Tenth E CHAPTER 11 VECTOR MECHANICS FOR ENGINEERS: DYNAMICS Ferdinand P. Beer E. Russell Johnston, Jr. Phillip J. Cornwell Lecture Notes: Brian P. Self California Polytechnic State Uniersity Kinematics of Particles 13 The McGraw-Hill Companies, Inc. All rights resered.
Contents Introduction Rectilinear Motion: Position, Velocity & Acceleration Determination of the Motion of a Particle Sample Problem 11. Sample Problem 11.3 Uniform Rectilinear-Motion Uniformly Accelerated Rectilinear- Motion Motion of Seeral Particles: Relatie Motion Sample Problem 11.4 Motion of Seeral Particles: Dependent Motion Sample Problem 11.5 Graphical Solution of Rectilinear- Motion Problems Other Graphical Methods Curilinear Motion: Position, Velocity & Acceleration Deriaties of Vector Functions Rectangular Components of Velocity and Acceleration Motion Relatie to a Frame in Translation Tangential and Normal Components Radial and Transerse Components Sample Problem 11.1 Sample Problem 11.1 13 The McGraw-Hill Companies, Inc. All rights resered. 11 -
Kinematic relationships are used to help us determine the trajectory of a golf ball, the orbital speed of a satellite, and the accelerations during acrobatic flying. 13 The McGraw-Hill Companies, Inc. All rights resered. 11-3
Introduction Dynamics includes: Kinematics: study of the geometry of motion. Relates displacement, elocity, acceleration, and time without reference to the cause of motion. F thrust F drag F lift Kinetics: study of the relations existing between the forces acting on a body, the mass of the body, and the motion of the body. Kinetics is used to predict the motion caused by gien forces or to determine the forces required to produce a gien motion. 13 The McGraw-Hill Companies, Inc. All rights resered. 11-4
Introduction Particle kinetics includes: Rectilinear motion: position, elocity, and acceleration of a particle as it moes along a straight line. Curilinear motion: position, elocity, and acceleration of a particle as it moes along a cured line in two or three dimensions. 13 The McGraw-Hill Companies, Inc. All rights resered. 11-5
Rectilinear Motion: Position, Velocity & Acceleration Rectilinear motion: particle moing along a straight line Position coordinate: defined by positie or negatie distance from a fixed origin on the line. The motion of a particle is known if the position coordinate for particle is known for eery alue of time t. May be expressed in the form of a function, e.g., 3 6t t or in the form of a graph x s. t. x 13 The McGraw-Hill Companies, Inc. All rights resered. 11-6
Rectilinear Motion: Position, Velocity & Acceleration 13 The McGraw-Hill Companies, Inc. All rights resered. Consider particle which occupies position P at time t and P at t+dt, Dx Aerage elocity Dt Instantaneous elocity Dx Dt Instantaneous elocity may be positie or negatie. Magnitude of elocity is referred to as particle speed. lim Dt From the definition of a deriatie, Dx dx lim Dt Dt dt e.g., x 6t dx dt t 3 1t 3t 11-7
Rectilinear Motion: Position, Velocity & Acceleration Consider particle with elocity at time t and at t+dt, D Instantaneous acceleration a lim Dt Dt Instantaneous acceleration may be: - positie: increasing positie elocity or decreasing negatie elocity - negatie: decreasing positie elocity or increasing negatie elocity. 13 The McGraw-Hill Companies, Inc. All rights resered. From the definition of a deriatie, D d d x a lim Dt Dt dt dt e.g. 1t 3t a d dt 1 6t 11-8
Concept Quiz What is true about the kinematics of a particle? a) The elocity of a particle is always positie b) The elocity of a particle is equal to the slope of the position-time graph c) If the position of a particle is zero, then the elocity must zero d) If the elocity of a particle is zero, then its acceleration must be zero 13 The McGraw-Hill Companies, Inc. All rights resered. 11-9
Rectilinear Motion: Position, Velocity & Acceleration From our example, x 6t t dx dt 3 1t 3t a d d x 1 t dt dt 6 What are x,, and a at t = s? - at t = s, x = 16 m, = max = 1 m/s, a = Note that max occurs when a=, and that the slope of the elocity cure is zero at this point. What are x,, and a at t = 4 s? - at t = 4 s, x = x max = 3 m, =, a = -1 m/s 13 The McGraw-Hill Companies, Inc. All rights resered. 11-1
Determination of the Motion of a Particle We often determine accelerations from the forces applied (kinetics will be coered later) Generally hae three classes of motion - acceleration gien as a function of time, a = f(t) - acceleration gien as a function of position, a = f(x) - acceleration gien as a function of elocity, a = f() Can you think of a physical example of when force is a function of position? When force is a function of elocity? a spring 13 The McGraw-Hill Companies, Inc. All rights resered. drag 11-11
Acceleration as a function of time, position, or elocity If. Kinematic relationship Integrate a a t d () dt at d t a t dt a a x dt dx d and a dt d a x dx d x x a x dx a a d () dt a d a dx 13 The McGraw-Hill Companies, Inc. All rights resered. d a x x dx t dt d a 11-1
Sample Problem 11. SOLUTION: Integrate twice to find (t) and y(t). Sole for t when elocity equals zero (time for maximum eleation) and ealuate corresponding altitude. Ball tossed with 1 m/s ertical elocity from window m aboe ground. Sole for t when altitude equals zero (time for ground impact) and ealuate corresponding elocity. Determine: elocity and eleation aboe ground at time t, highest eleation reached by ball and corresponding time, and time when ball will hit the ground and corresponding elocity. 13 The McGraw-Hill Companies, Inc. All rights resered. 11-13
Sample Problem 11. SOLUTION: Integrate twice to find (t) and y(t). d a 9.81m dt t t d 9.81dt dy 1 9.81t dt yt t dy 1 9.81t y s t t 9.81t m 1 s m 9.81 t s 1 dt y t y 1t 9.81t y m s t m 1 t 4.95 t m s 13 The McGraw-Hill Companies, Inc. All rights resered. 11-14
Sample Problem 11. Sole for t when elocity equals zero and ealuate corresponding altitude. m s m s t 1 9.81 t t 1.19 s Sole for t when altitude equals zero and ealuate corresponding elocity. y t y m 1 m 1 m s m s t m 4.95 t s m s 1.19s 4.95 1.19s y 5.1m 13 The McGraw-Hill Companies, Inc. All rights resered. 11-15
Sample Problem 11. Sole for t when altitude equals zero and ealuate corresponding elocity. y m s m s t m 1 t 4.95 t t m 1 s m s m 9.81 t s t 1.43s t 3.8s m s 3.8s 1 9.81 3.8s m. s meaningless 13 The McGraw-Hill Companies, Inc. All rights resered. 11-16
Sample Problem 11.3 SOLUTION: a k Integrate a = d/dt = -k to find (t). Integrate (t) = dx/dt to find x(t). Brake mechanism used to reduce gun recoil consists of piston attached to barrel moing in fixed cylinder filled with oil. As barrel recoils with initial elocity, piston moes and oil is forced through orifices in piston, causing piston and cylinder to decelerate at rate proportional to their elocity. Integrate a = d/dx = -k to find (x). Determine (t), x(t), and (x). 13 The McGraw-Hill Companies, Inc. All rights resered. 11-17
Sample Problem 11.3 SOLUTION: Integrate a = d/dt = -k to find (t). t d d t a k k dt ln kt dt t e kt Integrate (t) = dx/dt to find x(t). t dx e dt kt x t kt 1 dx e dt x t e k kt x t k kt t 1 e 13 The McGraw-Hill Companies, Inc. All rights resered. 11-18
Sample Problem 11.3 Integrate a = d/dx = -k to find (x). a d dx k kx d k dx d k x dx kx Alternatiely, with and then x x t t k kt t 1 e e or kt 1 k e t kt t kx 13 The McGraw-Hill Companies, Inc. All rights resered. 11-19
Group Problem Soling +y A bowling ball is dropped from a boat so that it strikes the surface of a lake with a speed of 5 m/s. Assuming the ball experiences a downward acceleration of a =1 -.1 when in the water, determine the elocity of the ball when it strikes the bottom of the lake. Which integral should you choose? (a) t d a t dt (c) d a x dx x x (b) x x dx d a (d) d a t dt 13 The McGraw-Hill Companies, Inc. All rights resered. 11 -
Concept Question When will the bowling ball start slowing down? +y A bowling ball is dropped from a boat so that it strikes the surface of a lake with a speed of 5 m/s. Assuming the ball experiences a downward acceleration of a =1 -.1 when in the water, determine the elocity of the ball when it strikes the bottom of the lake. The elocity would hae to be high enough for the.1 term to be bigger than 1 13 The McGraw-Hill Companies, Inc. All rights resered. 11-1
Group Problem Soling SOLUTION: Determine the proper kinematic relationship to apply (is acceleration a function of time, elocity, or position? The car starts from rest and accelerates according to the relationship a 3.1 Determine the total distance the car traels in one-half lap Integrate to determine the elocity after one-half lap It traels around a circular track that has a radius of meters. Calculate the elocity of the car after it has traelled halfway around the track. What is the car s maximum possible speed? 13 The McGraw-Hill Companies, Inc. All rights resered. 11 -
Group Problem Soling Gien: a 3.1 o =, r = m Find: after ½ lap Maximum speed Choose the proper kinematic relationship Acceleration is a function of elocity, and we also can determine distance. Time is not inoled in the problem, so we choose: d dx a x x dx d a Determine total distance traelled x r 3.14() 68.3 m 13 The McGraw-Hill Companies, Inc. All rights resered. 11-3
Group Problem Soling Determine the full integral, including limits x x dx d a 68.3 dx 3.1 d Ealuate the interal and sole for 1 68.3 ln 3.1. 68.3(.) ln 3.1 ln 3.1() ln 3.1 1.566 1.986=.158 Take the exponential of each side 3.1 e.158 13 The McGraw-Hill Companies, Inc. All rights resered. 11-4
Group Problem Soling Sole for 3.1 e.158.158 e 146. 46.368 m/s.1 3 How do you determine the maximum speed the car can reach? Velocity is a maximum when acceleration is zero a 3.1 This occurs when.1 3 max 3.1 max 54.77 m/s 13 The McGraw-Hill Companies, Inc. All rights resered. 11-5
Uniform Rectilinear Motion During free-fall, a parachutist reaches terminal elocity when her weight equals the drag force. If motion is in a straight line, this is uniform rectilinear motion. For a particle in uniform rectilinear motion, the acceleration is zero and the elocity is constant. dx dt x x x x dx x x t dt t t constant Careful these only apply to uniform rectilinear motion! 13 The McGraw-Hill Companies, Inc. All rights resered. 11-6
Uniformly Accelerated Rectilinear Motion If forces applied to a body are constant (and in a constant direction), then you hae uniformly accelerated rectilinear motion. Another example is freefall when drag is negligible 13 The McGraw-Hill Companies, Inc. All rights resered. 11-7
Uniformly Accelerated Rectilinear Motion For a particle in uniformly accelerated rectilinear motion, the acceleration of the particle is constant. You may recognize these constant acceleration equations from your physics courses. t d constant dt a d a dt at x t 1 x dx at dx at dt x x t at dt d a constant d a dx a x x dx x x Careful these only apply to uniformly accelerated rectilinear motion! 13 The McGraw-Hill Companies, Inc. All rights resered. 11-8
Motion of Seeral Particles We may be interested in the motion of seeral different particles, whose motion may be independent or linked together. 13 The McGraw-Hill Companies, Inc. All rights resered. 11-9
Motion of Seeral Particles: Relatie Motion For particles moing along the same line, time should be recorded from the same starting instant and displacements should be measured from the same origin in the same direction. x x B B A x A x B x x B A A relatie position of B with respect to A B B A A B B A A relatie elocity of B with respect to A a a B B A a a A B a a B A A relatie acceleration of B with respect to A 13 The McGraw-Hill Companies, Inc. All rights resered. 11-3
Sample Problem 11.4 SOLUTION: Substitute initial position and elocity and constant acceleration of ball into general equations for uniformly accelerated rectilinear motion. Ball thrown ertically from 1 m leel in eleator shaft with initial elocity of 18 m/s. At same instant, open-platform eleator passes 5 m leel moing upward at m/s. Determine (a) when and where ball hits eleator and (b) relatie elocity of ball and eleator at contact. 13 The McGraw-Hill Companies, Inc. All rights resered. Substitute initial position and constant elocity of eleator into equation for uniform rectilinear motion. Write equation for relatie position of ball with respect to eleator and sole for zero relatie position, i.e., impact. Substitute impact time into equation for position of eleator and relatie elocity of ball with respect to eleator. 11-31
Sample Problem 11.4 SOLUTION: Substitute initial position and elocity and constant acceleration of ball into general equations for uniformly accelerated rectilinear motion. y B B y at t m 18 s 1 at m 9.81 t s 1m 18 m s m t 4.95 t s Substitute initial position and constant elocity of eleator into equation for uniform rectilinear motion. y E E m s y E t 5m m s t 13 The McGraw-Hill Companies, Inc. All rights resered. 11-3
Sample Problem 11.4 Write equation for relatie position of ball with respect to eleator and sole for zero relatie position, i.e., impact. y 1 18t 4.95t 5 t B E t.39s meaningless t 3.65s Substitute impact time into equations for position of eleator and relatie elocity of ball with respect to eleator. y E 5 3.65 y E 1.3m B E 18 9.81t 16 9.81 3.65 B E m 19.81 s 13 The McGraw-Hill Companies, Inc. All rights resered. 11-33
Motion of Seeral Particles: Dependent Motion Position of a particle may depend on position of one or more other particles. Position of block B depends on position of block A. Since rope is of constant length, it follows that sum of lengths of segments must be constant. x x constant (one degree of freedom) A B Positions of three blocks are dependent. x x x constant (two degrees of freedom) A B C For linearly related positions, similar relations hold between elocities and accelerations. dx dt d dt A A dx dt d dt B B dx dt d dt C C or or a A A a B B C a C 13 The McGraw-Hill Companies, Inc. All rights resered. 11-34
Sample Problem 11.5 SOLUTION: Define origin at upper horizontal surface with positie displacement downward. Collar A has uniformly accelerated rectilinear motion. Sole for acceleration and time t to reach L. Pulley D is attached to a collar which is pulled down at 75 mm/s. At t =, collar A starts moing down from K with constant acceleration and zero initial elocity. Knowing that elocity of collar A is 3 mm/s as it passes L, determine the change in eleation, elocity, and acceleration of block B when block A is at L. Pulley D has uniform rectilinear motion. Calculate change of position at time t. Block B motion is dependent on motions of collar A and pulley D. Write motion relationship and sole for change of block B position at time t. Differentiate motion relation twice to deelop equations for elocity and acceleration of block B. 13 The McGraw-Hill Companies, Inc. All rights resered. 11-35
Sample Problem 11.5 SOLUTION: Define origin at upper horizontal surface with positie displacement downward. Collar A has uniformly accelerated rectilinear motion. Sole for acceleration and time t to reach L. A A A A A a x x mm mm 3 aa mm aa 5 s s ( ) = + a t A A A mm mm 3 = 5 t t = 1.333 s s s 13 The McGraw-Hill Companies, Inc. All rights resered. 11-36
Sample Problem 11.5 Pulley D has uniform rectilinear motion. Calculate change of position at time t. x D = x D x D - x D ( ) + D t æ ( ) = ç 75 mm è s ö 1.333s ø ( ) =1 mm Block B motion is dependent on motions of collar A and pulley D. Write motion relationship and sole for change of block B position at time t. Total length of cable remains constant, x x x x x x A D B A D B x x x x x x A A D D B B x x mm 1mm B B x B ( x ) 4mm - = - B 13 The McGraw-Hill Companies, Inc. All rights resered. 11-37
Sample Problem 11.5 Differentiate motion relation twice to deelop equations for elocity and acceleration of block B. x x x constant A D B A D B mm mm 3 75 B s s a a a A D B mm mm B = - 45 = 45 s s mm 5 () a B s mm mm a B = - 5 = 5 s s 13 The McGraw-Hill Companies, Inc. All rights resered. 11-38
Group Problem Soling Slider block A moes to the left with a constant elocity of 6 m/s. Determine the elocity of block B. Solution steps Sketch your system and choose coordinate system Write out constraint equation Differentiate the constraint equation to get elocity 13 The McGraw-Hill Companies, Inc. All rights resered. 11-39
Group Problem Soling Gien: A = 6 m/s left Find: B x A This length is constant no matter how the blocks moe Sketch your system and choose coordinates y B Define your constraint equation(s) x 3y constants L A B Differentiate the constraint equation to get elocity 6 m/s + 3 B B m/s 13 The McGraw-Hill Companies, Inc. All rights resered. Note that as x A gets bigger, y B gets smaller. 11-4
Graphical Solution of Rectilinear-Motion Problems Engineers often collect position, elocity, and acceleration data. Graphical solutions are often useful in analyzing these data. Data Fideltity / Highest Recorded Punch 18 Acceleration (g) 16 14 1 1 8 6 4 47.76 47.77 47.78 47.79 47.8 47.81 Time (s) Acceleration data from a head impact during a round of boxing. 13 The McGraw-Hill Companies, Inc. All rights resered. 11-41
Graphical Solution of Rectilinear-Motion Problems Gien the x-t cure, the -t cure is equal to the x-t cure slope. Gien the -t cure, the a-t cure is equal to the -t cure slope. 13 The McGraw-Hill Companies, Inc. All rights resered. 11-4
Graphical Solution of Rectilinear-Motion Problems Gien the a-t cure, the change in elocity between t 1 and t is equal to the area under the a-t cure between t 1 and t. Gien the -t cure, the change in position between t 1 and t is equal to the area under the -t cure between t 1 and t. 13 The McGraw-Hill Companies, Inc. All rights resered. 11-43
Other Graphical Methods Moment-area method to determine particle position at time t directly from the a-t cure: x1 x area under t cure t 1 using d = a dt, x 1 1 x t x t 1 t ta dt 1 1 x abscissa t 1 of 1 1 t t 1 1 t t d a dt first moment of area under a-t cure with respect to t = t 1 line. under a-t cure t t area 1 centroid C 13 The McGraw-Hill Companies, Inc. All rights resered. 11-44
Other Graphical Methods Method to determine particle acceleration from -x cure: a d dx AB tan BC subnormal to -x cure 13 The McGraw-Hill Companies, Inc. All rights resered. 11-45
Curilinear Motion: Position, Velocity & Acceleration The softball and the car both undergo curilinear motion. A particle moing along a cure other than a straight line is in curilinear motion. 13 The McGraw-Hill Companies, Inc. All rights resered. 11-46
Curilinear Motion: Position, Velocity & Acceleration The position ector of a particle at time t is defined by a ector between origin O of a fixed reference frame and the position occupied by particle. Consider a particle which occupies position P defined by and P defined by at t + Dt, r r at time t 13 The McGraw-Hill Companies, Inc. All rights resered. 11-47
Curilinear Motion: Position, Velocity & Acceleration Instantaneous elocity (ector) Dr dr lim D t Dt dt Instantaneous speed (scalar) Ds ds lim D t Dt dt 13 The McGraw-Hill Companies, Inc. All rights resered. 11-48
Curilinear Motion: Position, Velocity & Acceleration Consider elocity of a particle at time t and elocity at t + Dt, a D d lim D t Dt dt instantaneous acceleration (ector) In general, the acceleration ector is not tangent to the particle path and elocity ector. 13 The McGraw-Hill Companies, Inc. All rights resered. 11-49
Deriaties of Vector Functions Let P u be a ector function of scalar ariable u, dp DP Pu Du Pu lim lim du Du Du Du Du Deriatie of ector sum, dp Q dp dq du du du Delete or put in bonus slides Deriatie of product of scalar and ector functions, d f P df dp P f du du du Deriatie of scalar product and ector product, dp Q dp dq Q P du du du dp Q dp dq Q P du du du 13 The McGraw-Hill Companies, Inc. All rights resered. 11-5
Rectangular Components of Velocity & Acceleration When position ector of particle P is gien by its rectangular components, r xi y j zk Velocity ector, dx dy dz i j k dt dt dt i j k x y z xi y j zk Acceleration ector, d x d y d a i j dt dt dt a i a j a k x y z z k xi y j zk 13 The McGraw-Hill Companies, Inc. All rights resered. 11-51
Rectangular Components of Velocity & Acceleration Rectangular components particularly effectie when component accelerations can be integrated independently, e.g., motion of a projectile, a x a y g a z x with initial cons, x y z y,, x y z Integrating twice yields x x x y y gt z t y y 1 gt z x Motion in horizontal direction is uniform. y Motion in ertical direction is uniformly accelerated. Motion of projectile could be replaced by two independent rectilinear motions. z 13 The McGraw-Hill Companies, Inc. All rights resered. 11-5
Sample Problem 11.7 SOLUTION: Consider the ertical and horizontal motion separately (they are independent) Apply equations of motion in y-direction Apply equations of motion in x-direction A projectile is fired from the edge of a 15-m cliff with an initial elocity of 18 m/s at an angle of 3 with the horizontal. Neglecting air resistance, find (a) the horizontal distance from the gun to the point where the projectile strikes the ground, (b) the greatest eleation aboe the ground reached by the projectile. 13 The McGraw-Hill Companies, Inc. All rights resered. Determine time t for projectile to hit the ground, use this to find the horizontal distance Maximum eleation occurs when y = 11-53
Sample Problem 11.7 SOLUTION: Gien: () o =18 m/s (y) o =15 m (a) y = - 9.81 m/s (a) x = m/s Vertical motion uniformly accelerated: Horizontal motion uniformly accelerated: Choose positie x to the right as shown 13 The McGraw-Hill Companies, Inc. All rights resered. 11-54
Sample Problem 11.7 SOLUTION: Horizontal distance Projectile strikes the ground at: Substitute into equation (1) aboe Soling for t, we take the positie root Substitute t into equation (4) Maximum eleation occurs when y = Maximum eleation aboe the ground = 13 The McGraw-Hill Companies, Inc. All rights resered. 11-55
Concept Quiz If you fire a projectile from 15 meters aboe the ground (see Ex Problem 11.7), what launch angle will gie you the greatest horizontal distance x? a) A launch angle of 45 b) A launch angle less than 45 c) A launch angle greater than 45 d) It depends on the launch elocity 13 The McGraw-Hill Companies, Inc. All rights resered. 11-56
Group Problem Soling SOLUTION: A baseball pitching machine throws baseballs with a horizontal elocity. If you want the height h to be 1 m, determine the alue of. Consider the ertical and horizontal motion separately (they are independent) Apply equations of motion in y-direction Apply equations of motion in x-direction Determine time t for projectile to fall to 1 m. Calculate = 13 The McGraw-Hill Companies, Inc. All rights resered. 11-57
Group Problem Soling Gien: x= 1 m, y o = 1.5 m, y f = 1 m. Find: o Analyze the motion in the y-direction 1 y f y () t gt 1 3.5 5 gt 1.5 m (9.81 m/s ) t Analyze the motion in the x-direction x ( x ) t t 1 m ( )(.3193 s) 37.6 m/s 135.3 km/h t.3193 s 13 The McGraw-Hill Companies, Inc. All rights resered. 11-58
Motion Relatie to a Frame in Translation A soccer player must consider the relatie motion of the ball and her teammates when making a pass. It is critical for a pilot to know the relatie motion of his aircraft with respect to the aircraft carrier to make a safe landing. 13 The McGraw-Hill Companies, Inc. All rights resered. 11-59
Motion Relatie to a Frame in Translation Designate one frame as the fixed frame of reference. All other frames not rigidly attached to the fixed reference frame are moing frames of reference. Position ectors for particles A and B with respect to the fixed frame of reference Oxyz are r A and r B. Vector r B A joining A and B defines the position of B with respect to the moing frame Ax y z and r r r B A B A Differentiating twice, elocity of B relatie to A. a B B a A A a B B A A B a B A A acceleration of B relatie to A. Absolute motion of B can be obtained by combining motion of A with relatie motion of B with respect to moing reference frame attached to A. 13 The McGraw-Hill Companies, Inc. All rights resered. 11-6
Sample Problem 11.9 Automobile A is traeling east at the constant speed of 36 km/h. As automobile A crosses the intersection shown, automobile B starts from rest 35 m north of the intersection and moes south with a constant acceleration of 1. m/s. Determine the position, elocity, and acceleration of B relatie to A 5 s after A crosses the intersection. 13 The McGraw-Hill Companies, Inc. All rights resered. SOLUTION: Define inertial axes for the system Determine the position, speed, and acceleration of car A at t = 5 s Determine the position, speed, and acceleration of car B at t = 5 s Using ectors (Eqs 11.31, 11.33, and 11.34) or a graphical approach, determine the relatie position, elocity, and acceleration 11-61
Sample Problem 11.9 SOLUTION: Define axes along the road Gien: A =36 km/h, a A =, (x A ) = ( B ) =, a B = - 1. m/s, (y A ) = 35 m Determine motion of Automobile A: We hae uniform motion for A so: At t = 5 s 13 The McGraw-Hill Companies, Inc. All rights resered. 11-6
Sample Problem 11.9 SOLUTION: Determine motion of Automobile B: We hae uniform acceleration for B so: At t = 5 s 13 The McGraw-Hill Companies, Inc. All rights resered. 11-63
Sample Problem 11.9 SOLUTION: We can sole the problems geometrically, and apply the arctangent relationship: Or we can sole the problems using ectors to obtain equialent results: rb ra rb/a a a a j 5i r B/A r j5 i (m) B/A B A B/A 6j 1i B/A 6j1 i (m/s) B/A B A B/A 1.j i a B/ A B/ A 1. j (m/s ) Physically, a rider in car A would see car B traelling south and west. a 13 The McGraw-Hill Companies, Inc. All rights resered. 11-64
Concept Quiz If you are sitting in train B looking out the window, it which direction does it appear that train A is moing? a) 5 o c) b) 5 o d) 13 The McGraw-Hill Companies, Inc. All rights resered. 11-65
Tangential and Normal Components If we hae an idea of the path of a ehicle, it is often conenient to analyze the motion using tangential and normal components (sometimes called path coordinates). 13 The McGraw-Hill Companies, Inc. All rights resered. 11-66
Tangential and Normal Components y r= the instantaneous radius of curature e t e n e t = t e t d a e t e dt r n The tangential direction (e t ) is tangent to the path of the particle. This elocity ector of a particle is in this direction The normal direction (e n ) is perpendicular to e t and points towards the inside of the cure. The acceleration can hae components in both the e n and e t directions 13 The McGraw-Hill Companies, Inc. All rights resered. x 11-67
Tangential and Normal Components To derie the acceleration ector in tangential and normal components, define the motion of a particle as shown in the figure. e t and e t are tangential unit ectors for the particle path at P and P. When drawn with respect to the same origin, Det et et and D is the angle between them. De t sin De lim t D de e t n d D lim sin D D D D e n e n 13 The McGraw-Hill Companies, Inc. All rights resered. 11-68
Tangential and Normal Components With the elocity ector expressed as e t the particle acceleration may be written as d d de d de d ds a et et dt dt dt dt d ds dt but det ds en r d ds d dt After substituting, d a et e dt r n a t d dt a n r The tangential component of acceleration reflects change of speed and the normal component reflects change of direction. The tangential component may be positie or negatie. Normal component always points toward center of path curature. 13 The McGraw-Hill Companies, Inc. All rights resered. 11-69
Tangential and Normal Components Relations for tangential and normal acceleration also apply for particle moing along a space cure. a d et dt e r n a t d dt a n r The plane containing tangential and normal unit ectors is called the osculating plane. The normal to the osculating plane is found from eb et en e principal normal e n b binormal Acceleration has no component along the binormal. 13 The McGraw-Hill Companies, Inc. All rights resered. 11-7
Sample Problem 11.1 SOLUTION: Define your coordinate system Calculate the tangential elocity and tangential acceleration Calculate the normal acceleration A motorist is traeling on a cured section of highway of radius 75 m at the speed of 9 km/h. The motorist suddenly applies the brakes, causing the automobile to slow down at a constant rate. Knowing that after 8 s the speed has been reduced to 7 km/h, determine the acceleration of the automobile immediately after the brakes hae been applied. 13 The McGraw-Hill Companies, Inc. All rights resered. Determine oerall acceleration magnitude after the brakes hae been applied 11-71
Sample Problem 11.8 e t e n SOLUTION: 13 The McGraw-Hill Companies, Inc. All rights resered. km 1 m 1 h 9 km/h = 9 5 m/s h 1 km 36 s 7 km/h = m/s a t Define your coordinate system Determine elocity and acceleration in the tangential direction The deceleration constant, therefore D m/s - 5 m/s = aerage at= = = - Dt 8s (5 m/s) an = = =.833 m/s r 75 m an.833 m/s a a (.65) (.833) tan a = = n at a.65 m/s a =1.41 m/s a = 53.1 t.65 m/s Immediately after the brakes are applied, the speed is still 5 m/s 11-7
Tangential and Normal Components In 1, a race scheduled at the Texas Motor Speedway was cancelled because the normal accelerations were too high and caused some driers to experience excessie g-loads (similar to fighter pilots) and possibly pass out. What are some things that could be done to sole this problem? Some possibilities: Reduce the allowed speed Increase the turn radius (difficult and costly) Hae the racers wear g-suits 13 The McGraw-Hill Companies, Inc. All rights resered. 11-73
Group Problem Soling SOLUTION: Define your coordinate system Calculate the tangential elocity and tangential acceleration Calculate the normal acceleration The tangential acceleration of the centrifuge cab is gien by at.5 t (m/s ) where t is in seconds and a t is in m/s. If the centrifuge starts from fest, determine the total acceleration magnitude of the cab after 1 seconds. Determine oerall acceleration magnitude 13 The McGraw-Hill Companies, Inc. All rights resered. 11-74
Group Problem Soling Define your coordinate system In the side iew, the tangential direction points into the page Determine the tangential elocity at.5t t t.5t dt.5t.5t t Top View e n t.5 1 5 m/s Determine the normal acceleration 5 a n t 78.15 m/s r 8 Determine the total acceleration magnitude e t e n a a a 78.15 + (.5)(1) mag n t 13 The McGraw-Hill Companies, Inc. All rights resered. amag 78.85 m/s 11-75
Group Problem Soling Notice that the normal acceleration is much higher than the tangential acceleration. What would happen if, for a gien tangential elocity and acceleration, the arm radius was doubled? a) The accelerations would remain the same b) The a n would increase and the a t would decrease c) The a n and a t would both increase d) The a n would decrease and the a t would increase 13 The McGraw-Hill Companies, Inc. All rights resered. 11-76
Radial and Transerse Components By knowing the distance to the aircraft and the angle of the radar, air traffic controllers can track aircraft. Fire truck ladders can rotate as well as extend; the motion of the end of the ladder can be analyzed using radial and transerse components. 13 The McGraw-Hill Companies, Inc. All rights resered. 11-77
Radial and Transerse Components The position of a particle P is expressed as a distance r from the origin O to P this defines the radial direction e r. The transerse direction e is perpendicular to e r r re r The particle elocity ector is = re r + rqe q The particle acceleration ector is a = ( r - rq )e r + ( rq + rq )e q 13 The McGraw-Hill Companies, Inc. All rights resered. 11-78
Radial and Transerse Components We can derie the elocity and acceleration relationships by recognizing that the unit ectors change direction. r re r de d de dt de dt de d r e e r r der d d e d dt dt de d d er d dt dt The particle elocity ector is d dr de dr re e r r r r e dt dt dt dt r e r e r r d r e dt Similarly, the particle acceleration ector is a d dt dr dt d r e r dt d er r e dt dr de r dt dt r r e r r e r dr dt d e dt d r e dt r d de dt dt 13 The McGraw-Hill Companies, Inc. All rights resered. 11-79
Concept Quiz If you are traelling in a perfect circle, what is always true about radial/transerse coordinates and normal/tangential coordinates? a) The e r direction is identical to the e n direction. b) The e direction is perpendicular to the e n direction. c) The e direction is parallel to the e r direction. 13 The McGraw-Hill Companies, Inc. All rights resered. 11-8
Radial and Transerse Components When particle position is gien in cylindrical coordinates, it is conenient to express the elocity and acceleration ectors using the unit ectors, e, and k. e R Position ector, r Re z k R Velocity ector, dr R er dt R e z k Acceleration ector, d a R R er R R dt e z k 13 The McGraw-Hill Companies, Inc. All rights resered. 11-81
Sample Problem 11.1 SOLUTION: Ealuate time t for = 3 o. Ealuate radial and angular positions, and first and second deriaties at time t. Rotation of the arm about O is defined by =.15t where is in radians and t in seconds. Collar B slides along the arm such that r =.9 -.1t where r is in meters. Calculate elocity and acceleration in cylindrical coordinates. Ealuate acceleration with respect to arm. After the arm has rotated through 3 o, determine (a) the total elocity of the collar, (b) the total acceleration of the collar, and (c) the relatie acceleration of the collar with respect to the arm. 13 The McGraw-Hill Companies, Inc. All rights resered. 11-8
Sample Problem 11.1 SOLUTION: Ealuate time t for = 3 o..15 t 3.54 rad t 1.869 s Ealuate radial and angular positions, and first and second deriaties at time t. r.9.1t r.4t r.4m s.481m.449m s.15t.3 t.3 rad.54 rad.561 rad s s 13 The McGraw-Hill Companies, Inc. All rights resered. 11-83
Sample Problem 11.1 Calculate elocity and acceleration. r r.449 m s r.481m.561rad s.7 m s a a r a r r r.391m s r r tan 1 r.54 m s 31..481m.561rad s.481m.3rad s.449 m s.561rad s a r.4 m.359 m a s s tan 1 a a r a.531m s 4. 6 13 The McGraw-Hill Companies, Inc. All rights resered. 11-84
Sample Problem 11.1 Ealuate acceleration with respect to arm. Motion of collar with respect to arm is rectilinear and defined by coordinate r. a B OA r.4m s 13 The McGraw-Hill Companies, Inc. All rights resered. 11-85
Group Problem Soling SOLUTION: Define your coordinate system Calculate the angular elocity after three reolutions Calculate the radial and transerse accelerations The angular acceleration of the centrifuge arm aries according to Determine oerall acceleration magnitude q =.5q (rad/s ) where is measured in radians. If the centrifuge starts from rest, determine the acceleration magnitude after the gondola has traelled two full rotations. 13 The McGraw-Hill Companies, Inc. All rights resered. 11-86
Group Problem Soling Define your coordinate system In the side iew, the transerse direction points into the page Determine the angular elocity q =.5q (rad/s ) Acceleration is a function of position, so use: qdq =qdq e r Top View e r Ealuate the integral e ()(p ) ò.5qdq = q dq.5q (p ) q ò = q 13 The McGraw-Hill Companies, Inc. All rights resered. q q =.5é ë (p) ù û 11-87
Group Problem Soling Determine the angular elocity q =.5é(p) ù ë û q =.899 rad/s e r Determine the angular acceleration q =.5q =.5()(p) =.683 rad/s Find the radial and transerse accelerations a = ( r - rq )e r + ( rq + rq )e q = ( - (8)(.899) ) e r + (8)(.683) + = -63.166 e r +5.65e q (m/s ) Magnitude: mag r 13 The McGraw-Hill Companies, Inc. All rights resered. ( )e q a a a ( 63.166) + 5.65 amag 63.365 m/s 11-88
Group Problem Soling What would happen if you designed the centrifuge so that the arm could extend from 6 to 1 meters? r You could now hae adal acceleration terms. This might gie you more control oer how quickly the acceleration of the gondola changes (this is known as the G-onset rate). a = ( r - rq )e + ( rq + rq )e r q 13 The McGraw-Hill Companies, Inc. All rights resered. 11-89