine the acceleration (a) of point, (b) of point. 18 in. 20 Fig. P In the planetar gear sstem shown the radius of gears,,, and is 3 in. and the radius of the outer gear E is 9 in. Knowing that gear has a constant angular velocit of 150 rpm clockwise and that the outer gear E is stationar, determine the magnitude of the acceleration of the tooth of gear that is in contact with (a) gear, (b) gear E The disk shown has a constant angular velocit of 500 rpm counterclockwise. Knowing that rod is 250 mm long, determine the acceleration of collar when (a) u 5 90, (b) u q 50 mm Fig. P E 90 E 150 mm q Fig. P In the two-clinder air compressor shown the connecting rods and E are each 190 mm long and crank rotates about the fied point with a constant angular velocit of 1500 rpm clockwise. etermine the acceleration of each piston when u 5 0. Fig. P mm 45

55 968 Kinematics of Rigid odies rm has a constant angular velocit of 16 rad/s counterclockwise. t the instant when u 5 0, determine the acceleration (a) of collar, (b) of the midpoint G of bar. 10 in. 3 in. q 6 in. P Fig. P15.122, P15.123, and P rm has a constant angular velocit of 16 rad/s counterclockwise. t the instant when u 5 90, determine the acceleration (a) of collar, (b) of the midpoint G of bar. 150 mm rm has a constant angular velocit of 16 rad/s counterclockwise. t the instant when u 5 60, determine the acceleration of collar. q Knowing that crank rotates about point with a constant angular velocit of 900 rpm clockwise, determine the acceleration of the piston P when u mm Fig. P and P Knowing that crank rotates about point with a constant angular velocit of 900 rpm clockwise, determine the acceleration of the piston P when u Knowing that at the instant shown rod has zero angular acceleration and an angular velocit of 15 rad/s counterclockwise, determine (a) the angular acceleration of arm E, (b) the acceleration of point. 4 in. 5 in. 5 in. 4 in. G E 3 in. Fig. P and P mm 90 mm E Knowing that at the instant shown rod has zero angular acceleration and an angular velocit of 15 rad/s counterclockwise, determine (a) the angular acceleration of member, (b) the acceleration of point G. 90 mm Knowing that at the instant shown rod has a constant angular velocit of 6 rad/s clockwise, determine the acceleration of point. 225 mm 225 mm Fig. P and P Knowing that at the instant shown rod has a constant angular velocit of 6 rad/s clockwise, determine (a) the angular acceleration of member E, (b) the acceleration of point E.

56 Knowing that at the instant shown rod has zero angular acceleration and an angular velocit v 0 clockwise, determine (a) the angular acceleration of arm E, (b) the acceleration of point t the instant shown rod has zero angular acceleration and an angular velocit of 8 rad/s clockwise. Knowing that l m, determine the acceleration of the midpoint of member and Knowing that at the instant shown bar has a constant angular velocit of 4 rad/s clockwise, determine the angular acceleration (a) of bar, (b) of bar E. l l Fig. P and P Problems l E mm E 7 in. 4 in. 500 mm 8 in. 400 mm 400 mm Fig. P and P in. E Fig. P and P and Knowing that at the instant shown bar has an angular velocit of 4 rad/s and an angular acceleration of 2 rad/s 2, both clockwise, determine the angular acceleration (a) of bar, (b) of bar E b using the vector approach as is done in Sample Prob enoting b r the position vector of a point of a rigid slab that is in plane motion, show that (a) the position vector r of the instantaneous center of rotation is r 5 r 1 V 3 v v 2 Where V is the angular velocit of the slab and v is the velocit of point, (b) the acceleration of the instantaneous center of rotation is zero if, and onl if, a 5 a v v 1 V 3 v v r r w a where 5 ak is the angular acceleration of the slab. Fig. P15.137

57 970 Kinematics of Rigid odies * The wheels attached to the ends of rod roll along the surfaces shown. Using the method of Sec. 15.9, derive an epression for the angular velocit of the rod in terms of v, u, l, and b. b d q l v Fig. P and P Fig. P b q * The wheels attached to the ends of rod roll along the surfaces shown. Using the method of Sec and knowing that the acceleration of wheel is zero, derive an epression for the angular acceleration of the rod in terms of v, u, l, and b. * The drive disk of the Scotch crosshead mechanism shown has an angular velocit V and an angular acceleration, both directed counterclockwise. Using the method of Sec. 15.9, derive epressions for the velocit and acceleration of point. * Rod moves over a small wheel at while end moves to the right with a constant velocit v. Using the method of Sec. 15.9, derive epressions for the angular velocit and angular acceleration of the rod. l b q Fig. P and P * Rod moves over a small wheel at while end moves to the right with a constant velocit v. Using the method of Sec. 15.9, derive epressions for the horizontal and vertical components of the velocit of point. * disk of radius r rolls to the right with a constant velocit v. enoting b P the point of the rim in contact with the ground at t 5 0, derive epressions for the horizontal and vertical components of the velocit of P at an time t.

58 * t the instant shown, rod rotates with an angular velocit V and an angular acceleration, both clockwise. Using the method of Sec. 15.9, derive epressions for the velocit and acceleration of point Rate of hange of a Vector with Respect to a Rotating Frame 971 l q l l q Fig. P and P r * t the instant shown, rod rotates with an angular velocit V and an angular acceleration, both clockwise. Using the method of Sec. 15.9, derive epressions for the horizontal and vertical components of the velocit and acceleration of point. * The position of rod is controlled b a disk of radius r which is attached to oke. Knowing that the oke moves verticall upward with a constant velocit v 0, derive an epression for the angular acceleration of rod. Fig. P * In Prob , derive an epression for the angular acceleration of rod. * wheel of radius r rolls without slipping along the inside of a fied clinder of radius R with a constant angular velocit V. enoting b P the point of the wheel in contact with the clinder at t 5 0, derive epressions for the horizontal and vertical components of the velocit of P at an time t. (The curve described b point P is a hpoccloid.) R r w * In Prob , show that the path of P is a vertical straight line when r 5 R/2. erive epressions for the corresponding velocit and acceleration of P at an time t. Fig. P P RTE F HNGE F VETR WITH RESPET T RTTING FRME We saw in Sec that the rate of change of a vector is the same with respect to a fied frame and with respect to a frame in translation. In this section, the rates of change of a vector Q with respect to a fied frame and with respect to a rotating frame of reference will be considered. You will learn to determine the rate of change of Q with respect to one frame of reference when Q is defined b its components in another frame. It is recalled that the selection of a fied frame of reference is arbitrar. n frame ma be designated as fied ; all others will then be considered as moving. Photo 15.7 geneva mechanism is used to convert rotar motion into intermittent motion.

59 972 Kinematics of Rigid odies onsider two frames of reference centered at, a fied frame XYZ and a frame z which rotates about the fied ais ; let Y V denote the angular velocit of the frame z at a given instant (Fig ). onsider now a vector function Q(t) represented b the vector Q attached at ; as the time t varies, both the direction and Q the magnitude of Q change. Since the variation of Q is viewed differentl b an observer using XYZ as a frame of reference and b Ω j i an observer using z, we should epect the rate of change of Q to depend upon the frame of reference which has been selected. Therefore, the rate of change of Q with respect to the fied frame XYZ X k will be denoted b ( Q) XYZ, and the rate of change of Q with respect z to the rotating frame z will be denoted b ( Q) z. We propose to determine the relation eisting between these two rates of change. Z Let us first resolve the vector Q into components along the,, Fig and z aes of the rotating frame. enoting b i, j, and k the corresponding unit vectors, we write Q 5 Q i 1 Q j 1 Q z k (15.27) ifferentiating (15.27) with respect to t and considering the unit vectors i, j, k as fied, we obtain the rate of change of Q with respect to the rotating frame z: ( Q) z 5 Q i 1 Q j 1 Q z k (15.28) To obtain the rate of change of Q with respect to the fied frame XYZ, we must consider the unit vectors i, j, k as variable when differentiating (15.27). We therefore write (Q. ) XYZ 5 Q. i 1 Q. j 1 Q. di zk 1 Q dt 1 Q dj dt 1 Q dk z (15.29) dt Recalling (15.28), we observe that the sum of the first three terms in the right-hand member of (15.29) represents the rate of change ( Q) z. We note, on the other hand, that the rate of change ( Q) XYZ would reduce to the last three terms in (15.29) if the vector Q were fied within the frame z, since ( Q) z would then be zero. ut in that case, ( Q) XYZ would represent the velocit of a particle located at the tip of Q and belonging to a bod rigidl attached to the frame z. Thus, the last three terms in (15.29) represent the velocit of that particle; since the frame z has an angular velocit V with respect to XYZ at the instant considered, we write, b (15.5), di Q dt 1 Q dj dt 1 Q dk z dt 5 V3Q (15.30) Substituting from (15.28) and (15.30) into (15.29), we obtain the fundamental relation ( Q) XYZ 5 ( Q) z 1 V 3 Q (15.31) We conclude that the rate of change of the vector Q with respect to the fied frame XYZ is made of two parts: The first part represents the rate of change of Q with respect to the rotating frame z; the second part, V 3 Q, is induced b the rotation of the frame z.

60 The use of relation (15.31) simplifies the determination of the rate of change of a vector Q with respect to a fied frame of reference XYZ when the vector Q is defined b its components along the aes of a rotating frame z, since this relation does not require the separate computation of the derivatives of the unit vectors defining the orientation of the rotating frame Plane Motion of a Particle Relative to a Rotating Frame. oriolis cceleration PLNE MTIN F PRTILE RELTIVE T RTTING FRME. RILIS ELERTIN onsider two frames of reference, both centered at and both in the plane of the figure, a fied frame XY and a rotating frame (Fig ). Let P be a particle moving in the plane of the figure. The position vector r of P is the same in both frames, but its rate of change depends upon the frame of reference which has been selected. The absolute velocit v P of the particle is defined as the velocit observed from the fied frame XY and is equal to the rate of change (ṙ) XY of r with respect to that frame. We can, however, epress v P in terms of the rate of change (ṙ) observed from the rotating frame if we make use of Eq. (15.31). enoting b V the angular velocit of the frame with respect to XY at the instant considered, we write Y r Ω Fig P X v P 5 (ṙ) XY 5 V 3 r 1 (ṙ) (15.32) ut (ṙ) defines the velocit of the particle P relative to the rotating frame. enoting the rotating frame b ^ for short, we represent the velocit (ṙ) of P relative to the rotating frame b v P/^. Let us imagine that a rigid slab has been attached to the rotating frame. Then v P/^ represents the velocit of P along the path that it describes on that slab (Fig ), and the term V 3 r in (15.32) represents the velocit v P9 of the point P9 of the slab or rotating frame which coincides with P at the instant considered. Thus, we have v P 5 v P9 1 v P/^ (15.33) Y v P' = Ω r v P/ r Ω. = (r) P P' X where v P 5 absolute velocit of particle P v P9 5 velocit of point P9 of moving frame ^ coinciding with P v P/^ 5 velocit of P relative to moving frame ^ The absolute acceleration a P of the particle is defined as the rate of change of v P with respect to the fied frame XY. omputing the rates of change with respect to XY of the terms in (15.32), we write a P 5 v P 5 V. 3 r 1 V3ṙ1 d dt [(ṙ) ] (15.34) Fig where all derivatives are defined with respect to XY, ecept where indicated otherwise. Referring to Eq. (15.31), we note that the last term in (15.34) can be epressed as d dt [(ṙ) ] 5 (r ) 1 V3(ṙ)

61 974 Kinematics of Rigid odies n the other hand, ṙ represents the velocit v P and can be replaced b the right-hand member of Eq. (15.32). fter completing these two substitutions into (15.34), we write a P 5 V 3 r 1 V 3 (V 3 r) 1 2V 3 (ṙ) 1 ( r) (15.35) Referring to the epression (15.8) obtained in Sec for the acceleration of a particle in a rigid bod rotating about a fied ais, we note that the sum of the first two terms represents the acceleration a P9 of the point P9 of the rotating frame which coincides with P at the instant considered. n the other hand, the last term defines the acceleration a P/f of P relative to the rotating frame. If it were not for the third term, which has not been accounted for, a relation similar to (15.33) could be written for the accelerations, and a P could be epressed as the sum of a P9 and a P/f. However, it is clear that such a relation would be incorrect and that we must include the additional term. This term, which will be denoted b a c, is called the complementar acceleration, or oriolis acceleration, after the French mathematician de oriolis ( ). We write a P 5 a P9 1 a P/^ 1 a c (15.36) where a P 5 absolute acceleration of particle P a P9 5 acceleration of point P9 of moving frame ^ coinciding with P a P/^ 5 acceleration of P relative to moving frame ^ a c 5 2V 3 ( r) 5 2V 3 v P/^ 5 complementar, or oriolis, acceleration We note that since point P9 moves in a circle about the origin, its acceleration a P9 has, in general, two components: a component (a P9 ) t tangent to the circle, and a component (a P9 ) n directed toward. Similarl, the acceleration a P/^ generall has two components: a component (a P/^) t tangent to the path that P describes on the rotating slab, and a component (a P/^) n directed toward the center of curvature of that path. We further note that since the vector V is perpendicular to the plane of motion, and thus to v P/^, the magnitude of the oriolis acceleration a c 5 2V 3 v P/^ is equal to 2Vv P/^, and its direction can be obtained b rotating the vector v P/^ through 90 in the sense of rotation of the moving frame (Fig ). The oriolis acceleration reduces to zero when either V or v P/^ is zero. The following eample will help in understanding the phsical meaning of the oriolis acceleration. onsider a collar P which is Y v P/ a c = 2 Ω v P/ P r Ω Fig X It is important to note the difference between Eq. (15.36) and Eq. (15.21) of Sec When we wrote a 5 a 1 a / (15.21) in Sec. 15.8, we were epressing the absolute acceleration of point as the sum of its acceleration a / relative to a frame in translation and of the acceleration a of a point of that frame. We are now tring to relate the absolute acceleration of point P to its acceleration a P/f relative to a rotating frame f and to the acceleration a P9 of the point P9 of that frame which coincides with P; Eq. (15.36) shows that because the frame is rotating, it is necessar to include an additional term representing the oriolis acceleration a c.

62 made to slide at a constant relative speed u along a rod rotating at a constant angular velocit V about (Fig a). ccording to formula (15.36), the absolute acceleration of P can be obtained b adding vectoriall the acceleration a of the point of the rod coinciding with P, the relative acceleration a P/ of P with respect to the rod, and the oriolis acceleration a c. Since the angular velocit V of the rod is constant, a reduces to its normal component (a ) n of magnitude rv 2 ; and since u is constant, the relative acceleration a P/ is zero. ccording to the definition given above, the oriolis acceleration is a vector perpendicular to, of magnitude 2v u, and directed as shown in the figure. The acceleration of the collar P consists, therefore, of the two vectors shown in Fig a. Note that the result obtained can be checked b appling the relation (11.44). To understand better the significance of the oriolis acceleration, let us consider the absolute velocit of P at time t and at time t 1 t (Fig b). The velocit at time t can be resolved into its components u and v ; the velocit at time t 1 t can be resolved into its components u9 and v 9. rawing these components from the same origin (Fig c), we note that the change in velocit during the time t can be represented b the sum of three vectors, RR, TT, and T T. The vector TT measures the change in direction of the velocit v, and the quotient TT / t represents the acceleration a when t approaches zero. We check that the direction of TT is that of a when t approaches zero and that TT lim t0 t 5 lim t0 v u t 5 rvv 5 rv2 5 a The vector RR measures the change in direction of u due to the rotation of the rod; the vector T T measures the change in magnitude of v due to the motion of P on the rod. The vectors RR and T T result from the combined effect of the relative motion of P and of the rotation of the rod; the would vanish if either of these two motions stopped. It is easil verified that the sum of these two vectors defines the oriolis acceleration. Their direction is that of a c when t approaches zero, and since RR9 5 u u and T 0T9 5 v 9 2 v 5 (r 1 r)v 2 rv 5 v r, we check that a c is equal to lim a RR t0 t 1 T T u b 5 lim au t t0 t 1 v r b 5 uv 1 v u 5 2v u t Formulas (15.33) and (15.36) can be used to analze the motion of mechanisms which contain parts sliding on each other. The make it possible, for eample, to relate the absolute and relative motions of sliding pins and collars (see Sample Probs and 15.10). The concept of oriolis acceleration is also ver useful in the stud of long-range projectiles and of other bodies whose motions are appreciabl affected b the rotation of the earth. s was pointed out in Sec. 12.2, a sstem of aes attached to the earth does not trul constitute a newtonian frame of reference; such a sstem of aes should actuall be considered as rotating. The formulas derived in this section will therefore facilitate the stud of the motion of bodies with respect to aes attached to the earth Plane Motion of a Particle Relative to a Rotating Frame. oriolis cceleration T' a c = 2wu a = rw 2 w v ' = (r + Δr)w T T" Fig v = rw Δq (b) v ' v ' r Δq P P u' (a) r u' ' (c) R' Δq u u Δr R u 975

63 isk S R R = 50 mm P f = 135 isk l = 2R SMPLE PRLEM 15.9 The Geneva mechanism shown is used in man counting instruments and in other applications where an intermittent rotar motion is required. isk rotates with a constant counterclockwise angular velocit V of 10 rad/s. pin P is attached to disk and slides along one of several slots cut in disk S. It is desirable that the angular velocit of disk S be zero as the pin enters and leaves each slot; in the case of four slots, this will occur if the distance between the centers of the disks is l 5 12 R. t the instant when f 5 150, determine (a) the angular velocit of disk S, (b) the velocit of pin P relative to disk S. SLUTIN isk S P isk We solve triangle P, which corresponds to the position f Using the law of cosines, we write R r 2 5 R 2 1 l 2 2 2Rl cos R 2 r R mm r b P' l = 2R f = 150 From the law of sines, sin b R 5 sin 30 r sin b 5 sin b Since pin P is attached to disk, and since disk rotates about point, the magnitude of the absolute velocit of P is v P 5 Rv 5 (50 mm)(10 rad/s) mm/s v P mm/s d 60 We consider now the motion of pin P along the slot in disk S. enoting b P9 the point of disk S which coincides with P at the instant considered and selecting a rotating frame S attached to disk S, we write v P 5 v P9 1 v P/S v P' v P 30 v P/ b = 42.4 Noting that v P9 is perpendicular to the radius P and that v P/S is directed along the slot, we draw the velocit triangle corresponding to the equation above. From the triangle, we compute g v P9 5 v P sin g 5 (500 mm/s) sin 17.6 v P mm/s f 42.4 v P/S 5 v P cos g 5 (500 mm/s) cos 17.6 v P/S 5 v P/S mm/s d 42.4 Since v P9 is perpendicular to the radius P, we write v P9 5 rv S mm/s 5 (37.1 mm)v S V S 5 V S rad/s i 976

64 SMPLE PRLEM In the Geneva mechanism of Sample Prob. 15.9, disk rotates with a constant counterclockwise angular velocit V of 10 rad/s. t the instant when f 5 150, determine the angular acceleration of disk S isk S P isk R r P' f = 150 b l = 2R (a P' ) n = 618 mm/s 2 a c = 3890 mm/s SLUTIN Referring to Sample Prob. 15.9, we obtain the angular velocit of the frame S attached to disk S and the velocit of the pin relative to S: v S rad/s i b v P/S mm/s d 42.4 Since pin P moves with respect to the rotating frame S, we write a P 5 a P9 1 a P/S 1 a c (1) Each term of this vector equation is investigated separatel. bsolute cceleration a P. Since disk rotates with a constant angular velocit, the absolute acceleration a P is directed toward. We have a P 5 Rv 2 5 (500 mm)(10 rad/s) mm/s 2 a P mm/s 2 c 30 cceleration a P9 of the oinciding Point P9. The acceleration a P9 of the point P9 of the frame S which coincides with P at the instant considered is resolved into normal and tangential components. (We recall from Sample Prob that r mm.) (a P9 ) n 5 rv 2 S 5 (37.1 mm)(4.08 rad/s) mm/s 2 (a P9 ) n mm/s 2 d 42.4 (a P9 ) t 5 ra S a S (a P9 ) t a S f 42.4 Relative cceleration a PS. Since the pin P moves in a straight slot cut in disk S, the relative acceleration a P/S must be parallel to the slot; i.e., its direction must be a z oriolis cceleration a c. Rotating the relative velocit v P/S through 90 in the sense of V S, we obtain the direction of the oriolis component of the acceleration: h We write a c 5 2v S v P/S 5 2(4.08 rad/s)(477 mm/s) mm/s 2 a c mm/s 2 h 42.4 We rewrite Eq. (1) and substitute the accelerations found above: 42.4 (a P' ) t = 37.1a a P = 5000 mm/s 2 30 a P/ 42.4 a P 5 (a P9 ) n 1 (a P9 ) t 1 a P/S 1 a c [5000 c 30 ] 5 [618 d 42.4 ] 1 [37.1a S f 42.4 ] 1 [a P/S a 42.4 ] 1 [3890 h 42.4 ] z Equating components in a direction perpendicular to the slot, 5000 cos a S S 5 S rad/s 2 i 977

65 SLVING PRLEMS N YUR WN In this lesson ou studied the rate of change of a vector with respect to a rotating frame and then applied our knowledge to the analsis of the plane motion of a particle relative to a rotating frame. 1. Rate of change of a vector with respect to a fied frame and with respect to a rotating frame. enoting b ( Q) XYZ the rate of change of a vector Q with respect to a fied frame XYZ and b ( Q) z its rate of change with respect to a rotating frame z, we obtained the fundamental relation ( Q) XYZ 5 ( Q) z 1 V 3 Q (15.31) where V is the angular velocit of the rotating frame. This fundamental relation will now be applied to the solution of two-dimensional problems. 2. Plane motion of a particle relative to a rotating frame. Using the above fundamental relation and designating b ^ the rotating frame, we obtained the following epressions for the velocit and the acceleration of a particle P: v P 5 v P9 1 v P/^ (15.33) a P 5 a P9 1 a P/^ 1 a c (15.36) In these equations: a. The subscript P refers to the absolute motion of the particle P, that is, to its motion with respect to a fied frame of reference XY. b. The subscript P9 refers to the motion of the point P9 of the rotating frame ^ which coincides with P at the instant considered. c. The subscript P/^ refers to the motion of the particle P relative to the rotating frame ^. d. The term a c represents the oriolis acceleration of point P. Its magnitude is 2Vv P/^, and its direction is found b rotating v P/^ through 90 in the sense of rotation of the frame ^. You should keep in mind that the oriolis acceleration should be taken into account whenever a part of the mechanism ou are analzing is moving with respect to another part that is rotating. The problems ou will encounter in this lesson involve collars that slide on rotating rods, booms that etend from cranes rotating in a vertical plane, etc. When solving a problem involving a rotating frame, ou will find it convenient to draw vector diagrams representing Eqs. (15.33) and (15.36), respectivel, and use these diagrams to obtain either an analtical or a graphical solution. 978

66 PRLEMS and Two rotating rods are connected b slider block P. The rod attached at rotates with a constant angular velocit v. For the given data, determine for the position shown (a) the angular velocit of the rod attached at, (b) the relative velocit of slider block P with respect to the rod on which it slides b 5 8 in., v 5 6 rad/s b mm, v 5 10 rad/s. E P P Fig. P and P b and Two rotating rods are connected b slider block P. The velocit v 0 of the slider block relative to the rod is constant and is directed outwards. For the given data, determine the angular velocit of each rod in the position shown b mm, v mm/s b 5 8 in., v in./s. Fig. P and P b P E 20 in. 30 Fig. P and P and Pin P is attached to the collar shown; the motion of the pin is guided b a slot cut in rod and b the collar that slides on rod E. Knowing that at the instant considered the rods rotate clockwise with constant angular velocities, determine for the given data the velocit of pin F v E 5 4 rad/s, v bd rad/s v ae rad/s, v rad/s. 979

67 980 Kinematics of Rigid odies and Two rods E and pass through holes drilled into a heagonal block. (The holes are drilled in different planes so that the rods will not touch each other.) Knowing that at the instant considered rod E rotates counterclockwise with a constant angular velocit V, determine, for the given data, the relative H q velocit of the block with respect to each rod. 60 E (a) u 5 90, (b) u u l Fig. P and P Four pins slide in four separate slots cut in a circular plate as shown. When the plate is at rest, each pin has a velocit directed as shown and of the same constant magnitude u. If each pin maintains the same velocit relative to the plate when the plate rotates about with a constant counterclockwise angular velocit V, determine the acceleration of each pin. P 2 u Solve Prob , assuming that the plate rotates about with a constant clockwise angular velocit V. u P 1 r r r r P 3 u t the instant shown the length of the boom is being decreased at the constant rate of 0.2 m/s and the boom is being lowered at the constant rate of 0.08 rad/s. etermine (a) the velocit of point, (b) the acceleration of point. P 4 u Fig. P m q = 30 Fig. P and P z E 12 in. u Fig. P and P in. F 5 in t the instant shown the length of the boom is being increased at the constant rate of 0.2 m/s and the boom is being lowered at the constant rate of 0.08 rad/s. etermine (a) the velocit of point, (b) the acceleration of point and The sleeve is welded to an arm that rotates about with a constant angular velocit V. In the position shown rod F is being moved to the left at a constant speed u 5 16 in./s relative to the sleeve. For the given angular velocit V, determine the acceleration (a) of point, (b) of the point of rod F that coincides with point E V 5 (3 rad/s) i V 5 (3 rad/s) j.

68 The cage of a mine elevator moves downward at a constant speed of 40 ft/s. etermine the magnitude and direction of the oriolis acceleration of the cage if the elevator is located (a) at the equator, (b) at latitude 40 north, (c) at latitude 40 south rocket sled is tested on a straight track that is built along a meridian. Knowing that the track is located at latitude 40 north, determine the oriolis acceleration of the sled when it is moving north at a speed of 900 km/h. w u (a) 4 in. Problems The motion of nozzle is controlled b arm. t the instant shown the arm is rotating counterclockwise at the constant rate v rad/s and portion is being etended at the constant rate u 5 10 in./s with respect to the arm. For each of the arrangements shown, determine the acceleration of the nozzle. w 8 in. 3 in. u 4 in Solve Prob , assuming that the direction of the relative velocit u is reversed so that portion is being retracted. (b) and chain is looped around two gears of radius 40 mm that can rotate freel with respect to the 320-mm arm. The chain moves about arm in a clockwise direction at the constant rate of 80 mm/s relative to the arm. Knowing that in the position shown arm rotates clockwise about at the constant rate v rad/s, determine the acceleration of each of the chain links indicated Links 1 and Links 3 and Rod of length R rotates about with a constant clockwise angular velocit V 1. t the same time, rod of length r rotates about with a constant counterclockwise angular velocit V 2 with respect to rod. Show that if V 2 5 2V 1 the acceleration of point passes through point. Further show that this result is independent of R, r, and u. Fig. P u mm 160 mm Fig. P and P w 2 r w 1 Fig. P and P R Rod of length R 5 15 in. rotates about with a constant clockwise angular velocit V 1 of 5 rad/s. t the same time, rod of length r 5 8 in. rotates about with a constant counterclockwise angular velocit V 2 of 3 rad/s with respect to rod. Knowing that u 5 60, determine for the position shown the acceleration of point.

69 982 Kinematics of Rigid odies The collar P slides outward at a constant relative speed u along rod, which rotates counterclockwise with a constant angular velocit of 20 rpm. Knowing that r mm when u 5 0 and that the collar reaches when u 5 90, determine the magnitude of the acceleration of the collar P just as it reaches. u w r q P 500 mm Fig. P Pin P slides in a circular slot cut in the plate shown at a constant relative speed u 5 90 mm/s. Knowing that at the instant shown the plate rotates clockwise about at the constant rate v 5 3 rad/s, determine the acceleration of the pin if it is located at (a) point, (b) point, (c) point. u P 100 mm w Fig. P and P Pin P slides in a circular slot cut in the plate shown at a constant relative speed u 5 90 mm/s. Knowing that at the instant shown the angular velocit V of the plate is 3 rad/s clockwise and is decreasing at the rate of 5 rad/s 2, determine the acceleration of the pin if it is located at (a) point, (b) point, (c) point and Knowing that at the instant shown the rod attached at rotates with a constant counterclockwise angular velocit V of 6 rad/s, determine the angular velocit and angular acceleration of the rod attached at. 0.4 m 0.4 m Fig. P Fig. P15.176

70 t the instant shown bar has an angular velocit of 3 rad/s and an angular acceleration of 2 rad/s 2, both counterclockwise, determine the angular acceleration of the plate. Problems in. 3 in. 4 in. 6 in. Fig. P and P t the instant shown bar has an angular velocit of 3 rad/s and an angular acceleration of 2 rad/s 2, both clockwise, determine the angular acceleration of the plate The Geneva mechanism shown is used to provide an intermittent rotar motion of disk S. isk rotates with a constant counterclockwise angular velocit V of 8 rad/s. pin P is attached to disk and can slide in one of the si equall spaced slots cut in disk S. It is desirable that the angular velocit of disk S be zero as the pin enters and leaves each of the si slots; this will occur if the distance between the centers of the disks and the radii of the disks are related as shown. etermine the angular velocit and angular acceleration of disk S at the instant when f isk S R S = 3R P R = 1.25 in. f 125 mm isk when f = 120 l = 2R Fig. P mm In Prob , determine the angular velocit and angular acceleration of disk S at the instant when f The disk shown rotates with a constant clockwise angular velo c it of 12 rad/s. t the instant shown, determine (a) the angular velocit and angular acceleration of rod, (b) the velocit and acceleration of the point of the rod coinciding with E. E Fig. P15.181

71 984 Kinematics of Rigid odies * Rod passes through a collar which is welded to link E. Knowing that at the instant shown block moves to the right at a constant speed of 75 in./s, determine (a) the angular velocit of rod, (b) the velocit relative to the collar of the point of the rod in contact with the collar, (c) the acceleration of the point of the rod in contact with the collar. (Hint: Rod and link E have the same V and the same.) 30 E 6 in. Fig. P * Solve Prob assuming block moves to the left at a constant speed of 75 in./s (a) 1 1 = 2 (b) Fig *15.12 MTIN UT FIXE PINT In Sec the motion of a rigid bod constrained to rotate about a fied ais was considered. The more general case of the motion of a rigid bod which has a fied point will now be eamined. First, it will be proved that the most general displacement of a rigid bod with a fied point is equivalent to a rotation of the bod about an ais through. Instead of considering the rigid bod itself, we can detach a sphere of center from the bod and analze the motion of that sphere. learl, the motion of the sphere completel characterizes the motion of the given bod. Since three points define the position of a solid in space, the center and two points and on the surface of the sphere will define the position of the sphere and thus the position of the bod. Let 1 and 1 characterize the position of the sphere at one instant, and let 2 and 2 characterize its position at a later instant (Fig a). Since the sphere is rigid, the lengths of the arcs of great circle 1 1 and 2 2 must be equal, but ecept for this requirement, the positions of 1, 2, 1, and 2 are arbitrar. We propose to prove that the points and can be brought, respectivel, from 1 and 1 into 2 and 2 b a single rotation of the sphere about an ais. For convenience, and without loss of generalit, we select point so that its initial position coincides with the final position of ; thus, (Fig b). We draw the arcs of great circle 1 2, 2 2 and the arcs bisecting, respectivel, 1 2 and 2 2. Let be the point of intersection of these last two arcs; we complete the construction b drawing 1, 2, and 2. s pointed out above, because of the rigidit of the sphere, Since is b construction equidistant from 1, 2, and 2, we also have This is known as Euler s theorem.

72 s a result, the spherical triangles 1 2 and 1 2 are congruent and the angles 1 2 and 1 2 are equal. enoting b u the common value of these angles, we conclude that the sphere can be brought from its initial position into its final position b a single rotation through u about the ais. It follows that the motion during a time interval t of a rigid bod with a fied point can be considered as a rotation through u about a certain ais. rawing along that ais a vector of magnitude u/t and letting t approach zero, we obtain at the limit the instantaneous ais of rotation and the angular velocit V of the bod at the instant considered (Fig ). The velocit of a particle P of the bod can then be obtained, as in Sec. 15.3, b forming the vector product of V and of the position vector r of the particle: v 5 dr dt 5 V 3 r (15.37) The acceleration of the particle is obtained b differentiating (15.37) with respect to t. s in Sec we have a 5 3 r 1 V 3 (V 3 r) (15.38) Motion about a Fied Point w a P r Fig where the angular acceleration is defined as the derivative 5 dv dt (15.39) of the angular velocit V. In the case of the motion of a rigid bod with a fied point, the direction of V and of the instantaneous ais of rotation changes from one instant to the net. The angular acceleration therefore reflects the change in direction of V as well as its change in magnitude and, in general, is not directed along the instantaneous ais of rotation. While the particles of the bod located on the instantaneous ais of rotation have zero velocit at the instant considered, the do not have zero acceleration. lso, the accelerations of the various particles of the bod cannot be determined as if the bod were rotating permanentl about the instantaneous ais. Recalling the definition of the velocit of a particle with position vector r, we note that the angular acceleration, as epressed in (15.39), represents the velocit of the tip of the vector V. This propert ma be useful in the determination of the angular acceleration of a rigid bod. For eample, it follows that the vector is tangent to the curve described in space b the tip of the vector V. We should note that the vector V moves within the bod, as well as in space. It thus generates two cones called, respectivel, the bod cone and the space cone (Fig ). It can be shown that at an given instant, the two cones are tangent along the instantaneous ais of rotation and that as the bod moves, the bod cone appears to roll on the space cone. Space cone w Fig a od cone It is recalled that a cone is, b definition, a surface generated b a straight line passing through a fied point. In general, the cones considered here will not be circular cones.

73 986 Kinematics of Rigid odies efore concluding our analsis of the motion of a rigid bod with a fied point, we should prove that angular velocities are actuall vectors. s indicated in Sec. 2.3, some quantities, such as the finite rotations of a rigid bod, have magnitude and direction but do not obe the parallelogram law of addition; these quantities cannot be considered as vectors. In contrast, angular velocities (and also infinitesimal rotations), as will be demonstrated presentl, do obe the parallelogram law and thus are trul vector quantities. Photo 15.8 When the ladder rotates about its fied base, its angular velocit can be obtained b adding the angular velocities which correspond to simultaneous rotations about two different aes. w 1 w 2 w 1 w 2 w (a) (b) Fig onsider a rigid bod with a fied point which at a given instant rotates simultaneousl about the aes and with angular velocities V 1 and V 2 (Fig a). We know that this motion must be equivalent at the instant considered to a single rotation of angular velocit V. We propose to show that V 5 V 1 1 V 2 (15.40) i.e., that the resulting angular velocit can be obtained b adding V 1 and v 2 b the parallelogram law (Fig b). onsider a particle P of the bod, defined b the position vector r. enoting, respectivel, b v 1, v 2, and v the velocit of P when the bod rotates about onl, about onl, and about both aes simultaneousl, we write v 5 V 3 r v 1 5 V 1 3 r v 2 5 V 2 3 r (15.41) ut the vectorial character of linear velocities is well established (since the represent the derivatives of position vectors). We therefore have v 5 v 1 1 v 2 where the plus sign indicates vector addition. Substituting from (15.41), we write V 3 r 5 V 1 3 r 1 V 2 3 r V 3 r 5 (V 1 1 V 2 ) 3 r where the plus sign still indicates vector addition. Since the relation obtained holds for an arbitrar r, we conclude that (15.40) must be true.

74 *15.13 GENERL MTIN General Motion 987 The most general motion of a rigid bod in space will now be considered. Let and be two particles of the bod. We recall from Sec that the velocit of with respect to the fied frame of reference XYZ can be epressed as v 5 v 1 v / (15.42) where v / is the velocit of relative to a frame X9Y9Z9 attached to and of fied orientation (Fig ). Since is fied in this frame, the motion of the bod relative to X9Y9Z9 is the motion of a bod with a fied point. The relative velocit v / can therefore be obtained from (15.37) after r has been replaced b the position vector r / of relative to. Substituting for v / into (15.42), we write Y Z' a r Y' w r / X' v 5 v 1 v 3 r / (15.43) where V is the angular velocit of the bod at the instant considered. The acceleration of is obtained b a similar reasoning. We first write a 5 a 1 a / Z Fig X and, recalling Eq. (15.38), a 5 a 1 3 r / 1 V 3 (V 3 r / ) (15.44) where is the angular acceleration of the bod at the instant considered. Equations (15.43) and (15.44) show that the most general motion of a rigid bod is equivalent, at an given instant, to the sum of a translation, in which all the particles of the bod have the same velocit and acceleration as a reference particle, and of a motion in which particle is assumed to be fied. It is easil shown, b solving (15.43) and (15.44) for v and a, that the motion of the bod with respect to a frame attached to would be characterized b the same vectors V and as its motion relative to X9Y9Z9. The angular velocit and angular acceleration of a rigid bod at a given instant are thus independent of the choice of reference point. n the other hand, one should keep in mind that whether the moving frame is attached to or to, it should maintain a fied orientation; that is, it should remain parallel to the fied reference frame XYZ throughout the motion of the rigid bod. In man problems it will be more convenient to use a moving frame which is allowed to rotate as well as to translate. The use of such moving frames will be discussed in Secs and It is recalled from Sec that, in general, the vectors V and are not collinear, and that the accelerations of the particles of the bod in their motion relative to the frame X9Y9Z9 cannot be determined as if the bod were rotating permanentl about the instantaneous ais through.

75 Y P SMPLE PRLEM Z w 2 w 1 q = 30 X The crane shown rotates with a constant angular velocit V 1 of 0.30 rad/s. Simultaneousl, the boom is being raised with a constant angular velocit V 2 of 0.50 rad/s relative to the cab. Knowing that the length of the boom P is l 5 12 m, determine (a) the angular velocit V of the boom, (b) the angular acceleration of the boom, (c) the velocit v of the tip of the boom, (d) the acceleration a of the tip of the boom. SLUTIN Z z Y m w 1 = 0.30j w 2 = 0.50k P 6 m X a. ngular Velocit of oom. dding the angular velocit V 1 of the cab and the angular velocit V 2 of the boom relative to the cab, we obtain the angular velocit V of the boom at the instant considered: V 5 V 1 1 V 2 V 5 (0.30 rad/s)j 1 (0.50 rad/s)k b. ngular cceleration of oom. The angular acceleration of the boom is obtained b differentiating V. Since the vector V 1 is constant in magnitude and direction, we have 5 V 5 V 1 1 V V 2 where the rate of change V 2 is to be computed with respect to the fied frame XYZ. However, it is more convenient to use a frame z attached to the cab and rotating with it, since the vector V 2 also rotates with the cab and therefore has zero rate of change with respect to that frame. Using Eq. (15.31) with Q 5 V 2 and V 5 V 1, we write ( Q) XYZ 5 ( Q) z 1 V 3 Q (V 2) XYZ 5 (V 2) z 1 V 1 3 V 2 5 (V 2) XYZ (0.30 rad/s)j 3 (0.50 rad/s)k 5 (0.15 rad/s 2 )i Y m w 1 = 0.30j P 6 m c. Velocit of Tip of oom. Noting that the position vector of point P is r 5 (10.39 m)i + (6 m)j and using the epression found for V in part a, we write i j k v 5 V 3 r rad/s 0.50 rad/s m 6 m 0 v 5 2(3 m/s)i 1 (5.20 m/s)j 2 (3.12 m/s)k Z a = 0.15i w 2 = 0.50k X d. cceleration of Tip of oom. Recalling that v 5 V 3 r, we write a 5 3 r 1 V 3 (V 3 r) 5 3 r 1 V 3 v i j k i j k a k i i j k a 5 2(3.54 m/s 2 )i 2 (1.50 m/s 2 )j 1 (1.80 m/s 2 )k 988

76 2 in. 3 in. w 1 q z SMPLE PRLEM The rod, of length 7 in., is attached to the disk b a ball-and-socket connection and to the collar b a clevis. The disk rotates in the z plane at a constant rate v rad/s, while the collar is free to slide along the horizontal rod. For the position u 5 0, determine (a) the velocit of the collar, (b) the angular velocit of the rod. z z 2 in. w 1 = w 1 i 6 in. v w1 = 12 i r = 2k r = 6i + 3j r / = 6i + 3j 2k E r E/ = 3j + 2k v 3 in. 3 in. 2 in. SLUTIN a. Velocit of ollar. Since point is attached to the disk and since collar moves in a direction parallel to the ais, we have v 5 V 1 3 r 5 12i 3 2k 5 224j v 5 v i enoting b V the angular velocit of the rod, we write v 5 v 1 v / 5 v 1 V 3 r / i j k v i 5224j 1 v v v z v i 5 224j 1 (22v 2 3v z )i 1 (6v z 1 2v )j 1 (3v 2 6v )k Equating the coefficients of the unit vectors, we obtain v 5 22v 23v z (1) v 16v z (2) 0 5 3v 26v (3) Multipling Eqs. (1), (2), (3), respectivel, b 6, 3, 22 and adding, we write 6v v v 5 2(12 in./s)i b. ngular Velocit of Rod. We note that the angular velocit cannot be determined from Eqs. (1), (2), and (3), since the determinant formed b the coefficients of v, v, and v z is zero. We must therefore obtain an additional equation b considering the constraint imposed b the clevis at. The collar-clevis connection at permits rotation of about the rod and also about an ais perpendicular to the plane containing and. It prevents rotation of about the ais E, which is perpendicular to and lies in the plane containing and. Thus the projection of V on r E/ must be zero and we write V? r E/ 5 0 (v i 1 v j 1 v z k)? (23j 1 2k) v 1 2v z 5 0 (4) Solving Eqs. (1) through (4) simultaneousl, we obtain v v v v z V 5 (3.69 rad/s)i 1 (1.846 rad/s)j 1 (2.77 rad/s)k We could also note that the direction of E is that of the vector triple product r / 3 (r / 3 r / ) and write V? [r / 3 (r / 3 r / )] 5 0. This formulation would be particularl useful if the rod were skew. 989

77 SLVING PRLEMS N YUR WN In this lesson ou started the stud of the kinematics of rigid bodies in three dimensions. You first studied the motion of a rigid bod about a fied point and then the general motion of a rigid bod.. Motion of a rigid bod about a fied point. To analze the motion of a point of a bod rotating about a fied point ou ma have to take some or all of the following steps. 1. etermine the position vector r connecting the fied point to point. 2. etermine the angular velocit V of the bod with respect to a fied frame of reference. The angular velocit V will often be obtained b adding two component angular velocities V 1 and V 2 [Sample Prob ]. 3. ompute the velocit of b using the equation v 5 V 3 r (15.37) Your computation will usuall be facilitated if ou epress the vector product as a determinant. 4. etermine the angular acceleration of the bod. The angular acceleration represents the rate of change (V ) XYZ of the vector V with respect to a fied frame of reference XYZ and reflects both a change in magnitude and a change in direction of the angular velocit. However, when computing ou ma find it convenient to first compute the rate of change (V ) z of V with respect to a rotating frame of reference z of our choice and use Eq. (15.31) of the preceding lesson to obtain. You will write 5 (V ) XYZ 5 (V ) z 1 V 3 V where V is the angular velocit of the rotating frame z [Sample Prob ]. 5. ompute the acceleration of b using the equation a 5 3 r 1 V 3 (V 3 r) (15.38) Note that the vector product (V 3 r) represents the velocit of point and was computed in step 3. lso, the computation of the first vector product in (15.38) will be facilitated if ou epress this product in determinant form. Remember that, as was the case with the plane motion of a rigid bod, the instantaneous ais of rotation cannot be used to determine accelerations. 990

78 . General motion of a rigid bod. The general motion of a rigid bod ma be considered as the sum of a translation and a rotation. Keep the following in mind: a. In the translation part of the motion, all the points of the bod have the same velocit v and the same acceleration a as the point of the bod that has been selected as the reference point. b. In the rotation part of the motion, the same reference point is assumed to be a fied point. 1. To determine the velocit of a point of the rigid bod when ou know the velocit v of the reference point and the angular velocit V of the bod, ou simpl add v to the velocit v / 5 V 3 r / of in its rotation about : v 5 v 1 V 3 r / (15.43) s indicated earlier, the computation of the vector product will usuall be facilitated if ou epress this product in determinant form. Equation (15.43) can also be used to determine the magnitude of v when its direction is known, even if V is not known. While the corresponding three scalar equations are linearl dependent and the components of V are indeterminate, these components can be eliminated and v can be found b using an appropriate linear combination of the three equations [Sample Prob , part a]. lternativel, ou can assign an arbitrar value to one of the components of V and solve the equations for v. However, an additional equation must be sought in order to determine the true values of the components of V [Sample Prob , part b]. 2. To determine the acceleration of a point of the rigid bod when ou know the acceleration a of the reference point and the angular acceleration of the bod, ou simpl add a to the acceleration of in its rotation about, as epressed b Eq. (15.38): a 5 a 1 3 r / 1 V 3 (V 3 r / ) (15.44) Note that the vector product (V 3 r / ) represents the velocit v / of relative to and ma alread have been computed as part of our calculation of v. We also remind ou that the computation of the other two vector products will be facilitated if ou epress these products in determinant form. The three scalar equations associated with Eq. (15.44) can also be used to determine the magnitude of a when its direction is known, even if V and are not known. While the components of V and are indeterminate, ou can assign arbitrar values to one of the components of V and to one of the components of and solve the equations for a. 991

79 PRLEMS Plate and rod are rigidl connected and rotate about the ball-and-socket joint with an angular velocit V 5 v i 1 v j 1 v z k. Knowing that v 5 (3 in./s)i 1 (14 in./s)j 1 (v ) z k and v rad/s, determine (a) the angular velocit of the assembl, (b) the velocit of point. 4 in. 8 in. 8 in. 6 in. 4 in. z Fig. P Solve Prob , assuming that v rad/s t the instant considered the radar antenna shown rotates about the origin of coordinates with an angular velocit V 5 v i 1 v j 1 v z k. Knowing that (v ) mm/s, (v ) mm/s, and (v ) z mm/s, determine (a) the angular velocit of the antenna, (b) the velocit of point. 0.3 m z Fig. P and P m 0.25 m t the instant considered the radar antenna shown rotates about the origin of coordinates with an angular velocit V 5 v i 1 v j 1 v z k. Knowing that (v ) mm/s, (v ) mm/s, and (v ) z mm/s, determine (a) the angular velocit of the antenna, (b) the velocit of point.

80 The blade assembl of an oscillating fan rotates with a constant angular velocit V 1 5 2(360 rpm)i with respect to the motor housing. etermine the angular acceleration of the blade assembl, knowing that at the instant shown the angular velocit and angular acceleration of the motor housing are, respectivel, V 2 5 2(2.5 rpm)j and Problems 993 w 2 w 1 z w 1 z w 2 Fig. P The rotor of an electric motor rotates at the constant rate v rpm. etermine the angular acceleration of the rotor as the motor is rotated about the ais with a constant angular velocit V 2 of 6 rpm counterclockwise when viewed from the positive ais. Fig. P In the sstem shown, disk is free to rotate about the horizontal rod. ssuming that disk is stationar (v 2 5 0), and that shaft rotates with a constant angular velocit V 1, determine (a) the angular velocit of disk, (b) the angular acceleration of disk In the sstem shown, disk is free to rotate about the horizontal rod. ssuming that shaft and disk rotate with constant angular velocities V 1 and V 2, respectivel, both counterclockwise, determine (a) the angular velocit of disk, (b) the angular acceleration of disk The L-shaped arm rotates about the z ais with a constant angular velocit V 1 of 5 rad/s. Knowing that the 150-mm-radius disk rotates about with a constant angular velocit V 2 of 4 rad/s, determine the angular acceleration of the disk. 150 mm z w 1 r R w 2 Fig. P and P w mm z w 1 Fig. P In Prob , determine (a) the velocit of point, (b) the acceleration of point.

81 994 Kinematics of Rigid odies in.-radius disk spins at the constant rate v rad/s about an ais held b a housing attached to a horizontal rod that rotates at the constant rate v rad/s. For the position shown, determine (a) the angular acceleration of the disk, (b) the acceleration of point P on the rim of the disk if u 5 0, (c) the acceleration of point P on the rim of the disk if u z w 1 Fig. P and P in. q w 2 P in.-radius disk spins at the constant rate v rad/s about an ais held b a housing attached to a horizontal rod that rotates at the constant rate v rad/s. Knowing that u 5 30, determine the acceleration of point P on the rim of the disk gun barrel of length P 5 4 m is mounted on a turret as shown. To keep the gun aimed at a moving target the azimuth angle b is being increased at the rate db/dt 5 30 /s and the elevation angle g is being increased at the rate dg/dt 5 10 /s. For the position b 5 90 and g 5 30, determine (a) the angular velocit of the barrel, (b) the angular acceleration of the barrel, (c) the velocit and acceleration of point P. P g b z Fig. P In the planetar gear sstem shown, gears and are rigidl connected to each other and rotate as a unit about the inclined shaft. Gears and rotate with constant angular velocities of 30 rad/s and 20 rad/s, respectivel (both counterclockwise when viewed from the right). hoosing the ais to the right, the ais upward, and the z ais pointing out of the plane of the figure, determine (a) the common angular velocit of gears and, (b) the angular velocit of shaft FH, which is rigidl attached to the inclined shaft in in. E F G H Fig. P in. 4 in.

82 mm-radius wheel is mounted on an ale of length 100 mm. The wheel rolls without sliding on the horizontal floor, and the ale is perpendicular to the plane of the wheel. Knowing that the sstem rotates about the ais at a constant rate v rad/s, determine (a) the angular velocit of the wheel, (b) the angular acceleration of the wheel, (c) the acceleration of point located at the highest point on the rim of the wheel Several rods are brazed together to form the robotic guide arm shown which is attached to a ball-and-socket joint at. Rod slides in a straight inclined slot while rod slides in a slot parallel to the z-ais. Knowing that at the instant shown v 5 (180 mm/s)k, determine (a) the angular velocit of the guide arm, (b) the velocit of point, (c) the velocit of point. z Problems w Fig. P E z imensions in mm Fig. P z In Prob the speed of point is known to be constant. For the position shown, determine (a) the angular acceleration of the guide arm, (b) the acceleration of point. v The 45 sector of a 10-in.-radius circular plate is attached to a fied ball-and-socket joint at. s edge moves on the horizontal surface, edge moves along the vertical wall. Knowing that point moves at a constant speed of 60 in./s, determine for the position shown (a) the angular velocit of the plate, (b) the velocit of point Rod of length 275 mm is connected b ball-and-socket joints to collars and, which slide along the two rods shown. Knowing that collar moves toward the origin at a constant speed of 180 mm/s, determine the velocit of collar when c mm. Fig. P b c Rod of length 275 mm is connected b ball-and-socket joints to collars and, which slide along the two rods shown. Knowing that collar moves toward the origin at a constant speed of 180 mm/s, determine the velocit of collar when c 5 50 mm. 150 mm z Fig. P and P15.203

83 996 Kinematics of Rigid odies Rod is connected b ball-and-socket joints to collar and to the 16-in.-diameter disk. Knowing that disk rotates counterclockwise at the constant rate v rad/s in the z plane, determine the velocit of collar for the position shown. 20 in Rod of length 29 in. is connected b ball-and-socket joints to the rotating crank and to the collar. rank is of length 8 in. and rotates in the horizontal plane at the constant rate v rad/s. t the instant shown, when crank is parallel to the z ais, determine the velocit of collar. z 25 in. w 0 8 in. 8 in. Fig. P in. 8 in. z 12 in. w 0 Fig. P mm Rod of length 300 mm is connected b ball-and-socket joints to collars and, which slide along the two rods shown. Knowing that collar moves toward point at a constant speed of 50 mm/s, determine the velocit of collar when c 5 80 mm. c z Fig. P and P mm Rod of length 300 mm is connected b ball-and-socket joints to collars and, which slide along the two rods shown. Knowing that collar moves toward point at a constant speed of 50 mm/s, determine the velocit of collar when c mm.

84 Rod of length 25 in. is connected b ball-and-socket joints to collars and, which slide along the two rods shown. Knowing that collar moves toward point E at a constant speed of 20 in./s, determine the velocit of collar as collar passes through point. Problems Rod of length 25 in. is connected b ball-and-socket joints to collars and, which slide along the two rods shown. Knowing that collar moves toward point E at a constant speed of 20 in./s, determine the velocit of collar as collar passed through point. 20 in Two shafts and EG, which lie in the vertical z plane, are connected b a universal joint at. Shaft rotates with a constant angular velocit V 1 as shown. t a time when the arm of the crosspiece attached to shaft is vertical, determine the angular velocit of shaft EG. w 2 9 in. 12 in. 20 in. z E G Fig. P and P E 25 5 in. w 1 4 in. z 3 in. Fig. P Solve Prob , assuming that the arm of the crosspiece attached to the shaft is horizontal In Prob , the ball-and-socket joint between the rod and collar is replaced b the clevis shown. etermine (a) the angular velocit of the rod, (b) the velocit of collar. Fig. P In Prob , the ball-and-socket joint between the rod and collar is replaced b the clevis shown. etermine (a) the angular velocit of the rod, (b) the velocit of collar. Fig. P15.213

85 998 Kinematics of Rigid odies through For the mechanism of the problem indicated, determine the acceleration of collar Mechanism of Prob Mechanism of Prob Mechanism of Prob Mechanism of Prob Mechanism of Prob Mechanism of Prob *15.14 THREE-IMENSINL MTIN F PRTILE RELTIVE T RTTING FRME. RILIS ELERTIN Y We saw in Sec that given a vector function Q(t) and two frames of reference centered at a fied frame XYZ and a rotating frame z the rates of change of Q with respect to the two frames satisf the relation ( Q) XYZ 5 ( Q) z 1 V 3 Q (15.31) We had assumed at the time that the frame z was constrained to rotate about a fied ais. However, the derivation given in Sec remains valid when the frame z is constrained onl to have a fied point. Under this more general assumption, the ais represents the instantaneous ais of rotation of the frame z (Sec ) and the vector V, its angular velocit at the instant considered (Fig ). Let us now consider the three-dimensional motion of a particle P relative to a rotating frame z constrained to have a fied origin. Let r be the position vector of P at a given instant and V be the angular velocit of the frame z with respect to the fied frame XYZ at the same instant (Fig ). The derivations given in Sec for the two-dimensional motion of a particle can be readil etended to the three-dimensional case, and the absolute velocit v P of P (i.e., its velocit with respect to the fied frame XYZ) can be epressed as Ω Z Fig Ω j z Y k r i P Q X X v P 5 V 3 r 1 (ṙ) z (15.45) z Z Fig enoting b ^ the rotating frame z, we write this relation in the alternative form v P 5 v P9 1 v P/^ (15.46) where v P 5 absolute velocit of particle P v P9 5 velocit of point P9 of moving frame ^ coinciding with P v P/^ 5 velocit of P relative to moving frame ^ The absolute acceleration a P of P can be epressed as a P 5 V 3 r 1 V 3 (V 3 r) 1 2V 3 (ṙ) z 1 ( r) z (15.47)

86 n alternative form is Frame of Reference in General Motion 999 a P 5 a P9 1 a P/^ 1 a c (15.48) where a P 5 absolute acceleration of particle P a P9 5 acceleration of point P9 of moving frame ^ coinciding with P a P/^ 5 acceleration of P relative to moving frame ^ a c 5 2V 3 (ṙ) z 5 2V 3 v P/^ 5 complementar, or oriolis, acceleration We note that the oriolis acceleration is perpendicular to the vectors V and v P/^. However, since these vectors are usuall not perpendicular to each other, the magnitude of a c is in general not equal to 2Vv P/^, as was the case for the plane motion of a particle. We further note that the oriolis acceleration reduces to zero when the vectors V and v P/^ are parallel, or when either of them is zero. Rotating frames of reference are particularl useful in the stud of the three-dimensional motion of rigid bodies. If a rigid bod has a fied point, as was the case for the crane of Sample Prob , we can use a frame z which is neither fied nor rigidl attached to the rigid bod. enoting b V the angular velocit of the frame z, we then resolve the angular velocit V of the bod into the components V and V /^, where the second component represents the angular velocit of the bod relative to the frame z (see Sample Prob ). n appropriate choice of the rotating frame often leads to a simpler analsis of the motion of the rigid bod than would be possible with aes of fied orientation. This is especiall true in the case of the general three-dimensional motion of a rigid bod, i.e., when the rigid bod under consideration has no fied point (see Sample Prob ). *15.15 FRME F REFERENE IN GENERL MTIN onsider a fied frame of reference XYZ and a frame z which moves in a known, but arbitrar, fashion with respect to XYZ (Fig ). Let P be a particle moving in space. The position of P is defined at an instant b the vector r P in the fied frame, and b the vector r P/ in the moving frame. enoting b r the position vector of in the fied frame, we have r P 5 r 1 r P/ (15.49) The absolute velocit v P of the particle is obtained b writing v P 5 ṙ P 5 ṙ 1 ṙ P/ (15.50) where the derivatives are defined with respect to the fied frame XYZ. The first term in the right-hand member of (15.50) thus represents the velocit v of the origin of the moving aes. n the other hand, since the rate of change of a vector is the same with Y' Y r Z' z Z Fig r P/ r P P X' X It is important to note the difference between Eq. (15.48) and Eq. (15.21) of Sec See the footnote on page 974.

87 1000 Kinematics of Rigid odies respect to a fied frame and with respect to a frame in translation (Sec ), the second term can be regarded as the velocit v P/ of P relative to the frame X9Y9Z9 of the same orientation as XYZ and the same origin as z. We therefore have v P 5 v 1 v P/ (15.51) ut the velocit v P/ of P relative to X9Y9Z9 can be obtained from (15.45) b substituting r P/ for r in that equation. We write v P 5 v 1 V 3 r P/ 1 (ṙ P/ ) z (15.52) where V is the angular velocit of the frame z at the instant considered. The absolute acceleration a P of the particle is obtained b differentiating (15.51) and writing a P 5 v P 5 v 1 v P/ (15.53) where the derivatives are defined with respect to either of the frames XYZ or X9Y9Z9. Thus, the first term in the right-hand member of (15.53) represents the acceleration a of the origin of the moving aes and the second term represents the acceleration a P/ of P relative to the frame X9Y9Z9. This acceleration can be obtained from (15.47) b substituting r P/ for r. We therefore write a P 5 a 1 V 3 r P/ 1 V 3 (V 3 r P/ ) 1 2V 3 (ṙ P/ ) z 1 ( r P/ ) z (15.54) Photo 15.9 The motion of air particles in a hurricane can be considered as motion relative to a frame of reference attached to the Earth and rotating with it. Formulas (15.52) and (15.54) make it possible to determine the velocit and acceleration of a given particle with respect to a fied frame of reference, when the motion of the particle is known with respect to a moving frame. These formulas become more significant, and considerabl easier to remember, if we note that the sum of the first two terms in (15.52) represents the velocit of the point P9 of the moving frame which coincides with P at the instant considered, and that the sum of the first three terms in (15.54) represents the acceleration of the same point. Thus, the relations (15.46) and (15.48) of the preceding section are still valid in the case of a reference frame in general motion, and we can write v P 5 v P9 1 v P/^ (15.46) a P 5 a P9 1 a P/^ 1 a c (15.48) where the various vectors involved have been defined in Sec It should be noted that if the moving reference frame ^ (or z) is in translation, the velocit and acceleration of the point P9 of the frame which coincides with P become, respectivel, equal to the velocit and acceleration of the origin of the frame. n the other hand, since the frame maintains a fied orientation, a c is zero, and the relations (15.46) and (15.48) reduce, respectivel, to the relations (11.33) and (11.34) derived in Sec

88 Y 8 in. 30 SMPLE PRLEM The bent rod rotates about the vertical. t the instant considered, its angular velocit and angular acceleration are, respectivel, 20 rad/s and 200 rad/s 2, both clockwise when viewed from the positive Y ais. The collar moves along the rod, and at the instant considered, 5 8 in. The velocit and acceleration of the collar relative to the rod are, respectivel, 50 in./s and 600 in./s 2, both upward. etermine (a) the velocit of the collar, (b) the acceleration of the collar. X Z SLUTIN Y ' 8 in. 30 a / v / Frames of Reference. The frame XYZ is fied. We attach the rotating frame z to the bent rod. Its angular velocit and angular acceleration relative to XYZ are therefore V 5 (220 rad/s)j and V 5 (2200 rad/s 2 )j, respectivel. The position vector of is r 5 (8 in.)(sin 30 i 1 cos 30 j) 5 (4 in.)i 1 (6.93 in.)j a. Velocit v. enoting b 9 the point of the rod which coincides with and b ^ the rotating frame z, we write from Eq. (15.46) v 5 v 9 1 v /^ (1) Z z Ω = ( 20 rad/s)j. Ω = ( 200 rad/s 2 )j X where v 9 5 V 3 r 5 (220 rad/s)j 3 [(4 in.)i 1 (6.93 in.)j] 5 (80 in./s)k v /^ 5 (50 in./s)(sin 30 i 1 cos 30 j) 5 (25 in./s)i 1 (43.3 in./s)j Substituting the values obtained for v 9 and v /^ into (1), we find v 5 (25 in./s)i 1 (43.3 in./s)j 1 (80 in./s)k b. cceleration a. From Eq. (15.48) we write a 5 a 9 1 a /^ 1 a c (2) where a 9 5 V 3 r 1 V 3 (V 3 r) 5 (2200 rad/s 2 )j 3 [(4 in.)i 1 (6.93 in.)j] 2 (20 rad/s)j 3 (80 in./s)k 5 1(800 in./s 2 )k 2 (1600 in./s 2 )i a /^ 5 (600 in./s 2 )(sin 30 i 1 cos 30 j) 5 (300 in./s 2 )i 1 (520 in./s 2 )j a c 5 2V 3 v /^ 5 2(220 rad/s)j 3 [(25 in./s)i 1 (43.3 in./s)j] 5 (1000 in./s 2 )k Substituting the values obtained for a 9, a /^, and a c into (2), a 5 2(1300 in./s 2 )i 1 (520 in./s 2 )j 1 (1800 in./s 2 )k 1001

89 Y P SMPLE PRLEM w 1 q = 30 X The crane shown rotates with a constant angular velocit V 1 of 0.30 rad/s. Simultaneousl, the boom is being raised with a constant angular velocit V 2 of 0.50 rad/s relative to the cab. Knowing that the length of the boom P is l 5 12 m, determine (a) the velocit of the tip of the boom, (b) the acceleration of the tip of the boom. Z w 2 SLUTIN z Y m Ω = w 1 = 0.30j w / = w 2 = 0.50k P 6 m X Frames of Reference. The frame XYZ is fied. We attach the rotating frame z to the cab. Its angular velocit with respect to the frame XYZ is therefore V 5 V 1 5 (0.30 rad/s)j. The angular velocit of the boom relative to the cab and the rotating frame z (or ^, for short) is V /^ 5 V 2 5 (0.50 rad/s)k. a. Velocit v P. From Eq. (15.46) we write v P 5 v P9 1 v P/^ (1) where v P9 is the velocit of the point P9 of the rotating frame which coincides with P: v P9 5 V 3 r 5 (0.30 rad/s)j 3 [(10.39 m)i 1 (6 m)j] 5 2(3.12 m/s)k Z and where v P/^ is the velocit of P relative to the rotating frame z. ut the angular velocit of the boom relative to z was found to be V /^ 5 (0.50 rad/s)k. The velocit of its tip P relative to z is therefore v P/^ 5 V /^ 3 r 5 (0.50 rad/s)k 3 [(10.39 m)i 1 (6 m)j] 5 2(3 m/s)i 1 (5.20 m/s)j Substituting the values obtained for v P9 and v P/^ into (1), we find v P 5 2(3 m/s)i 1 (5.20 m/s)j 2 (3.12 m/s)k b. cceleration a P. From Eq. (15.48) we write a P 5 a P9 1 a P/^ 1 a c (2) Since V and V /^ are both constant, we have a P9 5 V 3 (V 3 r) 5 (0.30 rad/s)j 3 (23.12 m/s)k 5 2(0.94 m/s 2 )i a P/^ 5 V /^ 3 (V /^ 3 r) 5 (0.50 rad/s)k 3 [2(3 m/s)i 1 (5.20 m/s)j] 5 2(1.50 m/s 2 )j 2 (2.60 m/s 2 )i a c 5 2V 3 v P/^ 5 2(0.30 rad/s)j 3 [2(3 m/s)i 1 (5.20 m/s)j] 5 (1.80 m/s 2 )k Substituting for a P9, a P/^, and a c into (2), we find a P 5 2(3.54 m/s 2 )i 2 (1.50 m/s 2 )j 1 (1.80 m/s 2 )k 1002

90 w 1 L P R w 2 isk SMPLE PRLEM isk, of radius R, is pinned to end of the arm of length L located in the plane of the disk. The arm rotates about a vertical ais through at the constant rate v 1, and the disk rotates about at the constant rate v 2. etermine (a) the velocit of point P located directl above, (b) the acceleration of P, (c) the angular velocit and angular acceleration of the disk. SLUTIN Y L Ω = w 1 j P' P R X Frames of Reference. The frame XYZ is fied. We attach the moving frame z to the arm. Its angular velocit with respect to the frame XYZ is therefore V 5 v 1 j. The angular velocit of disk relative to the moving frame z (or ^, for short) is V /^ 5 v 2 k. The position vector of P relative to is r 5 Li + Rj, and its position vector relative to is r P/ 5 Rj. a. Velocit v P. enoting b P9 the point of the moving frame which coincides with P, we write from Eq. (15.46) v P 5 v P9 1 v P/^ (1) where v P9 5 V 3 r 5 v 1 j 3 (Li 1 Rj) 5 2v 1 Lk v P/^ 5 V /^ 3 r P/ 5 v 2 k 3 Rj 5 2v 2 Ri Substituting the values obtained for v P9 and v P/^ into (1), we find v P 5 2v 2 Ri 2 v 1 Lk b. cceleration a P. From Eq. (15.48) we write a P 5 a P9 1 a P/^ 1 a c (2) Since V and V /^ are both constant, we have a P9 5 V 3 (V 3 r) 5 v 1 j 3 (2v 1 Lk) 5 2v 2 1Li a P/^ 5 V /^ 3 (V /^ 3 r P/ ) 5 v 2 k 3 (2v 2 Ri) 5 2v 2 2Rj a c 5 2V 3 v P/^ 5 2v 1 j 3 (2v 2 Ri) 5 2v 1 v 2 Rk Substituting the values obtained into (2), we find a P 5 2v 2 1Li 2 v 2 2Rj 1 2v 1 v 2 Rk Z z w / = w 2 k c. ngular Velocit and ngular cceleration of isk. v 5 V 1 v /^ V 5 v 1 j 1 v 2 k Using Eq. (15.31) with Q 5 V, we write 5 (V ) XYZ 5 (V ) z 1 V 3 V v 1 j 3 (v 1 j 1 v 2 k) 5 v 1 v 2 i 1003

91 SLVING PRLEMS N YUR WN In this lesson ou concluded our stud of the kinematics of rigid bodies b learning how to use an auiliar frame of reference ^ to analze the threedimensional motion of a rigid bod. This auiliar frame ma be a rotating frame with a fied origin, or it ma be a frame in general motion.. Using a rotating frame of reference. s ou approach a problem involving the use of a rotating frame ^ ou should take the following steps. 1. Select the rotating frame ^ that ou wish to use and draw the corresponding coordinate aes,, and z from the fied point. 2. etermine the angular velocit V of the frame ^ with respect to a fied frame XYZ. In most cases, ou will have selected a frame which is attached to some rotating element of the sstem; V will then be the angular velocit of that element. 3. esignate as P9 the point of the rotating frame ^ that coincides with the point P of interest at the instant ou are considering. etermine the velocit v P9 and the acceleration a P9 of point P9. Since P9 is part of ^ and has the same position vector r as P, ou will find that v P9 5 V 3 r and a P9 5 3 r 1 V 3 (V 3 r) where is the angular acceleration of ^. However, in man of the problems that ou will encounter, the angular velocit of ^ is constant in both magnitude and direction, and etermine the velocit and acceleration of point P with respect to the frame ^. s ou are tring to determine v P/^ and a P/^ ou will find it useful to visualize the motion of P on frame ^ when the frame is not rotating. If P is a point of a rigid which has an angular velocit and an angular relative to ^ [Sample Prob ], ou will find that v P/^ 5 3 r and a P/^ 3 r r) In man of the problems that ou will encounter, the angular velocit of relative to frame ^ is constant in both magnitude and direction, etermine the oriolis acceleration. onsidering the angular velocit V of frame ^ and the velocit v P/^ of point P relative to that frame, which was computed in the previous step, ou write a c 5 2V 3 v P/^ 1004

92 6. The velocit and the acceleration of P with respect to the fied frame XYZ can now be obtained b adding the epressions ou have determined: v P 5 v P9 1 v P/^ (15.46) a P 5 a P9 1 a P/^ 1 a c (15.48). Using a frame of reference in general motion. The steps that ou will take differ onl slightl from those listed under. The consist of the following: 1. Select the frame ^ that ou wish to use and a reference point in that frame, from which ou will draw the coordinate aes,,, and z defining that frame. You will consider the motion of the frame as the sum of a translation with and a rotation about. 2. etermine the velocit v of point and the angular velocit V of the frame. In most cases, ou will have selected a frame which is attached to some element of the sstem; V will then be the angular velocit of that element. 3. esignate as P9 the point of frame ^ that coincides with the point P of interest at the instant ou are considering, and determine the velocit v P9 and the acceleration a P9 of that point. In some cases, this can be done b visualizing the motion of P if that point were prevented from moving with respect to ^ [Sample Prob ]. more general approach is to recall that the motion of P9 is the sum of a translation with the reference point and a rotation about. The velocit v P9 and the acceleration a P9 of P9, therefore, can be obtained b adding v and a, respectivel, to the epressions found in paragraph 3 and replacing the position vector r b the vector r P/ drawn from to P: v P9 5 v 1 V 3 r P/ a P9 5 a 1 3 r P/ 1 V 3 (V 3 r P/ ) Steps 4, 5, and 6 are the same as in Part, ecept that the vector r should again be replaced b r P/. Thus, Eqs. (15.46) and (15.48) can still be used to obtain the velocit and the acceleration of P with respect to the fied frame of reference XYZ. 1005

93 PRLEMS Rod is welded to the 12-in.-radius plate which rotates at the constant rate v rad/s. Knowing that collar moves toward end of the rod at a constant speed u 5 78 in./s, determine, for the position shown, (a) the velocit of, (b) the acceleration of. Y 8 in. u 12 in. 10 in. Z w 1 X Fig. P The bent rod shown rotates at the constant rate v rad/s. Knowing that collar moves toward point at a constant relative speed u 5 34 in./s, determine, for the position shown, the velocit and acceleration of if (a) 5 5 in., (b) 5 15 in. Y Y w 1 Z 8 in. u 200 mm u 30 Fig. P in. 15 in. E ω 1 8 in. X 1006 Z Fig. P X The circular plate shown rotates about its vertical diameter at the constant rate v rad/s. Knowing that in the position shown the disk lies in the XY plane and point of strap moves upward at a constant relative speed u m/s, determine (a) the velocit of, (b) the acceleration of.

94 Solve Prob , assuming that, at the instant shown, the angular velocit V 1 of the plate is 10 rad/s and is decreasing at the rate of 25 rad/s 2, while the relative speed u of point of strap is 1.5 m/s and is decreasing at the rate 3 m/s square plate of side 18 in. is hinged at and to a clevis. The plate rotates at the constant rate v rad/s with respect to the clevis, which itself rotates at the constant rate v rad/s about the Y ais. For the position shown, determine (a) the velocit of point, (b) the acceleration of point. Y Problems 1007 w 1 Z w 2 18 in in. 9 in. X Y Fig. P and P square plate of side 18 in. is hinged at and to a clevis. The plate rotates at the constant rate v rad/s with respect to the clevis, which itself rotates at the constant rate v rad/s about the Y ais. For the position shown, determine (a) the velocit of corner, (b) the acceleration of corner through The rectangular plate shown rotates at the constant rate v rad/s with respect to arm E, which itself rotates at the constant rate v rad/s about the Z ais. For the position shown, determine the velocit and acceleration of the point of the plate indicated orner Point orner. 90 mm E 135 mm w 1 Z w 2 X 135 mm Fig. P15.226, P15.227, and P Solve Prob , assuming that at the instant shown the angular velocit V 2 of the plate with respect to arm E is 12 rad/s and is decreasing at the rate of 60 rad/s 2, while the angular velocit V 1 of the arm about the Z ais is 9 rad/s and is decreasing at the rate of 45 rad/s Solve Prob , assuming that at the instant shown the angular velocit V 1 of the rod is 3 rad/s and is increasing at the rate of 12 rad/s 2, while the relative speed u of the collar is 34 in./s and is decreasing at the rate of 85 in./s Using the method of Sec , solve Prob Using the method of Sec , solve Prob

95 1008 Kinematics of Rigid odies Using the method of Sec , solve Prob The bod and rod of the robotic component shown rotate at the constant rate v rad/s about the Y ais. Simultaneousl a wire-and-pulle control causes arm to rotate about at the constant rate v 5 db/dt rad/s. Knowing b 5 120, determine (a) the angular acceleration of arm, (b) the velocit of, (c) the acceleration of. Y w 1 16 in. Z 20 in. X Fig. P disk of radius 120 mm rotates at the constant rate v rad/s with respect to the arm, which itself rotates at the constant rate v rad/s. For the position shown, determine the velocit and acceleration of point. Y ω 1 Y w 1 Z 140 mm 75 mm ω mm X u Fig. P and P disk of radius 120 mm rotates at the constant rate v rad/s with respect to the arm, which itself rotates at the constant rate v rad/s. For the position shown, determine the velocit and acceleration of point. Z Fig. P X w The crane shown rotates at the constant rate v rad/s; simultaneousl, the telescoping boom is being lowered at the constant rate v rad/s. Knowing that at the instant shown the length of the boom is 20 ft and is increasing at the constant rate u ft/s, determine the velocit and acceleration of point.

96 The arm of length 5 m is used to provide an elevated platform for construction workers. In the position shown, arm is being raised at the constant rate du/dt rad/s; simultaneousl, the unit is being rotated about the Y ais at the constant rate v rad/s. Knowing that u 5 20, determine the velocit and acceleration of point. Problems 1009 Y w 1 q 0.8 m X Z Fig. P Solve Prob , assuming that u disk of 180-mm radius rotates at the constant rate v rad/s with respect to arm, which itself rotates at the constant rate v rad/s about the Y ais. etermine at the instant shown the velocit and acceleration of point on the rim of the disk. Y 180 mm w 1 w 2 Z X 360 mm 150 mm Fig. P and P disk of 180-mm radius rotates at the constant rate v rad/s with respect to arm, which itself rotates at the constant rate v rad/s about the Y ais. etermine at the instant shown the velocit and acceleration of point on the rim of the disk.

97 1010 Kinematics of Rigid odies Y and In the position shown the thin rod moves at a constant speed u 5 3 in./s out of the tube. t the same time tube rotates at the constant rate v rad/s with respect to arm. Knowing that the entire assembl rotates about the X ais at the constant rate v rad/s, determine the velocit and acceleration of end of the rod. Y 9 in. w 1 X 6 in. u w 1 w 2 Z 6 in. u Z w 2 9 in. X Fig. P Fig. P Two disks, each of 130-mm radius, are welded to the 500-mm rod. The rod-and-disks unit rotates at the constant rate v rad/s with respect to arm. Knowing that at the instant shown v rad/s, determine the velocit and acceleration of (a) point E, (b) point F. Y G 3 in. u 5 in. Y 130 mm 250 mm Z H Fig. P mm w 1 w 2 F E 130 mm In Prob , determine the velocit and acceleration of (a) point G, (b) point H. X E 6 in. F 6 in. w 1 3 in. 10 in. G Z X Fig. P and P The vertical plate shown is welded to arm EFG, and the entire unit rotates at the constant rate v rad/s about the Y ais. t the same time, a continuous link belt moves around the perimeter of the plate at a constant speed u in./s. For the position shown, determine the acceleration of the link of the belt located (a) at point, (b) at point The vertical plate shown is welded to arm EFG, and the entire unit rotates at the constant rate v rad/s about the Y ais. t the same time, a continuous link belt moves around the perimeter of the plate at a constant speed u in./s. For the position shown, determine the acceleration of the link of the belt located (a) at point, (b) at point.

98 REVIEW N SUMMRY This chapter was devoted to the stud of the kinematics of rigid bodies. We first considered the translation of a rigid bod [Sec. 15.2] and observed that in such a motion, all points of the bod have the same velocit and the same acceleration at an given instant. Rigid bod in translation We net considered the rotation of a rigid bod about a fied ais [Sec. 15.3]. The position of the bod is defined b the angle u that the line P, drawn from the ais of rotation to a point P of the bod, forms with a fied plane (Fig ). We found that the magnitude of the velocit of P is v 5 ds dt 5 ru. sin f (15.4) where u is the time derivative of u. We then epressed the velocit of P as v 5 dr dt 5 V 3 r (15.5) Rigid bod in rotation about a fied ais q f r P ' z where the vector V 5 vk 5 u k (15.6) Fig is directed along the fied ais of rotation and represents the angular velocit of the bod. enoting b the derivative dv/dt of the angular velocit, we epressed the acceleration of P as a 5 3 r 1 V 3 (V 3 r) (15.8) ifferentiating (15.6), and recalling that k is constant in magnitude and direction, we found that 5 ak 5 v. k 5 ük (15.9) The vector represents the angular acceleration of the bod and is directed along the fied ais of rotation. 1011

99 1012 Kinematics of Rigid odies a t = a k r v = wk r P r P a n = w 2 r w = wk a = ak wk Fig Fig Rotation of a representative slab Tangential and normal components ngular velocit and angular acceleration of rotating slab Net we considered the motion of a representative slab located in a plane perpendicular to the ais of rotation of the bod (Fig ). Since the angular velocit is perpendicular to the slab, the velocit of a point P of the slab was epressed as v 5 vk 3 r (15.10) where v is contained in the plane of the slab. Substituting V 5 vk and 5 ak into (15.8), we found that the acceleration of P could be resolved into tangential and normal components (Fig ) respectivel equal to a t 5 ak 3 r a t 5 ra a n 5 2v 2 r a n 5 rv 2 (15.119) Recalling Eqs. (15.6) and (15.9), we obtained the following epressions for the angular velocit and the angular acceleration of the slab [Sec. 15.4]: or v 5 du dt a 5 dv dt 5 d2 u dt 2 dv a 5 v du (15.12) (15.13) (15.14) We noted that these epressions are similar to those obtained in hap. 11 for the rectilinear motion of a particle. Two particular cases of rotation are frequentl encountered: uniform rotation and uniforml accelerated rotation. Problems involving either of these motions can be solved b using equations similar to those used in Secs and 11.5 for the uniform rectilinear motion and the uniforml accelerated rectilinear motion of a particle, but where, v, and a are replaced b u, v, and a, respectivel [Sample Prob. 15.1].

100 v v ' Review and Summar 1013 v v = + (fied) wk r / ' v/ v v / v Fig Plane motion = Translation with + Rotation about v = v + v / The most general plane motion of a rigid slab can be considered as the sum of a translation and a rotation [Sec. 15.5]. For eample, the slab shown in Fig can be assumed to translate with point, while simultaneousl rotating about. It follows [Sec. 15.6] that the velocit of an point of the slab can be epressed as v 5 v 1 v / (15.17) where v is the velocit of and v / the relative velocit of with respect to or, more precisel, with respect to aes 99 translating with. enoting b r / the position vector of relative to, we found that v / 5 vk 3 r / v / 5 rv (15.18) The fundamental equation (15.17) relating the absolute velocities of points and and the relative velocit of with respect to was epressed in the form of a vector diagram and used to solve problems involving the motion of various tpes of mechanisms [Sample Probs and 15.3]. Velocities in plane motion nother approach to the solution of problems involving the velocities of the points of a rigid slab in plane motion was presented in Sec and used in Sample Probs and It is based on the determination of the instantaneous center of rotation of the slab (Fig ). Instantaneous center of rotation v v v v Fig (a) (b)

101 1014 Kinematics of Rigid odies ' a a = + wk (fied) ak r / a / ' a (a / ) n a / a (a / ) (a t / ) t a a (a / ) n Plane motion = Translation with + Rotation about Fig ccelerations in plane motion The fact that an plane motion of a rigid slab can be considered as the sum of a translation of the slab with a reference point and a rotation about was used in Sec to relate the absolute accelerations of an two points and of the slab and the relative acceleration of with respect to. We had a 5 a 1 a / (15.21) where a / consisted of a normal component (a / ) n of magnitude rv 2 directed toward, and a tangential component (a / ) t of magnitude ra perpendicular to the line (Fig ). The fundamental relation (15.21) was epressed in terms of vector diagrams or vector equations and used to determine the accelerations of given points of various mechanisms [Sample Probs through 15.8]. It should be noted that the instantaneous center of rotation considered in Sec cannot be used for the determination of accelerations, since point, in general, does not have zero acceleration. oordinates epressed in terms of a parameter Rate of change of a vector with respect to a rotating frame Ω Z Fig z Y j k i Q X In the case of certain mechanisms, it is possible to epress the coordinates and of all significant points of the mechanism b means of simple analtic epressions containing a single parameter. The components of the absolute velocit and acceleration of a given point are then obtained b differentiating twice with respect to the time t the coordinates and of that point [Sec. 15.9]. While the rate of change of a vector is the same with respect to a fied frame of reference and with respect to a frame in translation, the rate of change of a vector with respect to a rotating frame is different. Therefore, in order to stud the motion of a particle relative to a rotating frame we first had to compare the rates of change of a general vector Q with respect to a fied frame XYZ and with respect to a frame z rotating with an angular velocit V [Sec ] (Fig ). We obtained the fundamental relation ( Q) XYZ 5 ( Q) z 1 V 3 Q (15.31) and we concluded that the rate of change of the vector Q with respect to the fied frame XYZ is made of two parts: The first part represents the rate of change of Q with respect to the rotating frame z; the second part, V 3 Q, is induced b the rotation of the frame z.

102 The net part of the chapter [Sec ] was devoted to the twodimensional kinematic analsis of a particle P moving with respect to a frame ^ rotating with an angular velocit V about a fied ais (Fig ). We found that the absolute velocit of P could be epressed as v P 5 v P9 1 v P/^ (15.33) where v P 5 absolute velocit of particle P v P9 5 velocit of point P9 of moving frame ^ coinciding with P v P/^ 5 velocit of P relative to moving frame ^ We noted that the same epression for v P is obtained if the frame is in translation rather than in rotation. However, when the frame is in rotation, the epression for the acceleration of P is found to contain an additional term a c called the complementar acceleration or oriolis acceleration. We wrote a P 5 a P9 1 a P/^ 1 a c (15.36) where a P 5 absolute acceleration of particle P a P9 5 acceleration of point P9 of moving frame ^ coinciding with P a P/^ 5 acceleration of P relative to moving frame ^ a c 5 2V 3 (ṙ) 5 2V 3 v P/^ 5 complementar, or oriolis, acceleration Since V and v P/^ are perpendicular to each other in the case of plane motion, the oriolis acceleration was found to have a magnitude a c 5 2Vv P/^ and to point in the direction obtained b rotating the vector v P/^ through 90 in the sense of rotation of the moving frame. Formulas (15.33) and (15.36) can be used to analze the motion of mechanisms which contain parts sliding on each other [Sample Probs and 15.10] Plane motion of a particle relative to a rotating frame Y v P' = Ω r Fig v P/ r Ω P Review and Summar. = (r) P' X The last part of the chapter was devoted to the stud of the kinematics of rigid bodies in three dimensions. We first considered the motion of a rigid bod with a fied point [Sec ]. fter proving that the most general displacement of a rigid bod with a fied point is equivalent to a rotation of the bod about an ais through, we were able to define the angular velocit V and the instantaneous ais of rotation of the bod at a given instant. The velocit of a point P of the bod (Fig ) could again be epressed as v 5 dr dt 5 V 3 r (15.37) ifferentiating this epression, we also wrote a 5 3 r 1 V 3 (V 3 r) (15.38) However, since the direction of V changes from one instant to the net, the angular acceleration is, in general, not directed along the instantaneous ais of rotation [Sample Prob ]. Motion of a rigid bod with a fied point a Fig r w P

103 1016 Kinematics of Rigid odies General motion in space Y' w It was shown in Sec that the most general motion of a rigid bod in space is equivalent, at an given instant, to the sum of a translation and a rotation. onsidering two particles and of the bod, we found that v 5 v 1 v / (15.42) Y a r / X' where v / is the velocit of relative to a frame X9Y9Z9 attached to and of fied orientation (Fig ). enoting b r / the position vector of relative to, we wrote v 5 v 1 V 3 r / (15.43) Z' r where V is the angular velocit of the bod at the instant considered [Sample Prob ]. The acceleration of was obtained b a similar reasoning. We first wrote Z Fig X and, recalling Eq. (15.38), a 5 a 1 a / a 5 a 1 3 r / 1 V 3 (V 3 r / ) (15.44) Three-dimensional motion of a particle relative to a rotating frame In the final two sections of the chapter we considered the threedimensional motion of a particle P relative to a frame z rotating with an angular velocit V with respect to a fied frame XYZ (Fig ). In Sec we epressed the absolute velocit v P of P as v P 5 v P9 1 v P/^ (15.46) where v P 5 absolute velocit of particle P v P9 5 velocit of point P9 of moving frame ^ coinciding with P v P/^ 5 velocit of P relative to moving frame ^ Y Ω j r i P z k X Z Fig

104 The absolute acceleration a P of P was then epressed as a P 5 a P9 1 a P/^ 1 a c (15.48) where a P 5 absolute acceleration of particle P a P9 5 acceleration of point P9 of moving frame ^ coinciding with P a P/^ 5 acceleration of P relative to moving frame ^ a c 5 2V 3 (ṙ) z 5 2V 3 v P/^ 5 complementar, or oriolis, acceleration It was noted that the magnitude a c of the oriolis acceleration is not equal to 2Vv P/^ [Sample Prob ] ecept in the special case when V and v P/^ are perpendicular to each other. Review and Summar 1017 We also observed [Sec ] that Eqs. (15.46) and (15.48) remain valid when the frame z moves in a known, but arbitrar, fashion with respect to the fied frame XYZ (Fig ), provided that the motion of is included in the terms v P9 and a P9 representing the absolute velocit and acceleration of the coinciding point P9. Frame of reference in general motion Y Y' r P/ P X' Z' r z r P X Z Fig Rotating frames of reference are particularl useful in the stud of the three-dimensional motion of rigid bodies. Indeed, there are man cases where an appropriate choice of the rotating frame will lead to a simpler analsis of the motion of the rigid bod than would be possible with aes of fied orientation [Sample Probs and 15.15].

105 REVIEW PRLEMS Knowing that at the instant shown crank has a constant angular velocit of 45 rpm clockwise, determine the acceleration (a) of point, (b) of point. 8 in. 4 in The rotor of an electric motor has a speed of 1800 rpm when the power is cut off. The rotor is then observed to come to rest after eecuting 1550 revolutions. ssuming uniforml accelerated motion, determine (a) the angular acceleration of the rotor, (b) the time required for the rotor to come to rest. Fig. P in disk of 0.15-m radius rotates at the constant rate v 2 with respect to plate, which itself rotates at the constant rate v 1 about the ais. Knowing that v 1 5 v rad/s, determine, for the position shown, the velocit and acceleration (a) of point, (b) of point F m z ω m ω 2 F Fig. P The fan of an automobile engine rotates about a horizontal ais parallel to the direction of motion of the automobile. When viewed from the rear of the engine, the fan is observed to rotate clockwise at the rate of 2500 rpm. Knowing that the automobile is turning right along a path of radius 12 m at a constant speed of 12 km/h, determine the angular acceleration of the fan at the instant the automobile is moving due north.

106 drum of radius 4.5 in. is mounted on a clinder of radius 7.5 in. cord is wound around the drum, and its etremit E is pulled to the right with a constant velocit of 15 in./s, causing the clinder to roll without sliding on plate F. Knowing that plate F is stationar, determine (a) the velocit of the center of the clinder, (b) the acceleration of point of the clinder. 7.5 in. Review Problems 4.5 in Solve Prob , assuming that plate F is moving to the right with a constant velocit of 9 in./s Water flows through a curved pipe that rotates with a constant clockwise angular velocit of 90 rpm. If the velocit of the water relative to the pipe is 8 m/s, determine the total acceleration of a particle of water at point P. Fig. P E F P 0.5 m w Fig. P Rod of length 24 in. is connected b ball-and-socket joints to a rotating arm and to a collar that slides on the fied rod E. Knowing that the length of arm is 4 in. and that it rotates at the constant rate v rad/s, determine the velocit of collar when u 5 0. w 1 q 4 in. 16 in. E 4 in. z Fig. P Solve Prob , assuming that u 5 90.

107 1020 Kinematics of Rigid odies rank has a constant angular velocit of 1.5 rad/s counterclockwise. For the position shown, determine (a) the angular velocit of rod, (b) the velocit of collar mm 160 mm Fig. P and P rank has a constant angular velocit of 1.5 rad/s counterclockwise. For the position shown, determine (a) the angular acceleration of rod, (b) the acceleration of collar Rod of length 125 mm is attached to a vertical rod that rotates about the ais at the constant rate v rad/s. Knowing that the angle formed b rod and the vertical is increasing at the constant rate db/dt 5 3 rad/s, determine the velocit and acceleration of end of the rod when b w 1 b z Fig. P15.259

108 MPUTER PRLEMS 15.1 The disk shown has a constant angular velocit of 500 rpm counterclockwise. Knowing that rod is 250 mm long, use computational software to determine and plot for values of u from 0 to 360 and using 30 increments, the velocit of collar and the angular velocit of rod. etermine the two values of u for which the speed of collar is zero. q 50 mm 150 mm Fig. P Two rotating rods are connected b a slider block P as shown. Knowing that rod P rotates with a constant angular velocit of 6 rad/s counterclockwise, use computational software to determine and plot for values of u from 0 to 180 the angular velocit and angular acceleration of rod E. etermine the value of u for which the angular acceleration a E of rod E is maimum and the corresponding value of a E. E 15 in. P q 30 in. Fig. P

109 1022 Kinematics of Rigid odies P l 15.3 In the engine sstem shown, l mm and b 5 60 mm. Knowing that crank rotates with a constant angular velocit of 1000 rpm clockwise, use computational software to determine and plot for values of u from 0 to 180 and using 10 increments, (a) the angular velocit and angular acceleration of rod, (b) the velocit and acceleration of the piston P Rod moves over a small wheel at while end moves to the right with a constant velocit of 180 mm/s. Use computational software to determine and plot for values of u from 20 to 90 and using 5 increments, the velocit of point and the angular acceleration of the rod. etermine the value of u for which the angular acceleration a of the rod is maimum and the corresponding value of a. q b Fig. P mm 140 mm q Fig. P Rod of length 24 in. is connected b ball-and-socket joints to the rotating arm and to collar that slides on the fied rod E. rm of length 4 in. rotates in the XY plane with a constant angular velocit of 10 rad/s. Use computational software to determine and plot for values of u from 0 to 360 the velocit of collar. etermine the two values of u for which the velocit of collar is zero. w 1 q 4 in. 16 in. E 4 in. z Fig. P15.5

110 15.6 Rod of length 25 in. is connected b ball-and-socket joints to collars and, which slide along the two rods shown. ollar moves toward support E at a constant speed of 20 in./s. enoting b d the distance from point to collar, use computational software to determine and plot the velocit of collar for values of d from 0 to 15 in. omputer Problems in. 9 in. 12 in. 20 in. z E Fig. P15.6

111 Three-bladed wind turbines, similar to the ones shown in this picture of a wind farm, are currentl the most common design. In this chapter ou will learn to analze the motion of a rigid bod b considering the motion of its mass center, the motion relative to its mass center, and the eternal forces acting on it. 1024