Rotational Kinematics and Dynamics. UCVTS AIT Physics

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Rotational Kinematics and Dynamics. UCVTS AIT Physics"

Transcription

1 Rotational Kinematics and Dynamics UCVTS AIT Physics

2 Angular Position Axis of rotation is the center of the disc Choose a fixed reference line Point P is at a fixed distance r from the origin

3 Angular Position, 2 Point P will rotate about the origin in a circle of radius r Every particle on the disc undergoes circular motion about the origin, O Polar coordinates are convenient to use to represent the position of P (or any other point) P is located at (r, q) where r is the distance from the origin to P and q is the measured counterclockwise from the reference line

4 Angular Position, 3 As the particle moves, the only coordinate that changes is q As the particle moves through q, it moves though an arc length s. The arc length and r are related: s = q r

5 Radian This can also be expressed as q is a pure number, but commonly is given the artificial unit, radian One radian is the angle subtended by an arc length equal to the radius of the arc

6 Conversions Comparing degrees and radians 1 rad = = 57.3 Converting from degrees to radians θ [rad] = [degrees]

7 Angular Displacement The angular displacement is defined as the angle the object rotates through during some time interval q q q This is the angle that the reference line of length r sweeps out f i

8 Average Angular Speed The average angular speed, ω, of a rotating rigid object is the ratio of the angular displacement to the time interval q q q f i t t t f i

9 Instantaneous Angular Speed The instantaneous angular speed is defined as the limit of the average speed as the time interval approaches zero lim q dq t 0 t dt

10 Angular Speed, final Units of angular speed are radians/sec rad/s or s -1 since radians have no dimensions Angular speed will be positive if θ is increasing (counterclockwise) Angular speed will be negative if θ is decreasing (clockwise)

11 Average Angular Acceleration The average angular acceleration, a, of an object is defined as the ratio of the change in the angular speed to the time it takes for the object to undergo the change: a f i t t t f i

12 Instantaneous Angular Acceleration The instantaneous angular acceleration is defined as the limit of the average angular acceleration as the time goes to 0 a lim t 0 t d dt

13 Directions, details Strictly speaking, the speed and acceleration (, a) are the magnitudes of the velocity and acceleration vectors The directions are actually given by the right-hand rule

14 Rotational Kinematic Equations with constant angular acceleration at f i q q t 1 f i i 2 at 2 a ( q q ) 2 2 f i f i 1 q q 2 2 ) f i i f t

15 Comparison Between Rotational and Linear Equations

16 Relationship Between Angular and Linear Quantities Displacements s Speeds v q r Accelerations a r a r Every point on the rotating object has the same angular motion Every point on the rotating object does not have the same linear motion

17 Centripetal Acceleration An object traveling in a circle, even though it moves with a constant speed, will have an acceleration Therefore, each point on a rotating rigid object will experience a centripetal acceleration a C v r 2 2 r

18 Rotational Motion Example For a compact disc player to read a CD, the angular speed must vary to keep the tangential speed constant (v t = r) At the inner sections, the angular speed is faster than at the outer sections

19 Inertia of Rotation Consider Newton s second law for the inertia of rotation to be patterned after the law for translation. F = 20 N a = 4 m/s 2 F = 20 N R = 0.5 m a = 2 rad/s 2 Linear Inertia, m 24 N m = 4 m/s 2 = 5 kg Rotational Inertia, I t (20 N)(0.5 m) I = = = 2.5 kg m a 4 m/s 2 2 Force does for translation what torque does for rotation:

20 Common Rotational Inertias L L I 1 3 ml 2 I 1 12 ml 2 R R R I = mr 2 I = ½mR 2 I 2 5 mr Hoop Disk or cylinder Solid sphere 2

21 Moments of Inertia of Various Rigid Objects

22 Rotational Kinetic Energy Consider tiny mass m: v = R K = ½mv 2 K = ½m(R) 2 m 1 m m 4 m 3 K = ½(mR 2 ) 2 axis m 2 Sum to find K total: K = ½(SmR 2 ) 2 (½ 2 same for all m ) Object rotating at constant. Rotational Inertia Defined: I = SmR 2

23 Example 1: What is the rotational kinetic energy of the device shown if it rotates at a constant speed of 600 rpm? First: I = SmR 2 I = (3 kg)(1 m) 2 + (2 kg)(3 m) 2 + (1 kg)(2 m) 2 2 kg 3 m 2 m 3 kg 1 m 1 kg I = 25 kg m 2 = 600 rpm = 62.8 rad/s K = ½Iw 2 = ½(25 kg m 2 )(62.8 rad/s) 2 K = 49,300 J

24 Example 2: A circular hoop and a disk each have a mass of 3 kg and a radius of 30 cm. Compare their rotational inertias. I mr 2 (3 kg)(0.2 m) 2 R I = kg m 2 I = mr 2 Hoop R I mr (3 kg)(0.2 m) I = ½mR 2 I = kg m 2 Disk

25 Parallel-Axis Theorem In the previous examples, the axis of rotation coincided with the axis of symmetry of the object For an arbitrary axis, the parallel-axis theorem often simplifies calculations The theorem states I = I CM + MD 2 I is about any axis parallel to the axis through the center of mass of the object I CM is about the axis through the center of mass D is the distance from the center of mass axis to the arbitrary axis

26 Parallel-Axis Theorem Example The axis of rotation goes through O The axis through the center of mass is shown The moment of inertia about the axis through O would be I O = I CM + MD 2

27 Moment of Inertia for a Rod Rotating Around One End The moment of inertia of the rod about its center is ICM D is ½ L Therefore, I I MD CM ML 2 2 L I ML M ML

28 Definition of Torque Torque is defined as the tendency to produce a change in rotational motion. Examples:

29 Torque Torque, t, is the tendency of a force to rotate an object about some axis Torque is a vector t = r F sin f = F d F is the force f is the angle the force makes with the horizontal d is the moment arm (or lever arm)

30 Torque is Determined by Three Factors: The magnitude of the applied force. The direction of the applied force. The location of the applied force. Each The forces 40-N of the force nearer 20-N the produces forces end of has the twice wrench a different the torque have greater as due does to torques. the the 20-N direction force. of force. Direction Magnitude Location of of Force of force force 20 N q q N 20 N N 20 N 40 N 20 N 20 N

31 Torque, cont The moment arm, d, is the perpendicular distance from the axis of rotation to a line drawn along the direction of the force d = r sin Φ

32 Torque, final The horizontal component of F (F cos f) has no tendency to produce a rotation Torque will have direction If the turning tendency of the force is counterclockwise, the torque will be positive If the turning tendency is clockwise, the torque will be negative

33 Sign Convention for Torque By convention, counterclockwise torques are positive and clockwise torques are negative. Positive torque: Counter-clockwise, out of page ccw cw Negative torque: clockwise, into page

34 Net Torque The force F 1 will tend to cause a counterclockwise rotation about O The force F 2 will tend to cause a clockwise rotation about O St t 1 t 2 F 1 d 1 F 2 d 2

35 Torque vs. Force Forces can cause a change in linear motion Described by Newton s Second Law Forces can cause a change in rotational motion The effectiveness of this change depends on the force and the moment arm The change in rotational motion depends on the torque

36 Torque Units The SI units of torque are N. m Although torque is a force multiplied by a distance, it is very different from work and energy The units for torque are reported in N. m and not changed to Joules

37 The Moment Arm The moment arm of a force is the perpendicular distance from the line of action of a force to the axis of rotation. F 1 r F 2 r r F 3

38 Example 1: An 80-N force acts at the end of a 12-cm wrench as shown. Find the torque. Extend line of action, draw, calculate r. r = 12 cm sin 60 0 = 10.4 cm t = (80 N)(0.104 m) = 8.31 N m

39 Alternate: An 80-N force acts at the end of a 12-cm wrench as shown. Find the torque. positive 12 cm Resolve 80-N force into components as shown. Note from figure: r x = 0 and r y = 12 cm t = (69.3 N)(0.12 m) t = 8.31 N m as before

40 Example 2: Find resultant torque about axis A for the arrangement shown below: Find t due to each force. Consider 20-N force first: 30 N m 2 m 40 N negative r A m 20 N r = (4 m) sin 30 0 = 2.00 m t = Fr = (20 N)(2 m) = 40 N m, cw The torque about A is clockwise and negative. t 20 = -40 N m

41 Example 2 (Cont.): Next we find torque due to 30-N force about same axis A. Find t due to each force. Consider 30-N force next. 30 N m 2 m 40 N r negative A m 20 N r = (8 m) sin 30 0 = 4.00 m t = Fr = (30 N)(4 m) = 120 N m, cw The torque about A is clockwise and negative. t 30 = -120 N m

42 Example 2 (Cont.): Finally, we consider the torque due to the 40-N force. Find t due to each force. Consider 40-N force next: 30 N m 2 m 40 N positive r A m 20 N r = (2 m) sin 90 0 = 2.00 m t = Fr = (40 N)(2 m) = 80 N m, ccw The torque about A is CCW and positive. t 40 = +80 N m

43 Example 2 (Conclusion): Find resultant torque about axis A for the arrangement shown below: Resultant torque is the sum of individual torques. 30 N m 2 m 40 N A m 20 N t R = t 20 + t 20 + t 20 = -40 N m -120 N m + 80 N m t R = - 80 N m Clockwise

44 Torque and Angular Acceleration Consider a particle of mass m rotating in a circle of radius r under the influence of tangential force F t The tangential force provides a tangential acceleration: F t = ma t

45 Torque and Angular Acceleration, Particle cont. The magnitude of the torque produced by F t around the center of the circle is t = F t r = (ma t ) r The tangential acceleration is related to the angular acceleration t = (ma t ) r = (mra) r = (mr 2 ) a Since mr 2 is the moment of inertia of the particle, t = Ia The torque is directly proportional to the angular acceleration and the constant of proportionality is the moment of inertia

46 Torque and Angular Acceleration, Extended cont. From Newton s Second Law df t = (dm) a t The torque associated with the force and using the angular acceleration gives dt = r df t = a t r dm = ar 2 dm Finding the net torque 2 2 t ar dm ar dm This becomes St Ia

47 Torque and Angular Acceleration, Extended final This is the same relationship that applied to a particle The result also applies when the forces have radial components The line of action of the radial component must pass through the axis of rotation These components will produce zero torque about the axis

48 Important Analogies For many problems involving rotation, there is an analogy to be drawn from linear motion. x f m A resultant force F produces negative acceleration a for a mass m. t I R 4 kg o 50 rad/s t = 40 N m A resultant torque t produces angular acceleration a of disk with rotational inertia I. F ma t I a

49 Newton s 2nd Law for Rotation How many revolutions required to stop? t = Ia F R 4 kg o 50 rad/s R = 0.20 m F = 40 N a FR = (½mR 2 )a 2F 2(40N) mr (4 kg)(0.2 m) q 0 2aq 2 f - 2 o (50 rad/s) 2 2a 2(100 rad/s ) a = 100 rad/s 2 q = 12.5 rad = 1.99 rev

50 Example 3: What is the linear acceleration of the falling 2-kg mass? Apply Newton s 2nd law to rotating disk: t Ia TR = (½MR 2 )a T = ½MRa a T = ½MR( ) ; R but a = ar; a = and T = ½Ma Apply Newton s 2nd law to falling mass: mg - T = ma mg - ½Ma T = ma (2 kg)(9.8 m/s 2 ) - ½(6 kg) a = (2 kg) a 19.6 N - (3 kg) a = (2 kg) a a = 3.92 m/s 2 a R R = 50 cm M 6 kg a =? R = 50 cm 6 kg +a 2 kg T T mg 2 kg

51 Torque and Angular Acceleration, Wheel Example The wheel is rotating and so we apply St Ia The tension supplies the tangential force The mass is moving in a straight line, so apply Newton s Second Law SF y = ma y = mg - T

52 Torque and Angular Acceleration, Multi-body Ex., 1 Both masses move in linear directions, so apply Newton s Second Law Both pulleys rotate, so apply the torque equation

53 Torque and Angular Acceleration, Multi-body Ex., 2 The mg and n forces on each pulley act at the axis of rotation and so supply no torque Apply the appropriate signs for clockwise and counterclockwise rotations in the torque equations

54 Work in Rotational Motion Find the work done by F on the object as it rotates through an infinitesimal distance ds = r dq dw = F. d s = (F sin f) r dq dw = t dq The radial component of F does no work because it is perpendicular to the displacement

55 Power in Rotational Motion The rate at which work is being done in a time interval dt is Power dw dq t t dt dt This is analogous to P = Fv in a linear system

56 Work-Kinetic Energy Theorem in Rotational Motion The work-kinetic energy theorem for rotational motion states that the net work done by external forces in rotating a symmetrical rigid object about a fixed axis equals the change in the object s rotational kinetic energy f W I d I f Ii i 2 2

57 Work and Power for Rotation Work = Fs = FRq Work = tq t FR q s F Work Power = = t tq t = q t s = Rq F Power = t Power = Torque x average angular velocity

58 Example 4: The rotating disk has a radius of 40 cm and a mass of 6 kg. Find the work and power if the 2-kg mass is lifted 20 m in 4 s. Work = tq = FR q s 20 m q = = = 50 rad R 0.4 m F = mg = (2 kg)(9.8 m/s 2 ); F = 19.6 N Work = (19.6 N)(0.4 m)(50 rad) s q 2 kg 6 kg F F=W s = 20 m Work = 392 J Work Power = = t 392 J 4s Power = 98 W

59 The Work-Energy Theorem Recall for linear motion that the work done is equal to the change in linear kinetic energy: 2 ½ 2 f 0 Fx ½mv mv Using angular analogies, we find the rotational work is equal to the change in rotational kinetic energy: 2 ½I 2 f 0 tq ½I

60 Two Kinds of Kinetic Energy Kinetic Energy of Translation: K = ½mv 2 Kinetic Energy of Rotation: K = ½I 2 R P v Total Kinetic Energy of a Rolling Object: T K mv I

61 Angular/Linear Conversions In many applications, you must solve an equation with both angular and linear parameters. It is necessary to remember the bridges: Displacement: sqr s q R Velocity: vr v R Acceleration: v ar a a R

62 Example 5: A circular hoop and a circular disk, each of the same mass and radius, roll at a linear speed v. Compare the kinetic energies. Two kinds of energy: K T = ½mv 2 K r = ½I 2 v v Total energy: E = ½mv 2 + ½I 2 = v R ½ ½ ½ ) v E mv mr 2 R ½ ½ ) v E mv mr 2 R Disk: E = ¾mv 2 Hoop: E = mv 2

63 Conservation of Energy The total energy is still conserved for systems in rotation and translation. However, rotation must now be considered. Begin: (U + K t + K R ) o = End: (U + K t + K R ) f Height? Rotation? velocity? mgh o ½I o 2 ½mv o 2 = mgh f ½I f 2 ½mv f 2 Height? Rotation? velocity?

64 Example 6: Find the velocity of the 2-kg mass just before it strikes the floor. R = 50 cm mgh o ½I o 2 ½mv o 2 = mgh f ½I f 2 ½mv f 2 6 kg 2 kg h = 10 m mgh mv I v mgh0 2 mv 2 ( 2 MR ) 2 R (2)(9.8)(10) (2) v (6) v I MR 2 2.5v 2 = 196 m 2 /s 2 v = 8.85 m/s

65 Applying the Work-Energy Theorem: What work is needed to stop wheel rotating: Work = K r F R 4 kg o 60 rad/s R = 0.30 m F = 40 N First find I for wheel: I = mr 2 = (4 kg)(0.3 m) 2 = 0.36 kg m 2 0 tq ½I ½I Work = -½I 2 o 2 2 f 0 Work = -½(0.36 kg m 2 )(60 rad/s) 2 Work = -648 J

66 Summary of Useless Equations

67 Pure Rolling Motion In pure rolling motion, an object rolls without slipping In such a case, there is a simple relationship between its rotational and translational motions

68 Rolling Object, Center of Mass The velocity of the center of mass is ds dq v R R CM dt dt The acceleration of the center of mass is dv dt d dt CM acm R R a

69 Rolling Object, Other Points A point on the rim, P, rotates to various positions such as Q and P At any instant, the point on the rim located at point P is at rest relative to the surface since no slipping occurs

70 Combined Rotation and Translation v cm v cm v cm Now consider a ball rolling without slipping. The angular velocity about the point P is same as for disk, so that we write: v R First consider a disk sliding without friction. The velocity of any part is equal to velocity v cm of the center of mass. R Or v R P v

71 Translation or Rotation? If you are to solve for a linear parameter, you must convert all angular terms to linear terms: q s R v R a a R I (?) mr 2 If you are to solve for an angular parameter, you must convert all linear terms to angular terms: s qr v R var

72 Example (a): Find velocity v of a disk if given its total kinetic energy E. Total energy: E = ½mv 2 + ½I 2 E mv I ; I mr ; v E mv mr E mv mv R ) ; 2 4 E 2 3mv 4 E or v 4 3m v R

73 Example (b) Find angular velocity of a disk given its total kinetic energy E. Total energy: E = ½mv 2 + ½I 2 E mv I ; I mr ; v R ) E m( R) mr ; E mr mr E 2 2 3mR 4 E or 4 3mR 2

74 Example 7: A hoop and a disk roll from the top of an incline. What are their speeds at the bottom if the initial height is 20 m? mgh o = ½mv 2 + ½I 2 Hoop: I = mr v mgh0 ½mv ½( mr ) 2 R mgh o = ½mv 2 + ½mv 2 ; mgh o = mv 2 20 m 2 v gh 0 (9.8 m/s )(20 m) Hoop: v = 14 m/s Disk: I = ½mR 2 ; mgh o = ½mv 2 + ½I v mgh0 ½mv ½(½ mr ) 2 R v gh v = 16.2 m/s

75 Angular Momentum Defined Consider a particle m moving with velocity v in a circle of radius r. Define angular momentum L: L = mvr Substituting v= r, gives: L = m(r) r = mr 2 For extended rotating body: L = (Smr 2 ) v = r axis m 1 m m 4 m 3 m 2 Object rotating at constant. Since I = Smr 2, we have: L = I Angular Momentum

76 Example 8: Find the angular momentum of a thin 4-kg rod of length 2 m if it rotates about its midpoint at a speed of 300 rpm. L = 2 m m = 4 kg For rod: I = ml 2 = (4 kg)(2 m) 2 I = 1.33 kg m 2 rev 2 rad 1 min rad/s min 1 rev 60 s L = I (1.33 kg m 2 )(31.4 rad/s) 2 L = 1315 kg m 2 /s

77 Angular Momentum Consider a particle of mass m located at the vector position r and moving with linear momentum p r F t r Adding the term t d( r p) dt dr dt dp dt p

78 Angular Momentum, cont The instantaneous angular momentum L of a particle relative to the origin O is defined as the cross product of the particle s instantaneous position vector r and its instantaneous linear momentum p L = r x p

79 Torque and Angular Momentum The torque is related to the angular momentum Similar to the way force is related to linear momentum t d L dt This is the rotational analog of Newton s Second Law St and L must be measured about the same origin This is valid for any origin fixed in an inertial frame

80 More About Angular Momentum The SI units of angular momentum are (kg. m 2 )/ s Both the magnitude and direction of L depend on the choice of origin The magnitude of L = mvr sin f f is the angle between p and r The direction of L is perpendicular to the plane formed by r and p

81 Angular Momentum of a Particle, Example The vector L = r x p is pointed out of the diagram The magnitude is L = mvr sin 90 o = mvr sin 90 o is used since v is perpendicular to r A particle in uniform circular motion has a constant angular momentum about an axis through the center of its path

82 Angular Momentum of a System of Particles The total angular momentum of a system of particles is defined as the vector sum of the angular momenta of the individual particles L tot = L 1 + L L n = SL i Differentiating with respect to time dl dt tot i dl dt i i t i

83 Angular Momentum of a Rotating Rigid Object Each particle of the object rotates in the xy plane about the z axis with an angular speed of The angular momentum of an individual particle is L i = m i r i 2 L and are directed along the z axis

84 Angular Momentum of a Rotating Rigid Object, cont To find the angular momentum of the entire object, add the angular momenta of all the individual particles 2 Lz Li miri I i i This also gives the rotational form of Newton s Second Law t ext dl dt d dt z I Ia

85 Angular Momentum of a Rotating Rigid Object, final The rotational form of Newton s Second Law is also valid for a rigid object rotating about a moving axis provided the moving axis: (1) passes through the center of mass (2) is a symmetry axis If a symmetrical object rotates about a fixed axis passing through its center of mass, the vector form holds: L = I where L is the total angular momentum measured with respect to the axis of rotation

86 Angular Momentum of a Bowling Ball The momentum of inertia of the ball is 2/5MR 2 The angular momentum of the ball is L z = I The direction of the angular momentum is in the positive z direction

87 Impulse and Momentum Recall for linear motion the linear impulse is equal to the change in linear momentum: F t mv mv Using angular analogies, we find angular impulse to be equal to the change in angular momentum: f 0 t t I I f 0

88 Example 9: A sharp force of 200 N is applied to the edge of a wheel free to rotate. The force acts for s. What is the final angular velocity? I = mr 2 = (2 kg)(0.4 m) 2 I = 0.32 kg m 2 Applied torque t FR t = s R F 2 kg o 0 rad/s R = 0.40 m F = 200 N Impulse = change in angular momentum t t = I f I o 0 FR t = I f f = 0.5 rad/s

89 Conservation of Momentum In the absence of external torque the rotational momentum of a system is conserved (constant). 0 I f f I o o = t t I f f I o o I o = 2 kg m 2 ; o = 600 rpm I f = 6 kg m 2 ; o =? f I00 I f 2 (2 kg m )(600 rpm) 6 kg m 2 f = 200 rpm

90 Summary Rotational Analogies Quantity Linear Rotational Displacement Displacement x Radians q Inertia Mass (kg) I (kgm 2 ) Force Newtons N Torque N m Velocity v m/s Rad/s Acceleration a m/s 2 a Rad/s 2 Momentum mv (kg m/s) I (kgm 2 rad/s)

91 The Vector Product Torque can also be found by using the vector product of force F and position vector r. For example, consider the figure below. Torque r F Sin q q F The effect of the force F at angle q (torque) is to advance the bolt out of the page. Magnitude: (F Sin q)r Direction = Out of page (+).

92 Definition of a Vector Product The magnitude of the vector (cross) product of two vectors A and B is defined as follows: A x B = l A l l B l Sin q In our example, the cross product of F and r is: F x r = l F l l r l Sin q Magnitude only q F Sin q r F In effect, this becomes simply: (F Sin q) r or F (r Sin q)

93 Example: Find the magnitude of the cross product of the vectors r and F drawn below: Torque 12 lb r x F = l r l l F l Sin q 6 in r x F = (6 in.)(12 lb) Sin 60 0 r x F = 62.4 lb in. 6 in. r x F = l r l l F l Sin q 60 0 Torque 12 lb r x F = (6 in.)(12 lb) Sin r x F = 62.4 lb in. Explain difference. Also, what about F x r?

94 Direction of the Vector Product. The direction of a vector product is determined by the right hand rule. A x B = C (up) B x A = -C (Down) What is direction of A x C? A C B A -C Curl fingers of right hand in direction of cross pro-duct (A to B) or (B to A). Thumb will point in the direction of product C. B

95 Torque Example: What are the magnitude and direction of the cross product, r x F? 10 lb 50 0 r x F = l r l l F l Sin q r x F = (6 in.)(10 lb) Sin in. r x F = 38.3 lb in. Magnitude F r Direction by right hand rule: Out of paper (thumb) or +k Out r x F = (38.3 lb in.) k What are magnitude and direction of F x r?

96 Cross Products Using (i,j,k) y j k z Consider 3D axes (x, y, z) i x Define unit vectors, i, j, k Consider cross product: i x i i i i x i = (1)(1) Sin 0 0 = 0 Magnitudes are zero for parallel vector products. j x j = (1)(1) Sin 0 0 = 0 k x k = (1)(1)Sin 0 0 = 0

97 Vector Products Using (i,j,k) z k y j i j i x Consider 3D axes (x, y, z) Define unit vectors, i, j, k Consider dot product: i x j i x j = (1)(1) Sin 90 0 = 1 Magnitudes are 1 for perpendicular vector products. j x k = (1)(1) Sin 90 0 = 1 k x i = (1)(1) Sin 90 0 = 1

98 Vector Product (Directions) k y j i x Directions are given by the right hand rule. Rotating first vector into second. z j i x j = (1)(1) Sin 90 0 = +1 k k i j x k = (1)(1) Sin 90 0 = +1 i k x i = (1)(1) Sin 90 0 = +1 j

99 Vector Products Practice (i,j,k) z k y j j i x Directions are given by the right hand rule. Rotating first vector into second. i x k =? k x j =? - j (down) - i (left) j x -i =? + k (out) k i 2 i x -3 k =? + 6 j (up)

100 Using i,j Notation - Vector Products Consider: A = 2 i - 4 j and B = 3 i + 5 j A x B = (2 i - 4 j) x (3 i + 5 j) = 0 k -k 0 (2)(3) ixi + (2)(5) ixj + (-4)(3) jxi + (-4)(5) jxj A x B = (2)(5) k + (-4)(3)(-k) = +22 k Alternative: A = 2 i - 4 j B = 3 i + 5 j Evaluate determinant A x B = 10 - (-12) = +22 k

101 Conservation of Angular Momentum The total angular momentum of a system is constant in both magnitude and direction if the resultant external torque acting on the system is zero Net torque = 0 -> means that the system is isolated L tot = constant or L i = L f For a system of particles, L tot = SL n = constant

102 Conservation of Angular Momentum, cont If the mass of an isolated system undergoes redistribution, the moment of inertia changes The conservation of angular momentum requires a compensating change in the angular velocity I i i = I f f This holds for rotation about a fixed axis and for rotation about an axis through the center of mass of a moving system The net torque must be zero in any case

103 Conservation Law Summary For an isolated system - (1) Conservation of Energy: E i = E f (2) Conservation of Linear Momentum: p i = p f (3) Conservation of Angular Momentum: L i = L f

104 Conservation of Angular Momentum: The Merry-Go-Round The moment of inertia of the system is the moment of inertia of the platform plus the moment of inertia of the person Assume the person can be treated as a particle As the person moves toward the center of the rotating platform, the angular speed will increase To keep L constant

105 Motion of a Top The only external forces acting on the top are the normal force n and the gravitational force M g The direction of the angular momentum L is along the axis of symmetry The right-hand rule indicates that t = r F = r M g is in the xy plane

106 Motion of a Top, cont The direction of d L is parallel to that of t in part. The fact that L f = d L + L i indicates that the top precesses about the z axis. The precessional motion is the motion of the symmetry axis about the vertical The precession is usually slow relative to the spinning motion of the top

107 Gyroscope A gyroscope can be used to illustrate precessional motion The gravitational force Mg produces a torque about the pivot, and this torque is perpendicular to the axle The normal force produces no torque

108 Gyroscope, cont The torque results in a change in angular momentum d L in a direction perpendicular to the axle. The axle sweeps out an angle df in a time interval dt. The direction, not the magnitude, of L is changing The gyroscope experiences precessional motion

109 Gyroscope, final To simplify, assume the angular momentum due to the motion of the center of mass about the pivot is zero Therefore, the total angular momentum is L = I due to its spin This is a good approximation when is large

110 Precessional Frequency Analyzing the previous vector triangle, the rate at which the axle rotates about the vertical axis can be found p df dt Mgh I p is the precessional frequency

111 Gyroscope in a Spacecraft The angular momentum of the spacecraft about its center of mass is zero A gyroscope is set into rotation, giving it a nonzero angular momentum The spacecraft rotates in the direction opposite to that of the gyroscope So the total momentum of the system remains zero

Chapter 10. Rotation of a Rigid Object about a Fixed Axis

Chapter 10. Rotation of a Rigid Object about a Fixed Axis Chapter 10 Rotation of a Rigid Object about a Fixed Axis Angular Position Axis of rotation is the center of the disc Choose a fixed reference line. Point P is at a fixed distance r from the origin. A small

More information

Chapter 11. Angular Momentum

Chapter 11. Angular Momentum Chapter 11 Angular Momentum Angular Momentum Angular momentum plays a key role in rotational dynamics. There is a principle of conservation of angular momentum. In analogy to the principle of conservation

More information

Chapter 11. Angular Momentum

Chapter 11. Angular Momentum Chapter 11 Angular Momentum Angular Momentum Angular momentum plays a key role in rotational dynamics. There is a principle of conservation of angular momentum. In analogy to the principle of conservation

More information

Chap10. Rotation of a Rigid Object about a Fixed Axis

Chap10. Rotation of a Rigid Object about a Fixed Axis Chap10. Rotation of a Rigid Object about a Fixed Axis Level : AP Physics Teacher : Kim 10.1 Angular Displacement, Velocity, and Acceleration - A rigid object rotating about a fixed axis through O perpendicular

More information

Rotation. Kinematics Rigid Bodies Kinetic Energy. Torque Rolling. featuring moments of Inertia

Rotation. Kinematics Rigid Bodies Kinetic Energy. Torque Rolling. featuring moments of Inertia Rotation Kinematics Rigid Bodies Kinetic Energy featuring moments of Inertia Torque Rolling Angular Motion We think about rotation in the same basic way we do about linear motion How far does it go? How

More information

Rotational Motion About a Fixed Axis

Rotational Motion About a Fixed Axis Rotational Motion About a Fixed Axis Vocabulary rigid body axis of rotation radian average angular velocity instantaneous angular average angular Instantaneous angular frequency velocity acceleration acceleration

More information

= o + t = ot + ½ t 2 = o + 2

= o + t = ot + ½ t 2 = o + 2 Chapters 8-9 Rotational Kinematics and Dynamics Rotational motion Rotational motion refers to the motion of an object or system that spins about an axis. The axis of rotation is the line about which the

More information

Rotation. PHYS 101 Previous Exam Problems CHAPTER

Rotation. PHYS 101 Previous Exam Problems CHAPTER PHYS 101 Previous Exam Problems CHAPTER 10 Rotation Rotational kinematics Rotational inertia (moment of inertia) Kinetic energy Torque Newton s 2 nd law Work, power & energy conservation 1. Assume that

More information

Translational vs Rotational. m x. Connection Δ = = = = = = Δ = = = = = = Δ =Δ = = = = = 2 / 1/2. Work

Translational vs Rotational. m x. Connection Δ = = = = = = Δ = = = = = = Δ =Δ = = = = = 2 / 1/2. Work Translational vs Rotational / / 1/ Δ m x v dx dt a dv dt F ma p mv KE mv Work Fd / / 1/ θ ω θ α ω τ α ω ω τθ Δ I d dt d dt I L I KE I Work / θ ω α τ Δ Δ c t s r v r a v r a r Fr L pr Connection Translational

More information

Rolling, Torque & Angular Momentum

Rolling, Torque & Angular Momentum PHYS 101 Previous Exam Problems CHAPTER 11 Rolling, Torque & Angular Momentum Rolling motion Torque Angular momentum Conservation of angular momentum 1. A uniform hoop (ring) is rolling smoothly from the

More information

Rotation. Rotational Variables

Rotation. Rotational Variables Rotation Rigid Bodies Rotation variables Constant angular acceleration Rotational KE Rotational Inertia Rotational Variables Rotation of a rigid body About a fixed rotation axis. Rigid Body an object that

More information

Rotational Motion and Torque

Rotational Motion and Torque Rotational Motion and Torque Introduction to Angular Quantities Sections 8- to 8-2 Introduction Rotational motion deals with spinning objects, or objects rotating around some point. Rotational motion is

More information

Big Idea 4: Interactions between systems can result in changes in those systems. Essential Knowledge 4.D.1: Torque, angular velocity, angular

Big Idea 4: Interactions between systems can result in changes in those systems. Essential Knowledge 4.D.1: Torque, angular velocity, angular Unit 7: Rotational Motion (angular kinematics, dynamics, momentum & energy) Name: Big Idea 3: The interactions of an object with other objects can be described by forces. Essential Knowledge 3.F.1: Only

More information

Chapter 12: Rotation of Rigid Bodies. Center of Mass Moment of Inertia Torque Angular Momentum Rolling Statics

Chapter 12: Rotation of Rigid Bodies. Center of Mass Moment of Inertia Torque Angular Momentum Rolling Statics Chapter 1: Rotation of Rigid Bodies Center of Mass Moment of Inertia Torque Angular Momentum Rolling Statics Translational vs Rotational / / 1/ m x v dx dt a dv dt F ma p mv KE mv Work Fd P Fv / / 1/ I

More information

Rotation. EMU Physics Department. Ali ÖVGÜN.

Rotation. EMU Physics Department. Ali ÖVGÜN. Rotation Ali ÖVGÜN EMU Physics Department www.aovgun.com Rotational Motion Angular Position and Radians Angular Velocity Angular Acceleration Rigid Object under Constant Angular Acceleration Angular and

More information

Chapter 10. Rotation

Chapter 10. Rotation Chapter 10 Rotation Rotation Rotational Kinematics: Angular velocity and Angular Acceleration Rotational Kinetic Energy Moment of Inertia Newton s nd Law for Rotation Applications MFMcGraw-PHY 45 Chap_10Ha-Rotation-Revised

More information

Rotational Motion, Torque, Angular Acceleration, and Moment of Inertia. 8.01t Nov 3, 2004

Rotational Motion, Torque, Angular Acceleration, and Moment of Inertia. 8.01t Nov 3, 2004 Rotational Motion, Torque, Angular Acceleration, and Moment of Inertia 8.01t Nov 3, 2004 Rotation and Translation of Rigid Body Motion of a thrown object Translational Motion of the Center of Mass Total

More information

PS 11 GeneralPhysics I for the Life Sciences

PS 11 GeneralPhysics I for the Life Sciences PS 11 GeneralPhysics I for the Life Sciences ROTATIONAL MOTION D R. B E N J A M I N C H A N A S S O C I A T E P R O F E S S O R P H Y S I C S D E P A R T M E N T F E B R U A R Y 0 1 4 Questions and Problems

More information

Chapter 8 Lecture Notes

Chapter 8 Lecture Notes Chapter 8 Lecture Notes Physics 2414 - Strauss Formulas: v = l / t = r θ / t = rω a T = v / t = r ω / t =rα a C = v 2 /r = ω 2 r ω = ω 0 + αt θ = ω 0 t +(1/2)αt 2 θ = (1/2)(ω 0 +ω)t ω 2 = ω 0 2 +2αθ τ

More information

General Definition of Torque, final. Lever Arm. General Definition of Torque 7/29/2010. Units of Chapter 10

General Definition of Torque, final. Lever Arm. General Definition of Torque 7/29/2010. Units of Chapter 10 Units of Chapter 10 Determining Moments of Inertia Rotational Kinetic Energy Rotational Plus Translational Motion; Rolling Why Does a Rolling Sphere Slow Down? General Definition of Torque, final Taking

More information

Rigid Object. Chapter 10. Angular Position. Angular Position. A rigid object is one that is nondeformable

Rigid Object. Chapter 10. Angular Position. Angular Position. A rigid object is one that is nondeformable Rigid Object Chapter 10 Rotation of a Rigid Object about a Fixed Axis A rigid object is one that is nondeformable The relative locations of all particles making up the object remain constant All real objects

More information

Fundamentals Physics. Chapter 10 Rotation

Fundamentals Physics. Chapter 10 Rotation Fundamentals Physics Tenth Edition Halliday Chapter 10 Rotation 10-1 Rotational Variables (1 of 15) Learning Objectives 10.01 Identify that if all parts of a body rotate around a fixed axis locked together,

More information

APC PHYSICS CHAPTER 11 Mr. Holl Rotation

APC PHYSICS CHAPTER 11 Mr. Holl Rotation APC PHYSICS CHAPTER 11 Mr. Holl Rotation Student Notes 11-1 Translation and Rotation All of the motion we have studied to this point was linear or translational. Rotational motion is the study of spinning

More information

Angular velocity and angular acceleration CHAPTER 9 ROTATION. Angular velocity and angular acceleration. ! equations of rotational motion

Angular velocity and angular acceleration CHAPTER 9 ROTATION. Angular velocity and angular acceleration. ! equations of rotational motion Angular velocity and angular acceleration CHAPTER 9 ROTATION! r i ds i dθ θ i Angular velocity and angular acceleration! equations of rotational motion Torque and Moment of Inertia! Newton s nd Law for

More information

Test 7 wersja angielska

Test 7 wersja angielska Test 7 wersja angielska 7.1A One revolution is the same as: A) 1 rad B) 57 rad C) π/2 rad D) π rad E) 2π rad 7.2A. If a wheel turns with constant angular speed then: A) each point on its rim moves with

More information

Lecture PowerPoints. Chapter 10 Physics for Scientists and Engineers, with Modern Physics, 4 th edition Giancoli

Lecture PowerPoints. Chapter 10 Physics for Scientists and Engineers, with Modern Physics, 4 th edition Giancoli Lecture PowerPoints Chapter 10 Physics for Scientists and Engineers, with Modern Physics, 4 th edition Giancoli 2009 Pearson Education, Inc. This work is protected by United States copyright laws and is

More information

PLANAR KINETIC EQUATIONS OF MOTION (Section 17.2)

PLANAR KINETIC EQUATIONS OF MOTION (Section 17.2) PLANAR KINETIC EQUATIONS OF MOTION (Section 17.2) We will limit our study of planar kinetics to rigid bodies that are symmetric with respect to a fixed reference plane. As discussed in Chapter 16, when

More information

Chapter 12: Rotation of Rigid Bodies. Center of Mass Moment of Inertia Torque Angular Momentum Rolling Statics

Chapter 12: Rotation of Rigid Bodies. Center of Mass Moment of Inertia Torque Angular Momentum Rolling Statics Chapter 12: Rotation of Rigid Bodies Center of Mass Moment of Inertia Torque Angular Momentum Rolling Statics Translational vs Rotational 2 / / 1/ 2 m x v dx dt a dv dt F ma p mv KE mv Work Fd P Fv 2 /

More information

Rolling Motion and Angular Momentum

Rolling Motion and Angular Momentum P U Z Z L E R One of the most popular early bicycles was the penny farthing, introduced in 1870. The bicycle was so named because the size relationship of its two wheels was about the same as the size

More information

PHYS 111 HOMEWORK #11

PHYS 111 HOMEWORK #11 PHYS 111 HOMEWORK #11 Due date: You have a choice here. You can submit this assignment on Tuesday, December and receive a 0 % bonus, or you can submit this for normal credit on Thursday, 4 December. If

More information

Rolling, Torque, Angular Momentum

Rolling, Torque, Angular Momentum Chapter 11 Rolling, Torque, Angular Momentum Copyright 11.2 Rolling as Translational and Rotation Combined Motion of Translation : i.e.motion along a straight line Motion of Rotation : rotation about a

More information

31 ROTATIONAL KINEMATICS

31 ROTATIONAL KINEMATICS 31 ROTATIONAL KINEMATICS 1. Compare and contrast circular motion and rotation? Address the following Which involves an object and which involves a system? Does an object/system in circular motion have

More information

Chapter 8- Rotational Kinematics Angular Variables Kinematic Equations

Chapter 8- Rotational Kinematics Angular Variables Kinematic Equations Chapter 8- Rotational Kinematics Angular Variables Kinematic Equations Chapter 9- Rotational Dynamics Torque Center of Gravity Newton s 2 nd Law- Angular Rotational Work & Energy Angular Momentum Angular

More information

Handout 7: Torque, angular momentum, rotational kinetic energy and rolling motion. Torque and angular momentum

Handout 7: Torque, angular momentum, rotational kinetic energy and rolling motion. Torque and angular momentum Handout 7: Torque, angular momentum, rotational kinetic energy and rolling motion Torque and angular momentum In Figure, in order to turn a rod about a fixed hinge at one end, a force F is applied at a

More information

We define angular displacement, θ, and angular velocity, ω. What's a radian?

We define angular displacement, θ, and angular velocity, ω. What's a radian? We define angular displacement, θ, and angular velocity, ω Units: θ = rad ω = rad/s What's a radian? Radian is the ratio between the length of an arc and its radius note: counterclockwise is + clockwise

More information

Two-Dimensional Rotational Kinematics

Two-Dimensional Rotational Kinematics Two-Dimensional Rotational Kinematics Rigid Bodies A rigid body is an extended object in which the distance between any two points in the object is constant in time. Springs or human bodies are non-rigid

More information

Chap. 10: Rotational Motion

Chap. 10: Rotational Motion Chap. 10: Rotational Motion I. Rotational Kinematics II. Rotational Dynamics - Newton s Law for Rotation III. Angular Momentum Conservation (Chap. 10) 1 Newton s Laws for Rotation n e t I 3 rd part [N

More information

Rotational Kinetic Energy

Rotational Kinetic Energy Lecture 17, Chapter 10: Rotational Energy and Angular Momentum 1 Rotational Kinetic Energy Consider a rigid body rotating with an angular velocity ω about an axis. Clearly every point in the rigid body

More information

Webreview Torque and Rotation Practice Test

Webreview Torque and Rotation Practice Test Please do not write on test. ID A Webreview - 8.2 Torque and Rotation Practice Test Multiple Choice Identify the choice that best completes the statement or answers the question. 1. A 0.30-m-radius automobile

More information

Phys101 Lectures 19, 20 Rotational Motion

Phys101 Lectures 19, 20 Rotational Motion Phys101 Lectures 19, 20 Rotational Motion Key points: Angular and Linear Quantities Rotational Dynamics; Torque and Moment of Inertia Rotational Kinetic Energy Ref: 10-1,2,3,4,5,6,8,9. Page 1 Angular Quantities

More information

Chapter 9-10 Test Review

Chapter 9-10 Test Review Chapter 9-10 Test Review Chapter Summary 9.2. The Second Condition for Equilibrium Explain torque and the factors on which it depends. Describe the role of torque in rotational mechanics. 10.1. Angular

More information

Chapter 10: Dynamics of Rotational Motion

Chapter 10: Dynamics of Rotational Motion Chapter 10: Dynamics of Rotational Motion What causes an angular acceleration? The effectiveness of a force at causing a rotation is called torque. QuickCheck 12.5 The four forces shown have the same strength.

More information

Rotational Motion. Every quantity that we have studied with translational motion has a rotational counterpart

Rotational Motion. Every quantity that we have studied with translational motion has a rotational counterpart Rotational Motion & Angular Momentum Rotational Motion Every quantity that we have studied with translational motion has a rotational counterpart TRANSLATIONAL ROTATIONAL Displacement x Angular Displacement

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Exam Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Two men, Joel and Jerry, push against a wall. Jerry stops after 10 min, while Joel is

More information

Chapter 8. Rotational Equilibrium and Rotational Dynamics

Chapter 8. Rotational Equilibrium and Rotational Dynamics Chapter 8 Rotational Equilibrium and Rotational Dynamics Wrench Demo Torque Torque, τ, is the tendency of a force to rotate an object about some axis τ = Fd F is the force d is the lever arm (or moment

More information

PSI AP Physics I Rotational Motion

PSI AP Physics I Rotational Motion PSI AP Physics I Rotational Motion Multiple-Choice questions 1. Which of the following is the unit for angular displacement? A. meters B. seconds C. radians D. radians per second 2. An object moves from

More information

Rotational Motion. Lecture 17. Chapter 10. Physics I Department of Physics and Applied Physics

Rotational Motion. Lecture 17. Chapter 10. Physics I Department of Physics and Applied Physics Lecture 17 Chapter 10 Physics I 11.13.2013 otational Motion Torque Course website: http://faculty.uml.edu/andriy_danylov/teaching/physicsi Lecture Capture: http://echo360.uml.edu/danylov2013/physics1fall.html

More information

8 Rotational motion of solid objects

8 Rotational motion of solid objects 8 Rotational motion of solid objects Kinematics of rotations PHY166 Fall 005 In this Lecture we call solid objects such extended objects that are rigid (nondeformable) and thus retain their shape. In contrast

More information

FALL TERM EXAM, PHYS 1211, INTRODUCTORY PHYSICS I Thursday, 11 December 2014, 6 PM to 9 PM, Field House Gym

FALL TERM EXAM, PHYS 1211, INTRODUCTORY PHYSICS I Thursday, 11 December 2014, 6 PM to 9 PM, Field House Gym FALL TERM EXAM, PHYS 1211, INTRODUCTORY PHYSICS I Thursday, 11 December 2014, 6 PM to 9 PM, Field House Gym NAME: STUDENT ID: INSTRUCTION 1. This exam booklet has 13 pages. Make sure none are missing 2.

More information

Circular Motion, Pt 2: Angular Dynamics. Mr. Velazquez AP/Honors Physics

Circular Motion, Pt 2: Angular Dynamics. Mr. Velazquez AP/Honors Physics Circular Motion, Pt 2: Angular Dynamics Mr. Velazquez AP/Honors Physics Formulas: Angular Kinematics (θ must be in radians): s = rθ Arc Length 360 = 2π rads = 1 rev ω = θ t = v t r Angular Velocity α av

More information

Lecture 3. Rotational motion and Oscillation 06 September 2018

Lecture 3. Rotational motion and Oscillation 06 September 2018 Lecture 3. Rotational motion and Oscillation 06 September 2018 Wannapong Triampo, Ph.D. Angular Position, Velocity and Acceleration: Life Science applications Recall last t ime. Rigid Body - An object

More information

Chapter 8 continued. Rotational Dynamics

Chapter 8 continued. Rotational Dynamics Chapter 8 continued Rotational Dynamics 8.4 Rotational Work and Energy Work to accelerate a mass rotating it by angle φ F W = F(cosθ)x x = rφ = Frφ Fr = τ (torque) = τφ r φ s F to x θ = 0 DEFINITION OF

More information

AP Physics 1: Rotational Motion & Dynamics: Problem Set

AP Physics 1: Rotational Motion & Dynamics: Problem Set AP Physics 1: Rotational Motion & Dynamics: Problem Set I. Axis of Rotation and Angular Properties 1. How many radians are subtended by a 0.10 m arc of a circle of radius 0.40 m? 2. How many degrees are

More information

PSI AP Physics I Rotational Motion

PSI AP Physics I Rotational Motion PSI AP Physics I Rotational Motion Multiple-Choice questions 1. Which of the following is the unit for angular displacement? A. meters B. seconds C. radians D. radians per second 2. An object moves from

More information

Rotational Motion. Rotational Motion. Rotational Motion

Rotational Motion. Rotational Motion. Rotational Motion I. Rotational Kinematics II. Rotational Dynamics (Netwton s Law for Rotation) III. Angular Momentum Conservation 1. Remember how Newton s Laws for translational motion were studied: 1. Kinematics (x =

More information

AP Physics 1- Torque, Rotational Inertia, and Angular Momentum Practice Problems FACT: The center of mass of a system of objects obeys Newton s second law- F = Ma cm. Usually the location of the center

More information

Angular Displacement. θ i. 1rev = 360 = 2π rads. = "angular displacement" Δθ = θ f. π = circumference. diameter

Angular Displacement. θ i. 1rev = 360 = 2π rads. = angular displacement Δθ = θ f. π = circumference. diameter Rotational Motion Angular Displacement π = circumference diameter π = circumference 2 radius circumference = 2πr Arc length s = rθ, (where θ in radians) θ 1rev = 360 = 2π rads Δθ = θ f θ i = "angular displacement"

More information

Physics 131: Lecture 21. Today s Agenda

Physics 131: Lecture 21. Today s Agenda Physics 131: Lecture 1 Today s Agenda Rotational dynamics Torque = I Angular Momentum Physics 01: Lecture 10, Pg 1 Newton s second law in rotation land Sum of the torques will equal the moment of inertia

More information

Chapter 10 Rotational Kinematics and Energy. Copyright 2010 Pearson Education, Inc.

Chapter 10 Rotational Kinematics and Energy. Copyright 2010 Pearson Education, Inc. Chapter 10 Rotational Kinematics and Energy Copyright 010 Pearson Education, Inc. 10-1 Angular Position, Velocity, and Acceleration Copyright 010 Pearson Education, Inc. 10-1 Angular Position, Velocity,

More information

Rotational Kinematics

Rotational Kinematics Rotational Kinematics Rotational Coordinates Ridged objects require six numbers to describe their position and orientation: 3 coordinates 3 axes of rotation Rotational Coordinates Use an angle θ to describe

More information

Rotational Dynamics continued

Rotational Dynamics continued Chapter 9 Rotational Dynamics continued 9.4 Newton s Second Law for Rotational Motion About a Fixed Axis ROTATIONAL ANALOG OF NEWTON S SECOND LAW FOR A RIGID BODY ROTATING ABOUT A FIXED AXIS I = ( mr 2

More information

Slide 1 / 133. Slide 2 / 133. Slide 3 / How many radians are subtended by a 0.10 m arc of a circle of radius 0.40 m?

Slide 1 / 133. Slide 2 / 133. Slide 3 / How many radians are subtended by a 0.10 m arc of a circle of radius 0.40 m? 1 How many radians are subtended by a 0.10 m arc of a circle of radius 0.40 m? Slide 1 / 133 2 How many degrees are subtended by a 0.10 m arc of a circle of radius of 0.40 m? Slide 2 / 133 3 A ball rotates

More information

Slide 2 / 133. Slide 1 / 133. Slide 3 / 133. Slide 4 / 133. Slide 5 / 133. Slide 6 / 133

Slide 2 / 133. Slide 1 / 133. Slide 3 / 133. Slide 4 / 133. Slide 5 / 133. Slide 6 / 133 Slide 1 / 133 1 How many radians are subtended by a 0.10 m arc of a circle of radius 0.40 m? Slide 2 / 133 2 How many degrees are subtended by a 0.10 m arc of a circle of radius of 0.40 m? Slide 3 / 133

More information

Physics 101: Lecture 15 Torque, F=ma for rotation, and Equilibrium

Physics 101: Lecture 15 Torque, F=ma for rotation, and Equilibrium Physics 101: Lecture 15 Torque, F=ma for rotation, and Equilibrium Strike (Day 10) Prelectures, checkpoints, lectures continue with no change. Take-home quizzes this week. See Elaine Schulte s email. HW

More information

1 MR SAMPLE EXAM 3 FALL 2013

1 MR SAMPLE EXAM 3 FALL 2013 SAMPLE EXAM 3 FALL 013 1. A merry-go-round rotates from rest with an angular acceleration of 1.56 rad/s. How long does it take to rotate through the first rev? A) s B) 4 s C) 6 s D) 8 s E) 10 s. A wheel,

More information

Rotational Motion. Lecture 17. Chapter 10. Physics I Department of Physics and Applied Physics

Rotational Motion. Lecture 17. Chapter 10. Physics I Department of Physics and Applied Physics Lecture 17 Chapter 10 Physics I 04.0.014 otational Motion Torque Course website: http://faculty.uml.edu/andriy_danylov/teaching/physicsi Lecture Capture: http://echo360.uml.edu/danylov013/physics1spring.html

More information

Phys 106 Practice Problems Common Quiz 1 Spring 2003

Phys 106 Practice Problems Common Quiz 1 Spring 2003 Phys 106 Practice Problems Common Quiz 1 Spring 2003 1. For a wheel spinning with constant angular acceleration on an axis through its center, the ratio of the speed of a point on the rim to the speed

More information

Unit 8 Notetaking Guide Torque and Rotational Motion

Unit 8 Notetaking Guide Torque and Rotational Motion Unit 8 Notetaking Guide Torque and Rotational Motion Rotational Motion Until now, we have been concerned mainly with translational motion. We discussed the kinematics and dynamics of translational motion

More information

TutorBreeze.com 7. ROTATIONAL MOTION. 3. If the angular velocity of a spinning body points out of the page, then describe how is the body spinning?

TutorBreeze.com 7. ROTATIONAL MOTION. 3. If the angular velocity of a spinning body points out of the page, then describe how is the body spinning? 1. rpm is about rad/s. 7. ROTATIONAL MOTION 2. A wheel rotates with constant angular acceleration of π rad/s 2. During the time interval from t 1 to t 2, its angular displacement is π rad. At time t 2

More information

Physics 131: Lecture 21. Today s Agenda

Physics 131: Lecture 21. Today s Agenda Physics 131: Lecture 21 Today s Agenda Rotational dynamics Torque = I Angular Momentum Physics 201: Lecture 10, Pg 1 Newton s second law in rotation land Sum of the torques will equal the moment of inertia

More information

Exam 3 Practice Solutions

Exam 3 Practice Solutions Exam 3 Practice Solutions Multiple Choice 1. A thin hoop, a solid disk, and a solid sphere, each with the same mass and radius, are at rest at the top of an inclined plane. If all three are released at

More information

PHYSICS. Chapter 12 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc.

PHYSICS. Chapter 12 Lecture FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E RANDALL D. KNIGHT Pearson Education, Inc. PHYSICS FOR SCIENTISTS AND ENGINEERS A STRATEGIC APPROACH 4/E Chapter 12 Lecture RANDALL D. KNIGHT Chapter 12 Rotation of a Rigid Body IN THIS CHAPTER, you will learn to understand and apply the physics

More information

1301W.600 Lecture 16. November 6, 2017

1301W.600 Lecture 16. November 6, 2017 1301W.600 Lecture 16 November 6, 2017 You are Cordially Invited to the Physics Open House Friday, November 17 th, 2017 4:30-8:00 PM Tate Hall, Room B20 Time to apply for a major? Consider Physics!! Program

More information

Lesson 8. Luis Anchordoqui. Physics 168. Thursday, October 11, 18

Lesson 8. Luis Anchordoqui. Physics 168. Thursday, October 11, 18 Lesson 8 Physics 168 1 Rolling 2 Intuitive Question Why is it that when a body is rolling on a plane without slipping the point of contact with the plane does not move? A simple answer to this question

More information

Physics 201. Professor P. Q. Hung. 311B, Physics Building. Physics 201 p. 1/1

Physics 201. Professor P. Q. Hung. 311B, Physics Building. Physics 201 p. 1/1 Physics 201 p. 1/1 Physics 201 Professor P. Q. Hung 311B, Physics Building Physics 201 p. 2/1 Rotational Kinematics and Energy Rotational Kinetic Energy, Moment of Inertia All elements inside the rigid

More information

Chapter 8 continued. Rotational Dynamics

Chapter 8 continued. Rotational Dynamics Chapter 8 continued Rotational Dynamics 8.4 Rotational Work and Energy Work to accelerate a mass rotating it by angle φ F W = F(cosθ)x x = s = rφ = Frφ Fr = τ (torque) = τφ r φ s F to s θ = 0 DEFINITION

More information

CHAPTER 8: ROTATIONAL OF RIGID BODY PHYSICS. 1. Define Torque

CHAPTER 8: ROTATIONAL OF RIGID BODY PHYSICS. 1. Define Torque 7 1. Define Torque 2. State the conditions for equilibrium of rigid body (Hint: 2 conditions) 3. Define angular displacement 4. Define average angular velocity 5. Define instantaneous angular velocity

More information

Lecture Outline Chapter 10. Physics, 4 th Edition James S. Walker. Copyright 2010 Pearson Education, Inc.

Lecture Outline Chapter 10. Physics, 4 th Edition James S. Walker. Copyright 2010 Pearson Education, Inc. Lecture Outline Chapter 10 Physics, 4 th Edition James S. Walker Chapter 10 Rotational Kinematics and Energy Units of Chapter 10 Angular Position, Velocity, and Acceleration Rotational Kinematics Connections

More information

General Physics I. Lecture 8: Rotation of a Rigid Object About a Fixed Axis. Prof. WAN, Xin ( 万歆 )

General Physics I. Lecture 8: Rotation of a Rigid Object About a Fixed Axis. Prof. WAN, Xin ( 万歆 ) General Physics I Lecture 8: Rotation of a Rigid Object About a Fixed Axis Prof. WAN, Xin ( 万歆 ) xinwan@zju.edu.cn http://zimp.zju.edu.cn/~xinwan/ New Territory Object In the past, point particle (no rotation,

More information

Chapter 8. Rotational Equilibrium and Rotational Dynamics. 1. Torque. 2. Torque and Equilibrium. 3. Center of Mass and Center of Gravity

Chapter 8. Rotational Equilibrium and Rotational Dynamics. 1. Torque. 2. Torque and Equilibrium. 3. Center of Mass and Center of Gravity Chapter 8 Rotational Equilibrium and Rotational Dynamics 1. Torque 2. Torque and Equilibrium 3. Center of Mass and Center of Gravity 4. Torque and angular acceleration 5. Rotational Kinetic energy 6. Angular

More information

Chapter 8 Lecture. Pearson Physics. Rotational Motion and Equilibrium. Prepared by Chris Chiaverina Pearson Education, Inc.

Chapter 8 Lecture. Pearson Physics. Rotational Motion and Equilibrium. Prepared by Chris Chiaverina Pearson Education, Inc. Chapter 8 Lecture Pearson Physics Rotational Motion and Equilibrium Prepared by Chris Chiaverina Chapter Contents Describing Angular Motion Rolling Motion and the Moment of Inertia Torque Static Equilibrium

More information

Chapter 11. Angular Momentum

Chapter 11. Angular Momentum Chapter 11 Angular Momentum C H A P T E R O U T L I N E 11 1 The Vector Product and Torque 11 2 Angular Momentum 11 3 Angular Momentum of a Rotating Rigid Object 11 4 Conservation of Angular Momentum 11

More information

Chapter 10 Practice Test

Chapter 10 Practice Test Chapter 10 Practice Test 1. At t = 0, a wheel rotating about a fixed axis at a constant angular acceleration of 0.40 rad/s 2 has an angular velocity of 1.5 rad/s and an angular position of 2.3 rad. What

More information

CHAPTER 10 ROTATION OF A RIGID OBJECT ABOUT A FIXED AXIS WEN-BIN JIAN ( 簡紋濱 ) DEPARTMENT OF ELECTROPHYSICS NATIONAL CHIAO TUNG UNIVERSITY

CHAPTER 10 ROTATION OF A RIGID OBJECT ABOUT A FIXED AXIS WEN-BIN JIAN ( 簡紋濱 ) DEPARTMENT OF ELECTROPHYSICS NATIONAL CHIAO TUNG UNIVERSITY CHAPTER 10 ROTATION OF A RIGID OBJECT ABOUT A FIXED AXIS WEN-BIN JIAN ( 簡紋濱 ) DEPARTMENT OF ELECTROPHYSICS NATIONAL CHIAO TUNG UNIVERSITY OUTLINE 1. Angular Position, Velocity, and Acceleration 2. Rotational

More information

Name Student ID Score Last First. I = 2mR 2 /5 around the sphere s center of mass?

Name Student ID Score Last First. I = 2mR 2 /5 around the sphere s center of mass? NOTE: ignore air resistance in all Questions. In all Questions choose the answer that is the closest!! Question I. (15 pts) Rotation 1. (5 pts) A bowling ball that has an 11 cm radius and a 7.2 kg mass

More information

Dynamics of Rotational Motion

Dynamics of Rotational Motion Chapter 10 Dynamics of Rotational Motion To understand the concept of torque. To relate angular acceleration and torque. To work and power in rotational motion. To understand angular momentum. To understand

More information

Physics A - PHY 2048C

Physics A - PHY 2048C Physics A - PHY 2048C and 11/15/2017 My Office Hours: Thursday 2:00-3:00 PM 212 Keen Building Warm-up Questions 1 Did you read Chapter 12 in the textbook on? 2 Must an object be rotating to have a moment

More information

Chapter 8. Rotational Equilibrium and Rotational Dynamics

Chapter 8. Rotational Equilibrium and Rotational Dynamics Chapter 8 Rotational Equilibrium and Rotational Dynamics Force vs. Torque Forces cause accelerations Torques cause angular accelerations Force and torque are related Torque The door is free to rotate about

More information

DYNAMICS OF RIGID BODIES

DYNAMICS OF RIGID BODIES DYNAMICS OF RIGID BODIES Measuring angles in radian Define the value of an angle θ in radian as θ = s r, or arc length s = rθ a pure number, without dimension independent of radius r of the circle one

More information

Rotational Dynamics. Slide 2 / 34. Slide 1 / 34. Slide 4 / 34. Slide 3 / 34. Slide 6 / 34. Slide 5 / 34. Moment of Inertia. Parallel Axis Theorem

Rotational Dynamics. Slide 2 / 34. Slide 1 / 34. Slide 4 / 34. Slide 3 / 34. Slide 6 / 34. Slide 5 / 34. Moment of Inertia. Parallel Axis Theorem Slide 1 / 34 Rotational ynamics l Slide 2 / 34 Moment of Inertia To determine the moment of inertia we divide the object into tiny masses of m i a distance r i from the center. is the sum of all the tiny

More information

Handout 6: Rotational motion and moment of inertia. Angular velocity and angular acceleration

Handout 6: Rotational motion and moment of inertia. Angular velocity and angular acceleration 1 Handout 6: Rotational motion and moment of inertia Angular velocity and angular acceleration In Figure 1, a particle b is rotating about an axis along a circular path with radius r. The radius sweeps

More information

Chapter 8: Momentum, Impulse, & Collisions. Newton s second law in terms of momentum:

Chapter 8: Momentum, Impulse, & Collisions. Newton s second law in terms of momentum: linear momentum: Chapter 8: Momentum, Impulse, & Collisions Newton s second law in terms of momentum: impulse: Under what SPECIFIC condition is linear momentum conserved? (The answer does not involve collisions.)

More information

Mechanics 5 Dynamics of a rigid body. Basic phenomena

Mechanics 5 Dynamics of a rigid body. Basic phenomena Mechanics 5 Dynamics of a rigid body Torque Moment of Inertia Newton s laws for a rigid body Angular momentum Conservation law Basic phenomena In an empty space with no external forces acting on the body,

More information

Work and kinetic Energy

Work and kinetic Energy Work and kinetic Energy Problem 66. M=4.5kg r = 0.05m I = 0.003kgm 2 Q: What is the velocity of mass m after it dropped a distance h? (No friction) h m=0.6kg mg Work and kinetic Energy Problem 66. M=4.5kg

More information

Moment of Inertia Race

Moment of Inertia Race Review Two points, A and B, are on a disk that rotates with a uniform speed about an axis. Point A is closer to the axis than point B. Which of the following is NOT true? 1. Point B has the greater tangential

More information

PLANAR KINETICS OF A RIGID BODY FORCE AND ACCELERATION

PLANAR KINETICS OF A RIGID BODY FORCE AND ACCELERATION PLANAR KINETICS OF A RIGID BODY FORCE AND ACCELERATION I. Moment of Inertia: Since a body has a definite size and shape, an applied nonconcurrent force system may cause the body to both translate and rotate.

More information

Chapter 11 Rolling, Torque, and Angular Momentum

Chapter 11 Rolling, Torque, and Angular Momentum Prof. Dr. I. Nasser Chapter11-I November, 017 Chapter 11 Rolling, Torque, and Angular Momentum 11-1 ROLLING AS TRANSLATION AND ROTATION COMBINED Translation vs. Rotation General Rolling Motion General

More information