Torque and Rotational Dynamics Problems

1. 1A system of two wheels fixed to each other is free to rotate about a frictionless axis through the common center of the wheels and perpendicular to the page. Four forces are exerted tangentially to the rims of the wheels, as shown above. The magnitude of the net torque on the system about the axis is

   

   1. Aa. zero
   2. Bb. FR
   3. Cc. 2FR
   4. Dd. 5FR
   5. Ee. 14FR
2. 2A wheel of mass M and radius R rolls on a level surface without slipping. If the angular velocity of the wheel is ω, what is its linear momentum?

   1. Aa. MωR
   2. Bb. M(ω^2)R
   3. Cc. Mω(R^2)
   4. Dd. [M(ω^2)(R^2)]/2
   5. Ee. zero
3. 3For the wheel-and-axle system shown above, which of the following expresses the condition required for the system to be in static equilibrium?

   

   1. Aa. m1=m2
   2. Bb. am1=bm2
   3. Cc. am2=bm1
   4. Dd. a^2 m1=b^2 m2
   5. Ee. b^2 m1=a^2 m2
4. 4*A wheel with rotational inertia I is mounted on a fixed, frictionless axle. The angular speed ω of the wheel is increased from zero to ωf in a time interval T.*
   
   What is the average net torque on the wheel during this time interval?

   1. Aa. ωf/T
   2. Bb. ωf/T^2
   3. Cc. (Iωf^2)/T
   4. Dd. (Iωf)/T^2
   5. Ee. (Iωf)/T
5. 5What is the average power input to the wheel during this time interval?

   1. Aa. (Iωf) / 2T
   2. Bb. (Iωf^2) / 2T
   3. Cc. (Iωf^2) / (2T^2)
   4. Dd. (I^2)ωf / 2T^2
   5. Ee. (I^2)(ωf^2) / 2T^2
6. 6*An ant of mass m clings to the rim of a flywheel of radius r, as shown above. The flywheel rotates clockwise on a horizontal shaft S with constant angular velocity ω. As the wheel rotates, the ant revolves past the stationary points I, II, III, and IV. The ant can adhere to the wheel with a force much greater than its own weight.
   
   It will be most difficult for the ant to adhere to the wheel as it revolves past which of the four points?

   1. Aa. I
   2. Bb. II
   3. Cc. III
   4. Dd. IV
   5. Ee. It will be equally difficult for the ant to adhere to the wheel at all points.
7. 7What is the magnitude of the minimum adhesion force necessary for the ant to stay on the flywheel at point III?

   1. Aa. mg
   2. Bb. m(ω^2)r
   3. Cc. m(ω^2)r
   4. Dd. m(ω^2)r-mg
   5. Ee. m(ω^2)2+mg
8. 8A turntable that is initially at rest is set in motion with a constant angular acceleration α. What is the angular velocity of the turntable after it has made one complete revolution?

   1. Aa. √2α
   2. Bb. √2πα
   3. Cc. √4πα
   4. Dd. 2α
   5. Ee. 4πα
9. 9The rigid body shown in the diagram above consists of a vertical support post and two horizontal crossbars with spheres attached. The masses of the spheres and the lengths of the crossbars are indicated in the diagram. The body rotates about a vertical axis along the support post with constant angular speed ω. If the masses of the support post and the crossbars are negligible, what is the ratio of the angular momentum of the two upper spheres to that of the two lower spheres?

   1. Aa. 2/1
   2. Bb. 1/1
   3. Cc. 1/2
   4. Dd. 1/4
   5. Ee. 1/8
10. 10A 5-kilogram sphere is connected to a 10-kilogram sphere by a rigid rod of negligible mass, as shown above.
    
    Which of the five lettered points represents the center of mass of the sphere-rod combination?

    

    1. AA
    2. BB
    3. CC
    4. DD
    5. EE

11. 11The sphere-rod combination can be pivoted about an axis that is perpendicular to the plane of the page and that passes through one of the five lettered points. Through which point should the axis pass for the moment of inertia of the sphere-rod combination about this axis to be greatest?

    

    1. AA
    2. BB
    3. CC
    4. DD
    5. EE
12. 12A rod of negligible mass is pivoted at a point that is off-center, so that length l1 is different from length l2. The figures above show two cases in which masses are suspended from the ends of the rod. In each case the unknown mass m is balanced by a known mass, M1 or M2, so that th rod remains horizontal. What is the value of m in terms of the known masses?

    1. Aa. M1 + M2
    2. Bb. ½(M1 + M2)
    3. Cc. M1M2
    4. Dd. ½M1M2
    5. Ee. √M1M2
13. 13A figure skater is spinning on frictionless ice with her arms fully extended horizontally. She then drops her arms to her sides. Which of the following correctly describes her rotational kinetic energy and angular momentum as her arms fall?
    
    Rotational Kinetic Energy Angular momentum

    1. Aa. Remains constant Remains constant
    2. Bb. Decreases Increases
    3. Cc. Decreases Decreases
    4. Dd. Increases Decreases
    5. Ee. Increases Remains constant
14. 14A bowling ball of mass M and radius R whose moment of inertia about its center is (2/5)MR^2, 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

    1. Aa. v^2 / 5g
    2. Bb. (2v^2)/5g
    3. Cc. v^2 / 2g
    4. Dd. (7v^2) / 10g
    5. Ee. v^2 / g
15. 15In which of the following diagrams is the torque about point O equal in magnitude to the torque about point X in the diagram above? (All forces lie in the plane of the paper)

    

    1. A
    2. B
    3. C
    4. D
    5. E
16. 16*A cylinder rotates with constant angular acceleration about a fixed axis. The cylinder's moment of inertia about the axis is 4 kg m^2. At time t=2 seconds its angular velocity is 1 radian per second.*
    
    What is the angular acceleration of the cylinder between t=0 and t=2 seconds?

    1. Aa. 0.5 rad/s^2
    2. Bb. 1 rad/s^2
    3. Cc. 2 rad/s^2
    4. Dd. 4 rad/s^2
    5. Ee. 5 rad/s^2
17. 17*A cylinder rotates with constant angular acceleration about a fixed axis. The cylinder's moment of inertia about the axis is 4 kg m^2. At time t=2 seconds its angular velocity is 1 radian per second.*
    
    What is the kinetic energy of the cylinder at time t=2 seconds?

    1. Aa. 1 J
    2. Bb. 2 J
    3. Cc. 3 J
    4. Dd. 4 J
    5. Ee. cannot be determined without knowing the radius of the cylinder
18. 18*A cylinder rotates with constant angular acceleration about a fixed axis. The cylinder's moment of inertia about the axis is 4 kg m^2. At time t=2 seconds its angular velocity is 1 radian per second.*
    
    What is the angular momentum of the cylinder at time t=2 seconds?

    1. Aa. 1 kg m^2/s
    2. Bb. 2 kg m^2/s
    3. Cc. 3 kg m^2/s
    4. Dd. 4 kg m^2/s
    5. Ee. It cannot be determined without knowing the radius of the cylinder
19. 19A particle is moving in a circle of radius 2 meters according to the relation Ѳ=3(t^2) + 2t, where Ѳ is measured in radians and t in seconds. The speed of the particle at t=4 seconds is

    1. Aa. 13 m/s
    2. Bb. 16 m/s
    3. Cc. 26 m/s
    4. Dd. 52 m/s
    5. Ee. 338 m/s
20. 20A particle of mass m moves with a constant speed v along the dashed line y=a. When the x-coordinate of the particle is Xo, the magnitude of the angular momentum of the particle with respect to the origin of the system is

    

    1. Aa. zero
    2. Bb. mva
    3. Cc. mvxo
    4. Dd. mv √(x^2)+(a^2)
    5. Ee. (mva) / (√(x^2)+(a^2)
21. 21A square piece of plywood on a horizontal tabletop is subjected to the horizontal forces shown above right. Where should a third force of magnitude 5 newtons be applied to put the piece of plywood into equilibrium?

    

    1. A
    2. B
    3. C
    4. D
    5. E
22. 22A uniform rigid bar of weight W is supported in a horizontal orientation as shown above by a rope that makes a 30° angle with the horizontal. The force exerted on the bar at point O, where it is pivoted, is best represented by a vector whose direction is which of the following?

    

    1. A
    2. B
    3. C
    4. D
    5. E
23. 23A uniform stick has length L. The moment of inertia about the center of the stick is I₀. A particle of mass M is attached to one end of the stick. The moment of inertia of the combined system about the center of the stick is

    1. Aa. I₀ + ¼ML²
    2. Bb. I₀ + ½ML²
    3. Cc. I₀ + ½ML²
    4. Dd. I₀ + ML²
    5. Ee. I₀ + 5/4ML²
24. 24A light rigid rod with masses attached to its ends is pivoted about a horizontal axis as shown above. When released from rest in a horizontal orientation, the rod begins to rotate with an angular acceleration of magnitude

    

    1. Aa. g/7l
    2. Bb. g/5l
    3. Cc. g/4l
    4. Dd. 5g/7l
    5. Ee. g/l
25. 25A satellite of mass M moves in a circular orbit with radius R and a constant speed v. True statements about the satellite include which of the following?
    I. Its angular speed is v/R
    II. Its tangential acceleration is zero
    III. The magnitude of its centripetal acceleration is constant

    1. Aa. I only
    2. Bb. II only
    3. Cc. I and III only
    4. Dd. II and III only
    5. Ee. I, II, and III
26. 26A particle movies in the xy-plane with coordinates given by
    X= Acosωt and y=Asinωt,
    where A=1.5 meters and ω=2.0 radians per second. What is the magnitude of the particles acceleration?

    1. Aa. zero
    2. Bb. 1.3 m/s²
    3. Cc. 3.0 m/s²
    4. Dd. 4.5 m/s²
    5. Ee. 6.0 m/s²
27. 27A 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 momentum
    II. Linear momentum
    III. Kinetic energy

    1. Aa. I only
    2. Bb. II only
    3. Cc. I and II only
    4. Dd. II and III only
    5. Ee. I, II, and III
28. 28A bar shown above is pivoted about one end and is initially at rest in a vertical position. The bar is displaced slightly and as it falls it makes an angle Ѳ with the vertical at any given time, as shown above.
    
    Which of the following graphs best represents the bar's angular acceleration as a function of angle Ѳ?

    

    1. A
    2. B
    3. C
    4. D
    5. E
29. 29A bar shown above is pivoted about one end and is initially at rest in a vertical position. The bar is displaced slightly and as it falls it makes an angle Ѳ with the vertical at any given time, as shown above.
    
    Which of the following graphs best represents the bar's angular velocity ω as a function of time?

    

    1. A
    2. B
    3. C
    4. D
    5. E
30. 30A 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?

    

    1. A
    2. B
    3. C
    4. D
    5. E
31. 31*A uniform ladder of weight W leans without slipping against a wall making an angle Ѳ with a floor as shown above. There is friction between the ladder and the floor, but the friction between the ladder and the wall is negligible.*
    The magnitude of the normal force exerted by the floor on the ladder is

    1. Aa. W
    2. Bb. WsinѲ
    3. Cc. WcosѲ
    4. Dd. W/2 sinѲ
    5. Ee. W/2cosѲ
32. 32*A uniform ladder of weight W leans without slipping against a wall making an angle Ѳ with a floor as shown above. There is friction between the ladder and the floor, but the friction between the ladder and the wall is negligible.*
    The magnitude of the friction force exerted on the ladder by the floor is

    1. Aa. 2WtanѲ
    2. Bb. W
    3. Cc. WcotѲ
    4. Dd. W/2
    5. Ee. W/2 cotѲ
33. 33A 5-meter uniform plank of mass 100 kg rests on the top of a building with 2 m extended over the edge as shown above. How far can a 50-kg person venture past the edge of the building on the plank before the plank just begins to tip?

    

    1. Aa. ½m
    2. Bb. 1m
    3. Cc. 2/3 m
    4. Dd. 2m
    5. Ee. It is impossible to make the plank tip since the person would have to be more than 2 meters from the edge of the building
34. 34Which of the following quantities is a scalar that is always positive or zero?

    1. Aa. Power
    2. Bb. Work
    3. Cc. Kinetic Energy
    4. Dd. Linear momentum
    5. Ee. Angular momentum
35. 35To weigh a fish a person hangs a tackle box of mass 3.5 kg and a cooler of mass 5 kg from the ends of a uniform rigid pole that is suspended by a rope attached to its center. the system balances when the fish hangs at a point ¼ of the rod's length from the tackle box. What is the mass of the fish?

    1. Aa. 1.5 kg
    2. Bb. 2 kg
    3. Cc. 3 kg
    4. Dd. 6 kg
    5. Ee. 6.5 kg
36. 36Two objects, of masses 6 and 8 kg, are hung from the ends of a stick that is 70 cm long and has marks every 10 cm, as shown above. If the mass of the stick is negligible, at which of the points indicated should a cord be attached if the stick is to remain horizontal when suspended from the cord?

    

    1. AA
    2. BB
    3. CC
    4. DD
    5. EE
37. 37A 2 kg object moves in a circle of radius 4 m at a constant speed of 3 m/s. A net force of 4.5 N acts on the object. What is the angular momentum of the object with respect to an axis perpendicular to the circle and through its center?

    1. Aa. 9 N m/kg
    2. Bb. 12 m²/s
    3. Cc. 13.5 kg m²/s²
    4. Dd. 18 N m/kg
    5. Ee. 24 kg m²/s
38. 38A wheel of radius R and negligible mass is mounted on a horizontal frictionless axle so that the wheel is in a vertical plane. Three small objects having masses m, M, and 2M, respectively, are mounted on the rim of the wheel, as shown above. If the system is in static equilibrium, what is the value of m in terms of M?

    

    1. Aa. M/2
    2. Bb. M
    3. Cc. 3M/2
    4. Dd. 2M
    5. Ee. 5M/2
39. 39A rod on a horizontal tabletop is pivoted at one end and is free to rotate without friction about a vertical axis, as shown above. A force F is applied at the other end, at an angle Ѳ to the rod. If F were to be applied perpendicular to the rod, at what distance from the axis should it be applied in order to produce the same torque?

    1. Aa. L sin Ѳ
    2. Bb. L cos Ѳ
    3. Cc. L
    4. Dd. L tan Ѳ
    5. Ee. √2 L
40. 40A wheel rotating about a fixed axis has an angular position given by Ѳ=3.0 - 2.0t^3, where Ѳ is measured in radians and t in seconds. What is the angular acceleration of the wheel at t=2.0s?

    1. Aa. -1.0 rad/s²
    2. Bb. -24 rad/s²
    3. Cc. -2.0 rad/s²
    4. Dd. -4.0 rad/s²
    5. Ee. -3.5 rad/s²
41. 41The turntable of a record player has an angular velocity of 8.0 rad/s when it is turned off. The turntable comes to rest 2.5 s after being turned off. Through how many radians does the turntable rotate after being turned off? Assume constant angular acceleration

    1. Aa. 12 rad
    2. Bb. 8.0 rad
    3. Cc. 10 rad
    4. Dd. 16 rad
    5. Ee. 6.8 rad
42. 42The graphs below show angular velocity as a function of time. In which one is the magnitude of the angular acceleration constantly decreasing?

    

    1. AA
    2. BB
    3. CC
    4. DD
    5. EE
43. 43Two cylinders made of the same material roll down an inclined plane that is inclined at an angle q with the horizontal. They start from rest. They have the same mass, but the radius of cylinder B is twice the radius of cylinder A (B is a shorter cylinder). In what order do they reach the bottom?

    1. Aa. A reaches the bottom first because it's a longer cylinder
    2. Bb. A reaches the bottom first because it has a smaller moment of inertia
    3. Cc. B reaches the bottom first because it experiences a larger torque
    4. Dd. B reaches the bottom first because it travels a larger distance in one rotation
    5. Ee. They both reach the bottom at the same time, because each has the same linear velocity
44. 44A puck on a frictionless air hockey table has a mass of 5.0 g and is attached to a cord passing through a hole in the surface as in the figure. The puck is revolving at a distance 2.0 m from the hole with an angular velocity of 3.0 rad/s. The cord is them pulled from below, shortening the radius to 1.0 m. The new angular velocity (in rad/s) is

    

    1. Aa. 4.0
    2. Bb. 6.0
    3. Cc. 12
    4. Dd. 2.0
    5. Ee. 8.0
45. 45A cylinder of mass M and radius R whose moment of inertia about its center is ½MR², 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

    1. Aa. v²/2g
    2. Bb. 7v²/10g
    3. Cc. 3v²/4g
    4. Dd. 5v²/6g
    5. Ee. v²/g
46. 46A particle of mass m and speed v₀ collides and sticks to the end of a uniform solid cylinder of mass M and radius R (I=½MR²). If the cylinder is initially at rest and is pivoted about a frictionless axle through its center, what is the final angular velocity of the system after the collision?

    1. Aa. v₀/2R
    2. Bb. 2V₀/3R
    3. Cc. mv₀/(mR+MR)
    4. Dd. 2mv₀/(2mR+MR)
    5. Ee. 2mv₀/(2m+M)
47. 47Which arrangement of particles, all of mass m, has the smallest moment of inertia? They are all free to rotate in the plane of the page and the axis of rotation is perpendicular to the page at the point marked by the "x".

    

    1. AA
    2. BB
    3. CC
    4. DD
    5. EE
48. 48A force of 15 N acts at the end of a 2.0 m uniform rod that is pivoted about its center of mass. The rod starts from rest and attains an angular velocity of 10.0 rad/s in 2.0 seconds. What is the moment of inertia of the rod?

    

    1. Aa. 3
    2. Bb. 5
    3. Cc. 6
    4. Dd. 10
    5. Ee. 15
49. 49A disk is free to rotate about an axis perpendicular to the disk through its center. If the disk starts from rest and accelerates uniformly at the rate of 3 rad/s² for 4 s, its angular displacement during this time is

    1. Aa. 6 rad
    2. Bb. 12 rad
    3. Cc. 18 rad
    4. Dd. 24 rad
    5. Ee. 48 rad
50. 50A disk sliding on a horizontal surface that has negligible friction collides with a rod that is free to move and rotate on the surface, as shown in the top view above. Which of the following quantities must be the same for the disk-rod system before and after the collision
    
    I Linear momentum
    ii Angular momentum
    III Kinetic energy

    1. Aa. I only
    2. Bb. II only
    3. Cc. I and II only
    4. Dd. II and III only
    5. Ee. I, II, and III
51. 51*A solid cylinder of mass m and radius R has a string wound around it. A person holding the string pulls it vertically upward, as shown above, such that the cylinder is suspended in midair for a brief time interval Δt and its center of mass does not move. The tension in the string is T, and the rotational inertia of the cylinder about its axis is ½mR² *
    
    The net force on the cylinder during the time interval Δt is

    

    1. Aa. T
    2. Bb. mg
    3. Cc. T-mgR
    4. Dd. mgR-T
    5. Ee. zero
52. 52*A solid cylinder of mass m and radius R has a string wound around it. A person holding the string pulls it vertically upward, as shown above, such that the cylinder is suspended in midair for a brief time interval Δt and its center of mass does not move. The tension in the string is T, and the rotational inertia of the cylinder about its axis is ½mR² *
    
    The linear acceleration of the person's hand during the time interval Δt is

    

    1. Aa. (T-mg)/m
    2. Bb. 2g
    3. Cc. g/2
    4. Dd. T/m
    5. Ee. zero
53. 53*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.*
    
    If the plane is frictionless, what is the speed Vcm of the center of mass of the sphere at the bottom of the incline?

    

    1. Aa. √2gh
    2. Bb. (2Mgh)/I
    3. Cc. (2Mghr²)/I
    4. Dd. √2Mghr²/I
    5. Ee. √2Mghr²/(I+Mr²)
54. 54*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.*
    
    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?

    

    1. Aa. √2gh
    2. Bb. 2Mgh/I
    3. Cc. 2Mghr²/I
    4. Dd. √2Mghr²/I
    5. Ee. √2Mghr²/(I+Mr²)
55. 55A wheel of 0.5 m radius rolls without slipping on a horitontal 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 in radians per second about its point of contact on the surface is

    1. Aa. 2
    2. Bb. 4
    3. Cc. 8
    4. Dd. 16
    5. Ee. 32

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Quiz on Untitled, created by kristenwood22 on 14/04/2015.

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