Deck 13: Rotation II: a Conservation Approach

Question 1

A solid cylinder of radius R = 1.0 m and mass 10 kg rotates about its axis. When its angular velocity is 10 rad/s, its angular momentum (in kg⋅m²/s) is

A)50.
B)20.
C)40.
D)25.
E)70.

Answer 1

50.

Question 2

A wheel (radius = 12 cm) is mounted on a frictionless, horizontal axle that is perpendicular to the wheel and passes through the center of mass of the wheel. A light cord wrapped around the wheel supports a 0.40-kg object. If released from rest with the string taut, the object is observed to fall with a downward acceleration of 3.0 m/s². What is the rotational inertia (of the wheel) about the given axle?

A)0.023 kg⋅m²
B)0.013 kg⋅m²
C)0.020 kg⋅m²
D)0.016 kg⋅m²
E)0.035 kg⋅m²

Answer 2

0.013 kg⋅m²

Question 3

Three particles, each of which has a mass of 80 g, are positioned at the vertices of an equilateral triangle with sides of length 60 cm. The particles are connected by rods of negligible mass. What is the moment of inertia of this rigid body about an axis that is parallel to one side of the triangle and passes through the respective midpoints of the other two sides?

A)0.018 kg⋅m²
B)0.020 kg⋅m²
C)0.016 kg⋅m²
D)0.022 kg⋅m²
E)0.032 kg⋅m²

Answer 3

0.016 kg⋅m²

Question 4

Four identical particles (mass of each = 0.24 kg) are placed at the vertices of a rectangle (2.0 m × 3.0 m) and held in those positions by four light rods which form the sides of the rectangle. What is the rotational inertia of this rigid body about an axis that passes through the center of mass of the body and is parallel to the shorter sides of the rectangle?

A)2.4 kg⋅m²
B)2.2 kg⋅m²
C)1.9 kg⋅m²
D)2.7 kg⋅m²
E)8.6 kg⋅m²

Question 5

A uniform cylinder of radius R, mass M, and length L rotates freely about a horizontal axis parallel and tangent to the cylinder, as shown below. The rotational inertia of the cylinder about this axis is ​ ​

A) .
B) .
C)MR2.
D) .
E) .

Question 6

If M = 0.50 kg, L = 1.2 m, and the mass of each connecting rod shown is negligible, what is the rotational inertia about an axis perpendicular to the paper through the center of mass? Treat the mass as particles. ​ ​

A)3.7 kg⋅m²
B)2.8 kg⋅m²
C)3.2 kg⋅m²
D)2.3 kg⋅m²
E)3.9 kg⋅m²

Question 7

Four identical particles (mass of each = 0.40 kg) are placed at the vertices of a rectangle (2.5 m × 4.0 m) and held in those positions by four light rods which form the sides of the rectangle. What is the rotational inertia of this rigid body about an axis that passes through the mid-points of the shorter sides and is parallel to the longer sides?

A)2.2 kg⋅m²
B)2.8 kg⋅m²
C)2.5 kg⋅m²
D)3.1 kg⋅m²
E)1.6 kg⋅m²

Question 8

Four identical particles (mass of each = 0.40 kg) are placed at the vertices of a rectangle (2.0 m × 3.0 m) and held in those positions by four light rods which form the sides of the rectangle. What is the rotational inertia of this rigid body about an axis that passes through the mid-points of the longer sides and is parallel to the shorter sides?

A)2.7 kg⋅m²
B)3.6 kg⋅m²
C)3.1 kg⋅m²
D)4.1 kg⋅m²
E)1.6 kg⋅m²

Question 9

A car of mass 1000 kg moves with a speed of 50 m/s on a circular track of radius 100 m. What is the magnitude of its angular momentum (in kg⋅m²/s) relative to the center of the race track?

A)5.0 × 102
B)5.0 × 106
C)2.5 × 104
D)2.5 × 106
E)5.0 × 103

Question 10

The rigid body shown is rotated about an axis perpendicular to the paper and through the point P. If M = 0.40 kg, a = 30 cm, and b = 50 cm, how much work is required to take the body from rest to an angular speed of 5.0 rad/s? Neglect the mass of the connecting rods and treat the masses as particles.

A)2.9 J
B)2.6 J
C)3.1 J
D)3.4 J
E)1.6 J

Question 11

A particle whose mass is 2 kg moves in the xy plane with a constant speed of 3 m/s along the direction . What is its angular momentum (in kg⋅m²/s) relative to the origin?

A)0
B)
C)-
D)6
E)−6

Question 12

A mass (M1 = 5.0 kg) is connected by a light cord to a mass (m² = 4.0 kg) which slides on a smooth surface, as shown in the figure. The pulley (radius = 0.20 m) rotates about a frictionless axle. The acceleration of m² is 3.5 m/s². What is the rotational inertia of the pulley? ​ ​

A)0.29 kg⋅m²
B)0.42 kg⋅m²
C)0.20 kg⋅m²
D)0.62 kg⋅m²
E)0.60 kg⋅m²

Question 13

A uniform solid sphere rolls without slipping along a horizontal surface. What fraction of its total kinetic energy is in the form of rotational kinetic energy about the CM?

Question 14

A solid sphere, a solid cylinder, and a hoop all have the same mass and radius. Each are sent down identical inclined planes starting from rest. Their kinetic energies at the bottom of the incline are Ksphere, Kcylinder, and Khoop. Which of the following is true?

A)Ksphere > Kcylinder
B)Khoop > Ksphere
C)Khoop > Kcylinder
D)Kcylinder > Khoop
E)No answer above is correct.

Question 15

A wheel (radius = 0.20 m) is mounted on a frictionless, horizontal axis. A light cord wrapped around the wheel supports a 0.50-kg object, as shown in the figure. When released from rest the object falls with a downward acceleration of 5.0 m/s². What is the rotational inertia of the wheel? ​ ​

A)0.023 kg⋅m²
B)0.027 kg⋅m²
C)0.016 kg⋅m²
D)0.019 kg⋅m²
E)0.032 kg⋅m²

Question 16

The rigid body shown rotates about an axis through its center of mass and perpendicular to the paper. If M = 2.0 kg and L = 80 cm, what is the kinetic energy of this object when its angular speed about this axis is equal to 5.0 rad/s? Neglect the mass of the connecting rod and treat the masses as particles.

A)18 J
B)15 J
C)12 J
D)23 J
E)26 J

Question 17

A uniform sphere of radius R and mass M rotates freely about a horizontal axis that is tangent to an equatorial plane of the sphere, as shown below. The rotational inertia of the sphere about this axis is ​ ​

A) .
B) .
C) .
D) .
E) .

Question 18

A particle whose mass is 2.0 kg moves in the xy plane with a constant speed of 3.0 m/s along the direction . What is its angular momentum (in kg⋅m²/s) relative to the point (0, 5.0) meters?

A)12
B)11
C)13
D)14
E)21

Question 19

Two particles (m1 = 0.20 kg, m² = 0.30 kg) are positioned at the ends of a 2.0-m long rod of negligible mass. What is the rotational inertia of this rigid body about an axis perpendicular to the rod and through the center of mass?

A)0.48 kg⋅m²
B)0.50 kg⋅m²
C)1.2 kg⋅m²
D)0.80 kg⋅m²
E)0.70 kg⋅m²

Question 20

A particle whose mass is 2 kg moves in the xy plane with a constant speed of 3 m/s in the x direction along the line y = 5. What is its angular momentum (in kg⋅m²/s) relative to the origin?

A)−30
B)30
C)−15
D)15
E)45

Question 21

A 3.0-kg particle has a position vector given by , where is in meters and t is in seconds. What is the angular momentum of the particle, in kg⋅m²/s, about the origin at t = 2 s?

A)72
B)−72
C)24
D)−24
E)22

Question 22

The diagram below shows five thin cylindrical shells, each shell rotating with constant angular velocity about its central axis. The magnitude of the tangential speed of one point of each cylinder is shown, along with each cylinder's radius and mass. Which cylindrical shell has the largest angular momentum?

A)
B)
C)
D)
E)

Question 23

A space station out beyond the solar system is rotating with constant angular velocity. A spaceship heading into the station along a diameter of the station, uses its rockets to brake, and then docks inside the station at its center. When the spaceship docks, the angular momentum of the system consisting of the station and ship

A)is less than the original angular momentum of the station.
B)is the same as the original angular momentum of the station.
C)is greater than the original angular momentum of the station.
D)is less than the original angular momentum of the station, but the angular velocity increases.
E)is greater than the original angular momentum of the station, but the angular velocity decreases.

Question 24

Two blocks of masses m1 and m² are connected by a light cord that passes over a pulley of mass M, as shown. Block m² slides on a frictionless horizontal surface. The blocks and pulley are initially at rest. When m1 is released, the blocks accelerate and the pulley rotates.
​ ​
Use this exhibit to answer the following question(s).
The total angular momentum of the system of the two blocks and the pulley relative to the axis of rotation of the pulley is

A)the same at all times.
B)proportional to l1, the length of string from the pulley to m1.
C)proportional to l2, the length of string from the pulley to m².
D)conserved because the Earth doesn't move.
E)proportional to the speed of the blocks.

Question 25

In the figure, a 1.6-kg weight swings in a vertical circle at the end of a string having negligible weight. The string is 2 m long. If the weight is released with zero initial velocity from a horizontal position, its angular momentum (in kg⋅m²/s) at the lowest point of its path relative to the center of the circle is approximately

A)40
B)10
C)30
D)20
E)50

Question 26

Two blocks, m1 = 1.0 kg and m² = 2.0 kg, are connected by a light string as shown in the figure. If the radius of the pulley is 1.0 m and its rotational inertia is 5.0 kg⋅m², the acceleration of the system is ​ ​

A)(1/6)g
B)(3/8)g
C)(1/8)g
D)(1/2)g
E)(5/8)g

Question 27

Five objects of mass m move at velocity at a distance r from an axis of rotation perpendicular to the page through point A, as shown below. The one that has zero angular momentum about that axis is

A)
B)
C)
D)
E)

Question 28

A 0.5 kg fish, hooked as shown below, starts to swim away at a speed of 3 m/s. The angular momentum of the fish relative to the hand holding the fishing rod is about

A) .
B) .
C) .
D) .
E) .

Question 29

The diagram below shows five cylinders, each cylinder rotating with constant angular velocity about its central axis. The magnitude of the tangential speed of one point of each cylinder is shown, along with each cylinder's radius and mass. Which cylinder has the largest angular momentum?

A)
B)
C)
D)
E)

Question 30

A thin rod of mass M and length L is struck at one end by a ball of clay of mass m, moving with speed v as shown in the figure. The ball sticks to the rod. After the collision, the angular momentum of the clay-rod system about A, the midpoint of the rod, is

A)(m + M/3)(vL/2)
B)(m + M/12)(vL/2)
C)(m + M/6)(vL/2)
D)mvL/2
E)mvL

Question 31

Two blocks of masses m1 and m² are connected by a light cord that passes over a pulley of mass M, as shown. Block m² slides on a frictionless horizontal surface. The blocks and pulley are initially at rest. When m1 is released, the blocks accelerate and the pulley rotates.
​ ​
Use this exhibit to answer the following question(s).
The total angular momentum of the system of the two blocks and the pulley relative to the axis of rotation of the pulley is

A)proportional to the radius of the pulley.
B)proportional to the speed of the blocks.
C)proportional to the length of the string.
D)to all of the above.
E)only to (a) and (b) above.

Question 32

A pendulum bob of mass m is set into motion in a circular path in a horizontal plane as shown in the figure. The square of the angular momentum of the bob about the vertical axis through the point P is

A)m² gl3 sin4 θ/cos θ
B)m² gl3 sin3 θ/cos θ
C)m² gl3 sin2 θ/cos θ
D)m² gl3 sin θ/cos θ
E)m² gl3 sin2 θ

Question 33

When an object is effectively isolated from external torques, like an ice skater twirling on the tip of one skate, the angular momentum of the object

A)can be increased by shifting mass out away from the axis of rotation.
B)can be decreased by shifting mass out away from the axis of rotation.
C)can be increased by shifting mass in toward the axis of rotation.
D)can be decreased by shifting mass in toward the axis of rotation.
E)cannot be changed except by friction at the point of contact.

Question 34

A puck on a frictionless air hockey table has a mass of 5.0 kg 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 angular momentum of the puck (in kg⋅m²/s) is

A)80
B)20
C)30
D)60
E)50

Question 35

Stars originate as large bodies of slowly rotating gas. Because of gravity, these clumps of gas slowly decrease in size. The angular velocity of a star increases as it shrinks because of

A)conservation of angular momentum
B)conservation of linear momentum
C)conservation of energy
D)the law of universal gravitation
E)conservation of mass

Question 36

A hockey puck traveling at speed v on essentially frictionless ice collides with one end of a straight stick lying flat on the ice and sticks to that end. In this collision

A)momentum is conserved.
B)angular momentum is conserved.
C)energy is conserved.
D)all of the above are conserved.
E)only momentum and angular momentum are conserved.

Question 37

A force F is applied to a cylindrical roll of paper of radius R and mass M by pulling on the paper as shown. The acceleration of the center of mass of the roll of paper (when it rolls without slipping) is ​ ​

A) .
B) .
C) .
D) .
E) .

Question 38

The object shown below has mass m and velocity . The direction of its angular momentum vector with respect to an axis perpendicular to the page through point O is

A)downwards.
B)to the right.
C)into the page.
D)up out of the page.
E)counterclockwise.

Question 39

A hockey puck traveling at speed v on essentially frictionless ice collides elastically with one end of a straight stick lying flat on the ice. In this collision

A)momentum is conserved.
B)angular momentum is conserved.
C)energy is conserved.
D)all of the above are conserved.
E)only momentum and angular momentum are conserved.

Question 40

A mass m = 4.0 kg is connected, as shown, by a light cord to a mass M = 6.0 kg, which slides on a smooth horizontal surface. The pulley rotates about a frictionless axle and has a radius R = 0.12 m and a rotatinal inertia I = 0.090 kg⋅m². The cord does not slip on the pulley. What is the magnitude of the acceleration of m? ​ ​

A)​2.4 m/s²
B)​2.8 m/s²
C)​3.2 m/s²
D)​4.2 m/s²
E)​1.7 m/s²

Question 41

If L represents angular momentum, I represents rotational inertia, p represents linear momentum, m represents mass, and r represents a distance, which of the following can represent kinetic energy?

A)p2/2m
B)L2/2I
C)rpI
D)all of the above
E)both (a) and (b)

Question 42

What is the angular momentum of the Moon about the Earth? The mass of the Moon is 7.35 × 10²² kg, the center-to-center separation of the Earth and the Moon is 3.84 × 10⁵ km, and the orbital period of the Moon is 27.3 days. Ignore the small offset of the center of mass of the system from the center of the Earth in your calculation.