chapter 12 rolling- ap · web view2016 ap physics chapter 11 notes chapter 11 angular momentum +...

2016 AP Physics Chapter 11 Notes

Chapter 11 Angular Momentum + Rolling alongVector/ scalar

Translation Rotation (angular) Relate Translation and Rotational

Average Velocity

Average Acceleration

Mass

Momentum

Force

Work

Kinetic Energy

Power

Quantity Conditions for this Quantity to be conserved Rate of changeMechanical Energy

Momentum

Angular momentum

1. Rotational Inertia Equations

y


2. A particle of mass m moves with a constant speed v along the dashed line y=a. When the x-coordinate of the particle is x0, the magnitude of the angular momentum of the particle with respect to the origin of the system is

a. Zerob. mvac. mvxo

d. mv

e.

3. What would change the angular momentum of an object?

4. The angular momentum of a particle in four situations are (1) L=3t+4(2) L=-6t2

(3) L=2(4) L=4(t)-1.

a. In which situation is the net torque on the particle zero?

b. In which situation is the net torque on the particle positive and constant?

c. In which situation is the net torque on the particle negative and increasing in magnitude?

d. In which situation is the net torque on the particle negative and decreases in magnitude?

x

mv

(0,a)

Ox0


5. A stone falls from rest from the top of a building as shown above. Which of the following graphs best represents the stone's angular momentum L about the point P as a function of time? (now it is speeding up!)

6. Clues for conservationQuantity CluesMechanical Energy

Momentum

Angular momentum

7. A mass m on the end of a string moves in a circle on a horizontal frictionless table as shown to the right. The string is pulled through a hole in the table.

a. What is conserved when the string is pulled downwards? Why?


b. If the force of the hand is 3N what happens to the motion of the puck as the hand pulls downwards? Is this a net force or a net torque?

c. What is the work done by the hand?

8. An ice skater brings her arms in or a diver brings their legs in (negligible friction/air resistance).

a. What is conserved when they bring their limbs in? Why?

b. Why do they speed up? Is it due to a net force or a net torque?

c. What is the point of rotation?

9. A ball of mass m, suspended by a cord of length L, is displaced along its arc until it is elevated a vertical distance of L/2 above its lowest position at point P, as shown. The ball is then released from rest and swings like a simple pendulum. If air resistance and friction in the pivot are negligible, quantities that are conserved as the ball swings back and forth include which of the following?

I. Mechanical energy of the ball-Earth systemII. Linear momentum of the ballIII. Angular momentum of the ball about the pivot point

a. I onlyb. II only c. I and III onlyd. II and III onlye. I, II, and III

10. An ice skater is spinning about a vertical axis with arms fully extended. If the arms are pulled in closer to the body, in which of the following ways are the angular momentum and kinetic energy of the skater affected?

Angular Momentum Kinetic energy

YX Sun

Board

Skater

Top View


a. Increases Increasesb. Increases Remains Constantc. Remains Constant Increasesd. Remains Constant Remains Constante. Decreases Remains Constant

11. Assuming air resistance is negligible, how does a diver doing flips off a diving board change rotational speed in midair?

a. By changing rotational inertiab. By maintaining a constant linear speedc. By changing linear momentumd. By exerting a net torquee. By changing angular momentum

12. A satellite travels around the Sun in an elliptical orbit as shown below. As the satellite travels from point X to point Y, which of the following is true about its speed and angular momentum?

Speed Angular Momentuma. Remains Constant Remains Constantb. Increases Increasesc. Decreases Decreasesd. Increases Remains Constante. Decreases Remains Constant

13. A long board is free to slide on a sheet of frictionless ice. As shown in the top view, a skater skates to the board and hops onto one end, causing the board to slide and rotate. In this situation, which of the following occurs?

a. Linear momentum is converted to angular momentumb. Kinetic energy is converted to angular momentumc. Rotational kinetic energy is conservedd. Translational kinetic energy is conservede. Linear momentum and angular momentum are both conserved.

14. An arrow of mass m and speed vo strikes and sticks to one end of a meterstick of mass M as shown in the diagram. The meterstick is initially at rest on a horizontal surface and free to move without friction. The speed of the center of mass of the stick-arrow system after the arrow strikes


is given by which of the following expressions?

a. ½ (M+m)vo2

b. Mvo/M

c. mvo / (M+m)

d. vo/2

e. 0

15. A student initially stands on a circular platform that is free to rotate without friction about its center. The student jumps off tangentially, setting the platform spinning. Quantities that are conserved for the student-platform system as the student jumps include which of the following?

I. Angular momentumII. Linear momentumIII. Kinetic energy

(A)I only(B)II only(C)I and II only(D)II and III only(E) I, II, and III

16. Suppose you are standing on the center of a merry-go-round that is at rest. You are holding a rotating bicycle wheel over your head so that its rotation axis is pointing upward. The wheel is rotating counterclockwise when observed from above along your axis of rotation.

a. Suppose you now move the wheel so that its axis is horizontal. What happens to you?

b. What happens if you then point the axis of the wheel downward so that the wheel rotates clockwise as viewed from above?

17. Ballistic pendulum! The particle of mass m slides down the frictionless surface through height h and collides with the uniform vertical rod ( of Mass M and length d), sticking to it. The rod pivots about point O through the angle before momentarily stopping. Find


a. What is the velocity of the block before the collision?

b. What is the moment of inertia of the rod+mass?

c. Where is the center of mass of the system from the pivot point.

d. What is conserved as the rod+mass swings?

e. What is conserved in this collision? Why?

18. A circular platform has a radius R and rotational inertia I. The platform rotates around a fixed pivot at its center with negligible friction and an initial angular velocity ω. A child of mass m (represented by the small circle in the figure above) runs tangentially with speed v and jumps on


the outer edge of the platform. When the child is standing on the outer edge of the platform, the new angular velocity is

a. ω

b.Iω

I +m R2

c.Iω+mvRI +m R2

d. ( I ω2+m v2

I )1/2

e. ( I ω2+m v2

I +m R2 )1/2

19. A large circular disk of mass m and radius R is initially stationary on a horizontal icy surface. A person of mass m/2 stands on the edge of the disk. Without slipping on the disk, the person throws a large a stone of mass m/20 horizontally at initial speed vo from a height h above the ice in a radial direction, as shown in the figures above. All velocities are measured relative to the ground. The time it takes to throw the stone is negligible. Express all answers in terms of m, R, vo, and fundamental constants as appropriate.

a. Assuming that the disk is free to slide on the ice, derive an expression for the speed of the disk and person immediately after the stone is thrown.

b. The person now stands on a similar disk of mass m and radius R that has a fixed pole through its center so that it can only rotate on the ice. The person throws the same stone horizontally in a tangential direction at initial speed vo as shown in the figure. The rotational inertia of the disk is mR2/2. Derive an expression for the angular speed ω of the disk immediately after the stone is thrown.

c. The person now stands on the disk at rest R/2 from the center of the disk. The person now throws the stone horizontally with a speed vo in the same direction as in the last part. Is the angular speed of the disk immediately after throwing the stone from this new position greater


than, less than, or equal to the angular speed found in part (b)? Justify your answer.

20. A space shuttle astronaut in a circular orbit around the Earth has an assembly consisting of two small dense spheres, each of mass m, whose centers are connected by a rigid rod of length l and negligible mass. The astronaut also has a device that will launch a small lump of clay of mass m at speed v0 . Express your answers in terms of m, v0 l. and fundamental constants.

a. Initially, the assembly is "floating" freely at rest relative to the cabin, and the astronaut launches the clay lump so that it perpendicularly strikes and sticks to the midpoint of the rod, as shown above.

i. Determine the velocity of the system (assembly and clay lump) after the collision.

ii. Is mechanical energy conserved in this collision? Why or why not?

b. The assembly is brought to rest, the clay lump removed, and the experiment is repeated as shown, with the clay lump striking perpendicular to the rod but this time sticking to one of the spheres of the assembly.i. Determine the distance from the left end of the

rod to the center of mass of the system (assembly and clay lump) immediately after the collision. (Assume that the radii of the spheres and clay lump are much smaller than the separation of the spheres.)

ii. On the figure above, indicate the direction of the motion of the center of mass immediately after the collision.

iii. Determine the speed of the center of mass immediately after the collision.

iv. Determine the angular speed of the system (assembly and clay lump) immediately after the collision.

Translation + Rotation = Rolling


21. Where is the point of rotation for an object rolling?

22. The adjoining figure shows a disc of radius R rolling without slipping along a horizontal surface. Assume that the center of mass C of the disc is displaced along the positive x-direction. If the angular velocity of the disc is ω, what is the velocity VA

of the point A of the disc at the instant it is in contact with the

horizontal surface?(a) ωR directed along the positive x-direction(b) 2ωR directed along the positive x-direction(c) ωR directed along the negative x-direction(d) 2ωR directed along the negative x-direction(e) Zero

23. What is the velocity VB of the topmost point B of the disc?(a) ωR directed along the positive x-direction(b) 2ωR directed along the positive x-direction(c) Zero(d) ωR directed along the negative x-direction(e) 2ωR directed along the negative x-direction

24. What is the velocity VD of the point D (at the left edge) of the disc?(a) ωR directed along the positive x-direction(b) (√2)Rω directed along the positive x-direction(c) Zero(d) ωR directed vertically upwards(e) (√2)ωR directed at 45º to the horizontal

http://1.bp.blogspot.com/-kOsG14KFpGc/Tlt-UESTNEI/AAAAAAAABXs/ua3XykFeZK4/s1600/Rolling1-appr29-8-11.jpg


25. Consider any arbitrary point P on a thin disc rolling on any surface. If the center of mass is C and the point of contact with the surface is A at any instant, the instantaneous velocity of the point P is directed(a) along the radius through P(b) parallel to the surface(c) perpendicular to the line AP(d) along the line CP(e) perpendicular to the line CP

26. A wheel of mass M and radius R rolls on a level surface without slipping. If the angular velocity of the wheel is , what is its linear momentum?

a. MRb. M2Rc. MR2

d. M2R2/2e. Zero

27. A wheel of 0.5 m radius rolls without slipping on a horizontal surface. The axle of the wheel advances at constant velocity, moving a distance of 20 m in 5 s. The angular speed of the wheel about its point of contact on the surface is

a. 2 radians s-1

b. 4 radians s-1

c. 8 radians s-1

d. 16 radians s-1

e. 32 radians s-1

28. What is the kinetic energy of an object translating (sliding)?

What is the kinetic energy of an object rotating?

If rolling is a combination of rotation and translation, what is the kinetic energy for an object rolling?

What is the kinetic energy you consider an object rolling to be rotating about the bottom point?

http://2.bp.blogspot.com/-TBqACm6Ncus/Tlt99J2o6WI/AAAAAAAABXc/Gk-Ag1MEGfY/s1600/Rolling3-appr29-8-11.jpg


29. A 1000kg car has four 10kg wheels. When the car is moving, what fraction of the total kinetic energy of the car is due to rotation of the wheels about their axles? Assume that the wheels have the same rotational inertia as uniform disks of the same mass and size.

30. A ball initially at rest rolls without slipping down an inclined plane, as shown below. At the same time a block slides down an identical inclined plane. The block travels the same vertical distance as the ball before arriving at the bottom.

a. Draw a free body for the sphere and for the block.

b. Write an equation for acceleration of each.

c. Which arrives at the bottom with more total kinetic energy? Why?

h


A sphere of mass M, radius r, and rotational inertia I is released from rest at the top of an inclined plane of height h as shown above.31. If the plane is frictionless, what is the speed vcm of the center of mass of the sphere

at the bottom of the incline?

a.

b.

c.

d.

e.

32. If the plane has friction so that the sphere rolls without slipping, what is the speed vcm of the center of mass at the bottom of the incline?

a.

b.

c.

d.

e.

h


33. A bowling ball of mass M and radius R, whose moment of inertia about its center is 2/5 MR2, rolls without slipping along a level surface at speed v. The maximum vertical height to which it can roll if it ascends an incline is

a.

b.

c.

d.

e.

34. A hoop of mass m and radius r rolls with constant speed on a horizontal surface without slipping. What is the hoop’s translational kinetic energy divided by its rotational energy?

a. 4b. 2c. 1d. ½e. 1/4

35. Two wheels initially at rest roll the same distance without slipping down identical inclined planes starting from rest. Wheel B has twice the radius but the same mass as wheel A. All the mass is concentrated in their rims, so that the rotational inertias are I=mR2. Which has more translational kinetic energy when it gets to the bottom?

a. Wheel Ab. Wheel Bc. The translational kinetic energies are the samed. Need more information

36. Two cylinders of the same size and mass roll down an incline. Cylinder A has most of its weight concentrated at the rim, while cylinder B has most of its weight concentrated at the center. Which reaches the bottom of the incline first?

a. Ab. Bc. Both reach the bottom at the same time.


37. A solid disk and a ring roll down an incline. The ring is slower than the disk ifa. mring=mdisk, where m is the inertial massb. rring=rdisk, where r is the radiusc. mring=mdisk and rring=rdisk.d. The ring is always slower regardless of the relative values of m and r

38. A small marble of mass m and radius r will roll without slipping along a loop-the-loop track if it is released from rest somewhere on the straight section of track.a. From what initial height h above the bottom of the track must the marble be

released if it is to be on the verge of leaving the track at the top of the loop? (The radius of the loop-the-loop is R; assume R>>r.)

b. If the marble is released from height 6R above the bottom of the track, what is the horizontal component of the force acting on it at point Q?


39. A block of mass m slides up the incline shown above with an initial speed v0 in the position shown.

a. If the incline is frictionless, determine the maximum height H to which the block will rise, in terms of the given quantities and appropriate constants.

b. If the incline is rough with coefficient of sliding friction k, determine the maximum height to which the block will rise in terms of H and the given quantities.

A thin hoop of mass m and radius R moves up the incline shown above with an initial speed v0 in the position shown.

c. If the incline is rough and the hoop rolls up the incline without slipping, determine the maximum height to which the hoop will rise in terms of H and the given quantities.

d. If the incline is frictionless, determine the maximum height to which the hoop will rise in terms of H and the given quantities.


40. A cloth tape is wound around the outside of a uniform solid cylinder (mass M, radius R) and fastened to the ceiling as shown in the diagram above. The cylinder is held with the tape vertical and then released from rest. As the cylinder descends, it unwinds from the tape without slipping. The moment of inertia of a uniform solid cylinder about its center is 1/2MR2.

a. On the circle below draw vectors showing all the forces acting on the cylinder after it is released. Label each force clearly.

b. In terms of g, find the downward acceleration of the center of the cylinder as it unrolls from the tape.

c. While descending, does the center of the cylinder move toward the left, toward the right, or straight down? Explain.


41. A uniform cylinder of mass M, and radius R is initially at rest on a rough horizontal surface. The moment of inertia of a cylinder about its axis is ½ M R2 . A string, which is wrapped around the cylinder, is pulled upwards with a force T whose magnitude is 0.6Mg and whose direction is maintained vertically upward at all times. In consequence, the cylinder both accelerates horizontally and slips. The coefficient of kinetic friction is 0.5a. On the diagram below, draw vectors that represent each of the forces acting on the

cylinder identify and clearly label each force.

b. Determine the linear acceleration a of the center of the cylinder

c Calculate the angular acceleration ∝, of the cylinder.

d. Your results should show that a and αR are not equal. Explain


42. A student holds one end of a thread, which is wrapped around a cylindrical spool, as shown above. The student then drops the spool from a height h above the floor, and the thread unwinds as it falls. The spool has a mass M and a radius R, and the thread has negligible mass. The spool can be approximated as a solid cylinder of moment of inertial I=1/2MR2. Express your answer in terms of M, R, h, and fundamental constants.

a. Calculate the linear acceleration of the spool as it falls

b. Calculate the angular velocity of the spool just before it strikes the floor.

At time t=0, the spinning spool lands on the floor without bouncing and comes free from the thread. It continues to spin, but slips on the floor’s surface while doing so. Assume a constant coefficient of sliding friction µ.

c. Calculate the angular velocity of the spool as a function of time t.

d. Calculate the horizontal speed of the spool as a function of time, assuming the horizontal speed is zero at time t=0.


e. At what time does slipping between the spool and floor cease?

43. A ring of mass M, radius R, and rotational inertia MR2 is initially sliding on a frictionless surface at constant velocity Vo to the right, as shown above. At time t = 0 it encounters a surface with coefficient of friction and begins sliding and rotating. After traveling a distance L, the ring begins rolling without sliding. Express all answers to the following in terms of M, R, v0 , , and fundamental constants, as appropriate.(a) Starting from Newton’s second law in either translational or rotational form, as appropriate, derive a differential equation that can be used to solve for the magnitude of the following as the ring is sliding and rotating.i. The linear acceleration of the ring

ii. The angular acceleration of the ring

(b) Derive an expression for the magnitude of the following as the ring is sliding and rotating.i. The linear velocity v of the ring as a function of time t

ii. The angular velocity of the ring as a function of time t


(c) Derive an expression for the time it takes the ring to travel the distance L.

chapter 12 rolling- ap · web view2016 ap physics chapter 11 notes chapter 11 angular momentum +...

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