Physics 131: Test/Exam Problems: Rotations

Physics 131: Test/Exam Problems: Rotations

1) A uniform one-meter bar has a mass of 10 kg. One end of the bar is hinged to a building, and the other end is suspended by a cable that makes an angle of 45° above the horizontal as shown in the picture.

What is the tension T in the cable? (a)

(a) |T| = 69 N

(b) |T| = 98 N

(d) |T| = 129 N

(e) |T| = 132 N

2) What is the vertical component of the force exerted by the hinge? (c)

(a) | Fy | = 0

(b) | Fy | = 26 N

(d) | Fy | = 72 N

(e) | Fy | = 98 N

3) A three-meter plank has a mass of 15 kg and is supported by two saw-horses as shown below. The first saw-horse is at the right end, and the second saw-horse is two meters to the left of the first.

What is the force that the left saw-horse provides to support the plank? (b)

(a) F2 = 73.5 N

(b) F2 = 110 N

4) What is the force that the right saw-horse provides to support the plank? (a)

(a) F1 = 37 N

(b) F1 = 73.5 N

5) What is the largest distance d to the left of saw-horse 1 that a 10 kg mass can be placed without tipping the plank? (c)

(a) d = 2.35 m

(b) d = 2.5 m

6) When the 10-kg mass is at this position (at the largest distance from saw-horse 1 possible without tipping), what is the force that the left-saw horse provides to support the plank with the mass on it? (c)

(a) F2 = 73.5 N

(b) F2 = 123 N

7) A massless 100-cm stick has a 2-kg ball on one end, and an 8-kg ball on the other as shown below. What is the moment of inertia for this system if it is rotated about a point 50 cm to the right of the 2-kg ball? (a)

(a) I = 2.5 kg m2

(b) I = 5.0 kg m2

8) Four identical point masses are arranged at the corners of a rectangle. Rotation may occur about each of the two axes shown, both of which pass through the center of mass.

The relationship between the moments of inertia for rotations about the axes shown is: (c)

(a) IA > IB

(b) IA = IB

9) A block is attached to a disk-shaped pulley. The mass of the pulley is M = 0.10 kg and its radius is R = 0.4 m. The mass of the block is m = 0.25 kg. (Moment of Inertia of a disk I = ½MR2)

If the block starts a rest, what is the speed of the block after it drops by 5 m? (c)

(a) 11.3 m/s

(b) 9.9 m/s

(d) 8.1 m/s

(e) 7.2 m/s

10) A large crane used in construction uses a counter weight M to balance the weight m that is to be moved. The center of mass (CM) of the uniform beam is located 2.25 m from the support as shown. The mass of the beam is 300 kg,

If m = 1200 kg, what mass should M be such that the crane is in static equilibrium in the horizontal position? (c)

(a) 1500 kg

(b) 3340 kg

(d) 6680 kg

(e) 3900 kg

11) A larger mass m could be supported in equilibrium with the same value of M if (b)

(a) M was moved closer to the support point.

(b) M was moved further from the support point.

12) Two cylinders with identical dimensions are constrained to rotate about a vertical axis as shown.

The first has mass M and is acted on by a torque T. The second has a mass 2M and is acted on by a torque T/2. Both cylinders start from rest and after a certain amount of time have rotated through angles θ1 and θ2.

The relationship between θ1 and θ2 is: (a)

(a) θ1 = 4 θ2

(b) θ1 = 2 θ2

13) A solid cylinder (mass M = 0.50 kg, radius R = 2.5 cm) rolls without slipping down a plane inclined at 15° to the horizontal.

If the cylinder is released from rest, what is its speed after it travels 1.2 m down the plane? (b)

(a) 1.4 m/s

(b) 2.0 m/s

(d) 2.9 m/s

(e) 3.4 m/s

14) If a race (both objects are released at the same time, from rest at the top of the plane) was held between this cylinder and a solid sphere of the same mass and radius, (b)

(a) the cylinder would reach the bottom first.

(b) the sphere would reach the bottom first.

15) Two discs of identical size, but different masses (MA = 0.45 kg, MB = 0.25 kg) are shown below. Initially, disc A is suspended directly over disc B and is rotating with constant angular speed ωA = 12 rad/sec. Disc B is at rest on a frictionless surface. Viewed from above, A is rotating counterclockwise.

Then, A’s suspension breaks and it falls onto B. Later both disks move with the same angular velocity ωf. (b)

(a) ωf = 12 rad/s

(b) ωf = 7.7 rad/s

16) Compute the ratio of the total kinetic energy of the discs after A fell on B, KEafter, and the total kinetic energy of the discs before A fell on B, KEbefore. (c)

(a) KEafter / KEbefore = 0.37

(b) KEafter / KEbefore = 0.50

(d) KEafter / KEbefore = 1.0

(e) KEafter / KEbefore = 1.3

17) A uniform disc of mass M = 20 kg and radius R = 60 cm is rotating about an axis through the center as shown in the figure, with an angular velocity ω = 120 rad/s. A block of mass m = 3 kg falls vertically on the top surface of the disc at a distance r = 35 cm from the axis of rotation and sticks to the surface.

What is the angular velocity of the disc and the block after the block has fallen on the disc? (c)

(a) 69 rad/s

(b) 83 rad/s

18) Let ωf be the answer to the previous question. If the 3 kg block were to land on the disc at a distance r = 55 cm from the axis of rotation, the angular velocity of the disc with the block would be (a)

(a) smaller than ωf.

(b) equal to ωf.

19) A sphere of radius R = 30 cm is initially rotating at an angular speed of 250 revolutions per minute (rpm). A tangential force is applied to the sphere. As a result, the sphere slows down with a constant angular acceleration of 3 rad/s2.

How long does it take for the sphere to stop? (a)

(a) 8.7 s

(b) 12.1 s

20) How many revolutions does it take the sphere to stop? (b)

(a) 10.3 revolutions

(b) 18.1 revolutions

21) Let n be the answer to the previous question. If we were to use a sphere with a radius of 60 cm, but the initial angular speed and the angular acceleration stayed the same, the sphere would stop after (b)

(a) less than n revolutions.

(b) n revolutions.

22) Two solid cylinders have the same mass and radii, and are released at the same time at the top of a ramp. Cylinder 1 rolls down the ramp without slipping. Cylinder 2 slides down the ramp without friction. Which one of the two cylinders reaches the bottom of the ramp first? (b)

(a) Cylinder 1.

(b) Cylinder 2.

23) A ramp is 2 m long and makes an angle of 30° with respect to horizontal. A hollow cylinder is released from rest at the top of the ramp, and rolls to the bottom without slipping. As the cylinder rolls, compare the magnitude of its translational kinetic energy, KETRANS, to the magnitude of its rotational kinetic energy, KEROT. (c)

(a) KETRANS > KEROT

(b) KETRANS < KEROT

24) What is the velocity of the center of mass of the hollow cylinder at the bottom of the ramp? (d)

(a) 0.7 m/s

(b) 1.3 m/s

(d) 3.1 m/s

(e) 4.9 m/s

25) A circular merry-go-round of mass 200 kg and 2 m radius rotates freely with an initial angular velocity of 0.5 rad/s about a vertical axis passing through its center. A 50 kg student is initially standing on the rim of the merry-go-round. Treat the merry-go-round as a solid disk and the student as a point particle.

What is the magnitude of the angular momentum of the student plus merry-go-round? (c)

(a) 100 kg-m2/s

(b) 200 kg-m2/s

(d) 400 kg-m2/s

(e) 500 kg-m2/s

26) The student now moves toward the center of the merry-go-round. When she reaches the center, what is the angular velocity of the merry-go-round? (b)

(a) 0.50 rad/s

(b) 0.75 rad/s

(d) 1.50 rad/s

(e) 1.75 rad/s

27) Which of the following best describes what is conserved as the student walks from the rim of the merry-go-round to the center? (a)

(a) Only the angular momentum of the system is conserved.

(b) Only the kinetic energy of the system is conserved.

28) A cord passes over a pulley and suspends two blocks of different masses as shown in the figure below. The mass of the pulley is 4 kg and its radius is equal to 25 cm. The mass of the object on the left is m1 = 2 kg. The mass of the object on the right is m2 = 3 kg.

The blocks are initially at rest and they start to move. What is the speed of the second block after it has moved down by Δh = 3m . (Moment of inertia of the pulley I = ½ MpRp2). (a)

(a) 2.9 m/s

(b) 3.3 m/s

(d) 5.4 m/s

(e) 6.8 m/s

29) One end of a massless horizontal beam is attached to a vertical wall by a hinge, and the other end is supported by a string which is also attached to the wall. In case 1, as shown in the figure, a ball of mass M is hung from the end of the beam, while in case 2, an identical ball is hung from the middle of the beam.

How does the tension in the string attached to the wall compare between the two cases? (a)

(a) T1 > T2

(b) T1 < T2

30) What is the tension in the string attached to the wall in case 2?(b)

(a) T2 = Mg / 2

(b) T2 = Mg

31) A steam locomotive has two kinds of wheels: large, driving wheels with radius 0.5 m, and small guide wheels with radius 0.25 m (see the diagram). Starting from rest, the train undergoes constant linear acceleration for 30 s. At the end of 30 s the train is moving at 10 m/s.

What is the magnitude of the angular acceleration of the driving wheels? (b)

(a) 1/3 rad/s2

(b) 2/3 rad/s2

(d) 4/3 rad/s2

(e) 30 rad/s2

32) What is the angular velocity of the second hand of a clock? (a)

(a) 0.105 rad/s

(b) 22.7 rad/s

33) Two children ride on a merry-go-round of radius R = 3 m that is rotating with a constant angular velocity about its center. Child 1 is located on the rim of the merry-go-round and child 2 is located halfway between the center and the rim. Which child has the larger centripetal acceleration? (b)

(a) The one closer to the center.

(b) The one further from the center.

34) In the preceding problem, suppose that the angular velocity is 25.0 rad/s. Suppose that at t = 0, a brake is applied to the merry-go-round causing to decelerate with a constant angular acceleration of 5 rad/s2. How many revolutions does the merry-go-round make while coming to a stop? (a)

(a) 10

(b) 13

(d) 28

(e) 35

35) While the merry-go-round is slowing down, which of the two children has the larger magnitude of tangential acceleration? (b)

(a) The one closer to the center.

(b) The one further from the center.

36) A uniform disk of unknown mass M and radius R = 10 cm is free to rotate about its axis. A light cord is wrapped around the rim on the disk and then tied to a small can of mass m = 50 gm. The cord does not slip as it unwinds on the disk. When released the can moves down with acceleration 3.27 m/s2. Take g = 9.81 m/s2.

What is the angular acceleration of the disk? (e)

(a) 0 rad/s2

(b) 8.2 rad/s2

(d) 16.4 rad/s2

(e) 32.7 rad/s2

37) What is the tension in the string? (d)

(a) 0 N

(b) 0.164 N

(d) 0.327 N

(e) 0.654 N

38) If the can is released from rest, the kinetic energy of the can after it has dropped a distance h is (a)

(a) less than mgh.

(b) equal to mgh.

39) A student on a piano stool rotates freely with an angular velocity of 20 rad/s. The student holds a 1.25-kg weight in each outstretched arm, 0.759 m from the axis of rotation. The combined moment of inertia about the rotation axis of the student and stool, without the two weights, is 5.43 kg-m2, and this remains constant. Now the student pulls his arms inward to the rotation axis. What is the new angular velocity of the student? (c)

(a) 15 rad/s

(b) 20 rad/s

(d) 30 rad/s

(e) 35 rad/s

40) The work done by the student in pulling the weights inward is (a)

(a) greater than zero.

(b) less than zero.

41) When a turning figure skater draws her outstretched arms closer to her body, she finds her rotational speed increases. This is because (c)

(a) her moment of inertia becomes smaller while her rotational energy is conserved.

(b) her moment of inertia becomes smaller while energy and angular momentum are conserved.

(d) both her moment of inertia and her rotational energy are conserved.

(e) both her moment of inertia and her angular momentum are conserved.

42) The picture shows two different dumbbell shaped objects. Object A has two balls of mass m separated by a distance 2L, and object B has two balls of mass 2m separated by a distance L. Which of the objects has the largest moment of inertia for rotations around the x-axis? (Assume the balls can be treated as point masses). (a)

(a) object A

(b) object B

43) A uniform disk of mass M = 3 kg and radius R = 0.08 m spins with an initial angular velocity of w0 = 2000 rad/s.

An external torque t is applied to the disk that slows it with a constant angular acceleration of a = -65 rad/s2. What is the magnitude of the applied torque? (e)

(a) t = 1.94 N-m

(b) t = 0.26 N-m

(d) t = 1.35 N-m

(e) t = 0.62 N-m

44) A star is a uniform sphere of mass M and radius R. It rotates about its center with angular velocity  . The star then collapses to radius R/10 (with the mass still M). What is the angular velocity of the star after its collapse? (e)

(a)  / 100

(b)  / 10

 

(d) 10 

(e) 100 

45) What happens to the kinetic energy of the star as a result of its collapse? (a)

(a) It increases.

(b) It decreases.

46) A tightrope walker is better able to keep his balance by holding a long stick horizontally. Of the following, the best reason for this is to (b)

(a) shift his center of mass.

(b) increase his moment of inertia about a horizontal axis.