Chapter 1: Kinematics of Particles

Transcription

1 Chapter 1: Kinematics of Particles

2 1.1 INTRODUCTION Mechanics the state of rest of motion of bodies subjected to the action of forces Static equilibrium of a body that is either at rest or moes with constant elocity Dynamics deals with accelerated motion of a body 1) Kinematics geometric aspects of a motion ) Kinetics analysis of the forces causing the motion

3 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 downforce F drie F drag 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.

4 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.

5 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 x 6t t or in the form of a graph x s. t.

6 11-6 RECTILINEAR MOTION: POSITION, VELOCITY & ACCELERATION 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 Dt0 From the definition of a deriatie, Dx dx lim D t 0 Dt dt e.g., x 6t dx dt t 3 1t 3t

7 11-7 RECTILINEAR MOTION: POSITION, VELOCITY & ACCELERATION Consider particle with elocity at time t and at t+dt, D Instantaneous acceleration a lim Dt0 Dt Instantaneous acceleration may be: - positie: increasing positie elocity or decreasing negatie elocity - negatie: decreasing positie elocity or increasing negatie elocity. From the definition of a deriatie, D d d x a lim D t 0 Dt dt dt e.g. 1t 3t a d dt 1 6t

8 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

9 RECTILINEAR MOTION: POSITION, VELOCITY & ACCELERATION 11-9 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 = 0 Note that max occurs when a=0, 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, = 0, a = -1 m/s

10 11-10 DETERMINING 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 Drag

11 11-11 ACCELERATION AS A FUNCTION OF TIME, POSITION, OR VELOCITY If. Kinematic relationship Integrate a a t d () dt at d t 0 0 a t dt a a x dt dx d and a dt d a x dx d x x 0 0 a x dx a a d () dt a d a dx d a 0 0 x x dx 0 0 t dt d a

12 11-1 SAMPLE PROBLEM 11. STRATEGY: 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 10 m/s ertical elocity from window 0 m aboe ground. 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. Sole for t when altitude equals zero (time for ground impact) and ealuate corresponding elocity.

13 11-13 SAMPLE PROBLEM 11. MODELING and ANALYSIS: Integrate twice to find (t) and y(t). d a 9.81m dt t t d 9.81dt 0 0 s dy t dt yt t dy t y 0 t 0 t 9.81t m 10 s m 9.81 t s 1 dt y t y0 10t 9.81t 0 y m s t 0m 10 t t m s

14 11-14 SAMPLE PROBLEM 11. Sole for t when elocity equals zero and ealuate corresponding altitude. m s m s t t 0 t 1.019s Sole for t when altitude equals zero and ealuate corresponding elocity. y t y 0m 10 0m 10 m s m s t m t s m s 1.019s s y 5.1m

15 11-15 SAMPLE PROBLEM 11. Sole for t when altitude equals zero and ealuate corresponding elocity. y m s m s t 0m 10 t t 0 t m 10 s m s m 9.81 t s t 1.43s t 3.8s m s 3.8s s meaningless m. REFLECT and THINK: s When the acceleration is constant, the elocity changes linearly, and the position is a quadratic function of time.

16 11-16 UNIFORM RECTILINEAR MOTION Once a safe speed of descent for a ertical landing is reached, a Harrier jet pilot will adjust the ertical thrusters to equal the weight of the aircraft. The plane then traels at a constant elocity downward. 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 0 dx x x 0 0 t 0 dt t t constant Careful these only apply to uniform rectilinear motion!

17 11-17 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 0 dt 0 a d a dt at 0 x t x 0 dx at dx at dt x x t at dt 0 d a constant d a dx a x x dx x x Careful these only apply to uniformly accelerated rectilinear motion!

18 11-18 MOTION OF SEVERAL PARTICLES We may be interested in the motion of seeral different particles, whose motion may be independent or linked together.

19 MOTION OF SEVERAL PARTICLES: RELATIVE 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. A B A B x x x relatie position of B with respect to A A B A B x x x A B A B relatie elocity of B with respect to A A B A B A B A B a a a relatie acceleration of B with respect to A A B A B a a a

20 11-0 SAMPLE PROBLEM 11.5 STRATEGY: 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. 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.

21 11-1 SAMPLE PROBLEM 11.5 MODELING and ANALYSIS: Substitute initial position and elocity and constant acceleration of ball into general equations for uniformly accelerated rectilinear motion. y B B y 0 0 at 0 t m 18 s 1 at m 9.81 t s 1m 18 m s m t t s Substitute initial position and constant elocity of eleator into equation for uniform rectilinear motion. y E E m s y 0 E t 5m m s t

22 11 - SAMPLE PROBLEM 11.5 Write equation for relatie position of ball with respect to eleator and sole for zero relatie position, i.e., impact. y 1 18t 4.905t 5 t 0 B E t 0.39s meaningless t 3.65s Substitute impact time into equations for position of eleator and relatie elocity of ball with respect to eleator. y E y E 1.3m B E t B E m s

23 11-3 SAMPLE PROBLEM 11.5 REFLECT and THINK: The key insight is that, when two particles collide, their position coordinates must be equal. Also, although you can use the basic kinematic relationships in this problem, you may find it easier to use the equations relating a,, x, and t when the acceleration is constant or zero.

24 11-4 MOTION OF SEVERAL 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 0 0 or or a A A a B B C a C 0 0

25 11-5 SAMPLE PROBLEM 11.7 STRATEGY: 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 = 0, collar A starts moing down from K with constant acceleration and zero initial elocity. Knowing that elocity of collar A is 300 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.

26 11-6 SAMPLE PROBLEM 11.7 MODELING and ANALYSIS: 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.

27 11-7 SAMPLE PROBLEM 11.7 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. Total length of cable remains constant, x B ( x ) 0 400mm. - = - B

28 11-8 SAMPLE PROBLEM 11.7 Differentiate motion relation twice to deelop equations for elocity and acceleration of block B. B mm = 450 s a B mm =-5 s REFLECT and THINK: In this case, the relationship we needed was not between position coordinates, but between changes in position coordinates at two different times. The key step is to clearly define your position ectors. This is a two degree-of-freedom system, because two coordinates are required to completely describe it.

29 11-9 CURVILINEAR MOTION: POSITION, VELOCITY & ACCELERATION Curilinear motion occurs when the particle moes along a cured path The snowboarder and the train both undergo curilinear motion.

30 11-30 CURVILINEAR MOTION: POSITION, VELOCITY & ACCELERATION Position measured from a fixed point O, by the position ector r = r(t) Displacement For a particle trael a distance Δs along the cure to a new position P`, within a small time interal Δt, it is defined by r` = r + Δr The displacement Δr represents the change in the particle s position

31 11-31 CURVILINEAR MOTION: POSITION, VELOCITY & ACCELERATION Instantaneous elocity (ector) Dr dr lim D t 0 Dt dt Instantaneous speed (scalar) Ds ds lim D t 0 Dt dt

32 11-3 CURVILINEAR MOTION: POSITION, VELOCITY & ACCELERATION Consider elocity of a particle at time t and elocity at t + Dt, a D d lim D t 0 Dt dt instantaneous acceleration (ector) In general, the acceleration ector is not tangent to the particle path and elocity ector.

33 11-33 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

34 11-34 RECTANGULAR COMPONENTS OF VELOCITY & ACCELERATION Rectangular components particularly effectie when component accelerations can be integrated independently, e.g., motion of a projectile, a x 0 a y g a z 0 x with initial conditions, x y z y,, x 0 y 0 z 0 Integrating twice yields x x x 0 y y gt 0 z t y y 1 gt z 0 x 0 Motion in horizontal direction is uniform. y 0 Motion in ertical direction is uniformly accelerated. z 0 Motion of projectile could be replaced by two independent rectilinear motions.

35 11-35 SAMPLE PROBLEM STRATEGY: 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 150-m cliff with an initial elocity of 180 m/s at an angle of 30 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. Determine time t for projectile to hit the ground, use this to find the horizontal distance Maximum eleation occurs when y =0

36 11-36 SAMPLE PROBLEM MODELING and ANALYSIS: Gien: () o =180 m/s (y) o =150 m (a) y = m/s (a) x = 0 m/s Vertical motion uniformly accelerated: Horizontal motion uniformly accelerated: Choose positie x to the right as shown

37 11-37 SAMPLE PROBLEM MODELING and ANALYSIS: 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 =0 Maximum eleation aboe the ground =

38 11-38 SAMPLE PROBLEM REFLECT and THINK: Because there is no air resistance, you can treat the ertical and horizontal motions separately and can immediately write down the algebraic equations of motion. If you did want to include air resistance, you must know the acceleration as a function of speed (you will see how to derie this in Chapter 1), and then you need to use the basic kinematic relationships, separate ariables, and integrate.

39 11-39 CONCEPT QUIZ If you fire a projectile from 150 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

40 11-40 GROUP PROBLEM SOLVING A baseball pitching machine throws baseballs with a horizontal elocity 0. If you want the height h to be 1050 mm, determine the alue of 0. STRATEGY: 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 1050 mm Calculate 0 =0

41 11-41 GROUP PROBLEM SOLVING MODELING and ANALYSIS: Gien: x= 1. m, y o = 1.5 m, y f = 1050 mm. Find: o Analyze the motion in the y-direction 1 y f y0 (0) t gt gt m (9.81 m/s ) t t s Analyze the motion in the x-direction x 0 ( x) 0 t 0 t 1. m ( )( s) m/s 145 km/h REFLECT and THINK: Units are correct and magnitudes are reasonable

42 11-4 MOTION RELATIVE TO A FRAME IN TRANSLATION It is critical for a pilot to know the relatie motion of his helicopter with respect to the aircraft carrier to make a safe landing.

43 11-43 MOTION RELATIVE 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.

44 11-44 SAMPLE PROBLEM STRATEGY: Define inertial axes for the system Determine the position, speed, and acceleration of car A at t = 5 s 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. Determine the position, speed, and acceleration of car B at t = 5 s Using ectors (Eqs 11.30, 11.3, and 11.33) or a graphical approach, determine the relatie position, elocity, and acceleration

45 11-45 SAMPLE PROBLEM MODELING and ANALYSIS: Define axes along the road Gien: A =36 km/h, a A = 0, (x A ) 0 = 0 ( B ) 0 = 0, a B = - 1. m/s, (y A ) 0 = 35 m Determine motion of Automobile A: We hae uniform motion for A so: At t = 5 s

46 11-46 SAMPLE PROBLEM MODELING and ANALYSIS: Determine motion of Automobile B: We hae uniform acceleration for B so: At t = 5 s

47 11-47 SAMPLE PROBLEM 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 0j 50i r B/A r 0j50 i (m) B/A B A B/A 6j 10i B/A 6j10 i (m/s) B/A B A B/A 1.j 0i a a B/A B/A 1. j (m/s ) Physically, a rider in car A would see car B traelling south and west.

48 11-48 SAMPLE PROBLEM REFLECT and THINK: Note that the relatie position and elocity of B relatie to A change with time; the alues gien here are only for the moment t =5 s.

49 11-49 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) b) 5 o 5 o c) d)

50 11-50 TANGENTIAL AND NORMAL COMPONENTS If we hae an idea of the path of a ehicle or object, it is often conenient to analyze the motion using tangential and normal components (sometimes called path coordinates).

51 11-51 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 x

52 11-5 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 0 D 0 e n e n

53 11-53 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.

54 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.

55 11-55 SAMPLE PROBLEM STRATEGY: Define your coordinate system Calculate the tangential elocity and tangential acceleration A motorist is traeling on a cured section of highway of radius 750 m at the speed of 90 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. Calculate the normal acceleration Determine oerall acceleration magnitude after the brakes hae been applied

56 11-56 SAMPLE PROBLEM MODELING and ANALYSIS: Define your coordinate system Determine elocity and acceleration in the tangential direction The deceleration constant, therefore Immediately after the brakes are applied, the speed is still 5 m/s a a a n t

57 11-57 SAMPLE PROBLEM REFLECT and THINK: The tangential component of acceleration is opposite the direction of motion, and the normal component of acceleration points to the center of curature, which is what you would expect for slowing down on a cured path. Attempting to do the problem in Cartesian coordinates is quite difficult.

58 11-58 TANGENTIAL AND NORMAL COMPONENTS In 001, 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

59 11-59 GROUP PROBLEM SOLVING STRATEGY: 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 0.5 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 10 seconds. Determine oerall acceleration magnitude

60 11-60 GROUP PROBLEM SOLVING MODELING and ANALYSIS: Define your coordinate system In the side iew, the tangential direction points into the page Determine the tangential elocity at 0.5t t 0 t 0.5t dt 0.5t 0.5t t t m/s Determine the normal acceleration 5 a n t m/s r 8 0 Determine the total acceleration magnitude e t Top View e n e n a a a mag n t (0.5)(10) amag 78.3 m/s

61 11-61 GROUP PROBLEM SOLVING REFLECT and THINK: 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

62 11-6 RADIAL AND TRANSVERSE COMPONENTS The foot pedal on an elliptical machine rotates about and extends from a central piot point. This motion can be analyzed using radial and transerse components 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.

63 11-63 RADIAL AND TRANSVERSE 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 The particle elocity ector is r e r e r re r The particle acceleration ector is r a r r e r r e

64 11-64 RADIAL AND TRANSVERSE 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

65 11-65 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.

66 11-66 RADIAL AND TRANSVERSE 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

67 11-67 SAMPLE PROBLEM STRATEGY: Ealuate time t for = 30 o. Ealuate radial and angular positions, and first and second deriaties at time t. Rotation of the arm about O is defined by = 0.15t where is in radians and t in seconds. Collar B slides along the arm such that r = t where r is in meters. After the arm has rotated through 30 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. Calculate elocity and acceleration in cylindrical coordinates. Ealuate acceleration with respect to arm.

68 11-68 SAMPLE PROBLEM MODELING and ANALYSIS Ealuate time t for = 30 o. 0.15t rad t s Ealuate radial and angular positions, and first and second deriaties at time t. r t r 0.4t r 0.4m s 0.481m 0.449m s 0.15t 0.30t 0.30rad 0.54rad 0.561rad s s

69 11-69 SAMPLE PROBLEM Calculate elocity and acceleration. r a a r a r m r r r r s 0.481m0.561rad s 0.391m s r r tan 1 r 0.70m s 0.54m s m0.561rad s 0.481m 0.3rad s 0.449m s0.561rad s a r 0.40 m m a s s tan 1 a a r a 0.531m s 4. 6

70 11-70 SAMPLE PROBLEM Ealuate acceleration with respect to arm. Motion of collar with respect to arm is rectilinear and defined by coordinate r. a B OA r 0.40m s REFLECT and THINK: You should consider polar coordinates for any kind of rotational motion. They turn this problem into a straightforward solution, whereas any other coordinate system would make this problem much more difficult. One way to make this problem harder would be to ask you to find the radius of curature in addition to the elocity and acceleration. To do this, you would hae to find the normal component of the acceleration; that is, the component of acceleration that is perpendicular to the tangential direction defined by the elocity ector.

71 11-71 GROUP PROBLEM SOLVING STRATEGY: Define your coordinate system Calculate the angular elocity after three reolutions The angular acceleration of the centrifuge arm aries according to Calculate the radial and transerse accelerations Determine oerall acceleration magnitude 0.05 (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.

72 11-7 GROUP PROBLEM SOLVING MODELING and ANALYSIS: Define your coordinate system In the side iew, the transerse direction points into the page Determine the angular elocity 0.05 (rad/s ) e r Acceleration is a function of position, so use: d d e r Top View Ealuate the integral e ()( ) 0.05d d 0 0 ( ) ( )

73 11-73 GROUP PROBLEM SOLVING Determine the angular elocity 0.05 ( ).8099 rad/s e r Determine the angular acceleration 0.05 = 0.05()( ) rad/s Find the radial and transerse accelerations r 0 (8)(.8099) er (8)(0.683) 0 a r r e r r e Magnitude: er e (m/s ) r a a a ( ) mag e amag 63.4 m/s

74 11-74 GROUP PROBLEM SOLVING REFLECT and THINK: What would happen if you designed the centrifuge so that the arm could extend from 6 to 10 meters? r You could now hae additional 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). r a r r e r r e