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The next four questions refer to the following situation: A merry-go-round of radius R = 2 m is rotating with an initial angular velocity of i = 3 rad/s about a vertical frictionless axle through its center. A child of mass M = 25 kg is initially sitting at the edge of the merry-go-round. You may assume that the size of the child is small relative to that of the merry-go-round (i.e. you may treat the child as a point particle). The combined moment of inertia of the child plus the merry-go-round when the child is at the edge is 400 kg-m2.

1. Suppose the child crawls in to the center of the merry-go-round and sits right on top of the rotation axis. What is the final angular velocity of the merry-go-round f?

a. f = 2.0 rad/s b. f = 4.0 rad/s c. f = 6.0 rad/s d. f = 8.0 rad/s e. f = 9.0 rad/

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2. Compare the initial kinetic energy of the system Ki (child at edge) with the final kinetic energy of the system Kf (child at center).

a. Ki > Kf

b. Ki = Kf c. Ki < Kf


3. Now suppose the child crawls back out to the edge of the merry-go-round and then drags her foot along the ground until the merry-go-round stops. If the magnitude of the resulting external torque about the rotation axis is = 120 Nm, how long, t, does it take for the merry-go-round to come to a complete stop after she starts dragging her foot?

a. t = 10 s b. t = 20 s c. t = 30 s d. t = 40 s e. t = 50 s

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4. What is the magnitude of the work done on the system, W, as it slows to a stop due to the child’s foot dragging on the ground?

a. W = 500 J b. W = 800 J c. W = 1200 J d. W = 1500 J e. W = 1800 J

The next three questions refer to the following situation:

A stick having length L and a mass of M hangs vertically from a horizontal, frictionless nail through one end and is initially at rest. A small dart having mass M/3 and initial horizontal velocity V hits and sticks to the free (bottom) end of the stick. Immediately after the dart gets stuck in the end of the stick, the angular velocity of the stick-dart system about the nail is measured to be .

g

V

nail

5. What (which) quantity(ies) remain(s) unchanged during the collision between the dart and the stick? a. Only angular momentum about the nail b. Both angular momentum about the nail and linear momentum c. Both angular momentum about the nail and kinetic energy



g

V

nail

6. What is the initial speed of the dart, V? a. LV 02

b. LV 03

c. LV 02

1

d. LV 02

3

e. LV 02

5



g

V

nail

7. If the dart instead strikes and gets stuck in the middle of the stick, after the collision how would the new angular velocity of the stick-dart system about the nail, 1, compare with assuming the initial horizontal speed of the dart, V, is the same? a. 1 > 0 b. 1 = 0 c. 1 < 0

The next two questions refer to the following situation: A spring with force constant, k = 15 N/m, has one end attached to a wall and its other end attached to the axle of a disk of mass M = 2 Kg and radius R = 0.5 m. The disk is displaced 1 m to the right from the equilibrium position of the spring and released from rest. The disk rolls on the floor without slipping.

Spring stretched 1 m from equilibrium

kDisk of mass M,And radius R



8. What is the net torque, , about the contact point between the disk and the floor just after the disk is released? a. = 3.75 N-m (into the page) b. = 7.5 N-m (out of the page) c. = 11.25 N-m (out of the page) d. = 17.5 N-m (into the page) e. = 0


The next two questions refer to the following situation: A spring with force constant, k = 15 N/m, has one end attached to a wall and its other end attached to the axle of a disk of mass M = 2 Kg and radius R = 0.5 m. The disk is displaced 1 m to the right from the equilibrium position of the spring and released from rest. The disk rolls on the floor without slipping.






9. What is the speed, v, of the disk’s center of mass when it reaches the spring’s equilibrium position (that is, after the disk has moved a distance of 1 m to the left and the spring is not stretched)? a. v = 1.73 m/s b. v = 2.74 m/s c. v = 4.32 m/s d. v = 2.24 m/s e. v = 3.87 m/s

10. A hoop is hung on a nail, displaced to the position shown, and released from rest in the presence of the earth’s gravitational field. Ignoring any frictional effects, what quantity (ies) is (are) conserved after the hoop is released? a. Mechanical energy b. Angular momentum about the nail c. Momentum d. Kinetic energy e. Angular momentum and kinetic energy

g

11. A disk is rotating counterclockwise with an initial angular velocity, o. A force, F, is exerted on the rim of the disk in a direction vertically downward as shown. What is the direction of the angular acceleration, ? a. Out of the page b. Into the page c. =0 so it has no direction

Initial angular velocity,o, when Force is applied

F

Initial angular velocity,o, when Force is applied

F

Disk of radius R and mass M

String tied to ceiling and wound around rim of disk

Disk of radius R and mass M

String tied to ceiling and wound around rim of disk

g

12. A uniform disk of mass M, and radius R has string wound around its rim. One end of the string is tied to a ceiling. The disk is released from rest in the configuration shown. What is the magnitude of the disk’s acceleration, a, in terms of g? a. a = g b. a = (1/2)g c. a = (1/3)g d. a = (2/3)g e. a = (3/4)g

13. A uniform bar of mass M and length L is free to rotate about a pivot located (1/3)L from the left end as shown. Two forces of equal magnitude are applied to the ends of the bar as shown. What is the direction of the angular acceleration of the bar? a. Out of the page b. Into the page c. The angular acceleration is 0 so it has no direction.

L/3 2L/3

FF

30o

L/3 2L/3

FF

30o

14. A car is initially at rest, then undergoes acceleration and reaches speed of v = 15 m/s in 5 seconds. The radius of its tires is r = 0.30 m. What is the angular acceleration, , of the tires? a. = 10.0 rad/s2 b. = 15.0 rad/s2 c. = 20.0 rad/s2

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The next question refers to the following situation: 15. Three identical point particles, each of mass m, are shaped into a triangle by three identical massless rods. About which of the shown axes is the momentum of inertia smallest? a. A b. B c. C

A

B

C

L

L

The next question refers to the following situation: 16. A baton is made of a massless rod and two identical uniform, solid spheres. The mass and the radius of the spheres are m and R. The distance from the center of one sphere to the center of the other one is L. If the baton is rotating about an axis that is perpendicular to the rod and passing through the center of mass, what is the moment of inertia I? In this case the radii of the masses are not small compared with the separation of the masses. a. 22mRI

b. 222 LRmI

c.

22

2

1

5

4LRmI

d.

22

5

2LRmI

e.

22 2

5

2LRmI

L

m, R m, R

17. At B, what is the speed, v, of the sphere

a.

5

232ghv

b.

5

213ghv

g

A

B

C

v

L

h

g

A

B

C

v

L

h

18. Now, together with the solid uniform sphere, a hollow sphere with the same mass and radius rolls down the ramp. After they leave the ramp at B, they fall on the ground at different locations, CS and CH, for the solid and the hollow sphere, respectively. The horizontal distances from CS and CH to the ramp are LS and LH. Which distance is larger? a. LS > LH

b. LS < LH c. The distances are the same.

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The next two questions refer to the following situation: A mass, M = 1kg, is held by a massless beam connected to a wall by a hinge. The distance from the hinge to the center of mass of M is L = 1 m. In case 1 the angle between the beam and the wall is 90°, while in case 2 the angle is 35°.

M=1kg

90°

L=1m

M=1kg

35°

case 1 case 2

M=1kg

90°

L=1m

M=1kg

35°

case 1 case 2

19. In case 1, what is the magnitude of the torque around the hinge due to gravity? a. = 1096 Nm b. = 4.90 Nm c. = 9.81 Nm d. = 11.77 Nm e. = 14.72 Nm


The next two questions refer to the following situation: A mass, M = 1kg, is held by a massless beam connected to a wall by a hinge. The distance from the hinge to the center of mass of M is L = 1 m. In case 1 the angle between the beam and the wall is 90°, while in case 2 the angle is 35°.

M=1kg

90°

L=1m

M=1kg

35°

case 1 case 2

M=1kg

90°

L=1m

M=1kg

35°

case 1 case 2


20. Compare the magnitude of the torque in case 1, 1, to the magnitude of the torque in case 2, 2: a. 1 > 2 b. 1 <2 c. 1 = 2

m = 60 kg

M1=9 kg M2= 12kg

d1 = 1.7 m

d2 = ?

L/2 = 2m

m = 60 kg

M1=9 kg M2= 12kg

d1 = 1.7 m

d2 = ?

L/2 = 2m

The next two questions refer to the following situation: A uniform beam of mass m = 60 kg and length L = 4 m is supported at its center by a wedge shaped fulcrum. A mass, M1 = 9 kg, is located 1.7 meters to the left of the fulcrum, and a mass, M2 = 12 kg, is located an unknown distance, d2, to the right of the fulcrum. You may assume the size of the masses is small compared with d1 and d2 (i.e. you may treat M1 and M2 as point masses).

21. The system is balanced and is observed not to move. What is d2? a. d2 = 1.050 m b. d2 = 1.275 m c. d2 = 1.500 m d. d2 = 1.700 m e. d2 = 1.855 m


m = 60 kg

M1=9 kg M2= 12kg

d1 = 1.7 m

d2 = ?

L/2 = 2m

m = 60 kg

M1=9 kg M2= 12kg

d1 = 1.7 m

d2 = ?

L/2 = 2m

The next two questions refer to the following situation: A uniform beam of mass m = 60 kg and length L = 4 m is supported at its center by a wedge shaped fulcrum. A mass, M1 = 9 kg, is located 1.7 meters to the left of the fulcrum, and a mass, M2 = 12 kg, is located an unknown distance, d2, to the right of the fulcrum. You may assume the size of the masses is small compared with d1 and d2 (i.e. you may treat M1 and M2 as point masses).


22. Mass 2 is suddenly removed. At this point in time, what is the magnitude of the instantaneous acceleration, a, of mass 1? a. a = 2.41 m/s2 b. a = 5.66 m/s2 c. a = 7.83 m/s2 d. a = 9.81 m/s2 e. a = 12.41 m/s2

M

L

dtension = T

M

L

dtension = T

The next two questions refer to the following situation: A mass M hangs from a massless beam of length L. The beam is fixed to the wall with a hinge, and the beam is held up by a wire that is attached to the end of the beam and anchored in the wall a distance d above the hinge. The beam makes a right angle with the wall.

23. What is the tension, T, in the wire?

a. d

T MgL d

b. d

T MgL

c. 2 2

2

L dT Mg

d

d. L

T Mgd

e. 2

21

LT Mg

d

M

L

dtension = T

M

L

dtension = T

The next two questions refer to the following situation: A mass M hangs from a massless beam of length L. The beam is fixed to the wall with a hinge, and the beam is held up by a wire that is attached to the end of the beam and anchored in the wall a distance d above the hinge. The beam makes a right angle with the wall.

24. If d is changed and the length of the wire is adjusted to keep the beam at right angle to the wall (M and L remaining the same), what happens to the tension in the wire? a. If d increases the tension in the wire goes up. b. If d increases the tension in the wire goes down. c. The tension in the wire remains the same.

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