Deck 9: Energy and Momentum of Rotational Motion

Question 1

A gyroscope has a moment of inertia of 0.14 and an initial angular speed of 30 rad/s. Friction in the bearings causes its speed to reduce to zero in 30 s. What is the value of the average frictional torque?

A) 3.3 * 10^-2 N
M
B) 8.1 *10^-2 N
M
C) 14 * 10^-2 N
M
D) 7.0 * 10^-2 N
M

Answer 1

14 * 10^-2 N
M

Question 2

A non-uniform round object of mass M and radius R is found to have its total kinetic energy to be given by when rolling down an incline. What is this object's moment of inertia?

A)
B)
C)
D)

Answer 2

Question 3

Two spheres, one with the center core up to r = R/2 hollow and the other solid, have the same mass M and same outer radius R. If they are both rolling at the same linear speed, which one has the greater kinetic energy?

A) The both have the same kinetic energy.
B) The hollow one has the greater kinetic energy.
C) The solid one has the greater kinetic energy.
D) More information is needed to choose an answer.

Answer 3

The hollow one has the greater kinetic energy.

Question 4

An initially installed flywheel can store 10⁶ J of kinetic energy when rotating at 300 rad/s. It is replaced by another flywheel of the same size but made of a stronger material. If its mass is the same as that of the original, and it is now capable of achieving a rotational speed of 600 rad/s, what maximum energy can be stored?

A) 40 *10⁵ J
B) 20 * 10⁵ J
C) 10 * 10⁵ J
D) 5.0 * 10⁵ J

Question 5

A bucket of water with total mass 23 kg is attached to a rope, which in turn is wound around a 0.050-m radius cylinder, with crank, at the top of a well. The moment of inertia of the cylinder and crank is 0.21 kg m². The bucket is raised to the top of the well and released to fall back into the well. What is the kinetic energy of the cylinder and crank at the instant the bucket is moving with a speed of 6.0 m/s?

A) 2.1 * 10³ J
B) 1.5 * 10³ J
C) 0.86 * 10³ J
D) 0.40 * 10³ J

Question 6

A solid sphere of mass 2.5 kg and radius 0.12 m is at rest at the top of a ramp inclined 15 $\degree$ . It rolls to the bottom without slipping. The upper end of the ramp is 1.2 m higher than the lower end. Find the sphere's total kinetic energy when it reaches the bottom.

A) 70 J
B) 47 J
C) 29 J
D) 8.8 J

Question 7

A ventilation fan with a moment of inertia of 0.034 has a net torque of 0.11 N m applied to it. If it starts from rest, what kinetic energy will it have 10 s later?

A) 31 J
B) 18 J
C) 11 J
D) 6.6 J

Question 8

The total kinetic energy of a baseball thrown with a spinning motion is a function of:

A) its linear speed but not rotational speed.
B) its rotational speed but not linear speed.
C) both its linear and rotational speeds.
D) neither its linear nor rotational speed.

Question 9

A bus is designed to draw its power from a rotating flywheel that is brought up to its maximum speed (3000 rpm) by an electric motor. The flywheel is a solid cylinder of mass 250 kg and radius 0.500 m (I_cylinder = MR²/2). If the bus requires an average power of 10.0 kW, how long will the flywheel rotate?

A) 154 s
B) 308 s
C) 463 s
D) 617 s

Question 10

Three disks have the same mass and the same total kinetic energy. Disk #1 has radius R, disk #2 has radius 2R, and disk #3 has radius 3R. Which one has the highest ?

A) disk #1
B) disk #2
C) disk #3
D) The all have the same .

Question 11

A gyroscope has a moment of inertia of 0.140 kg m² and has an initial angular speed of 12.0 rad/s. If a lubricant is applied to the bearings of the gyroscope so that frictional torque is reduced to 2.00 *10^-2 N
M, then in what time interval will the gyroscope coast from 12.0 rad/s to zero?

A) 150 s
B) 105 s
C) 90.0 s
D) 84.0 s

Question 12

A solid sphere, a solid cylinder, and a hoop each have the same mass and radius. If they are spinning at the same angular velocity, which one has the greatest rotational kinetic energy?

A) the sphere
B) the cylinder
C) the hoop
D) They all have the same rotational kinetic energy.

Question 13

A rotating flywheel can be used as a method to store energy. If it has 2.0 * 10⁶ J of kinetic energy when rotating at 400 rad/s, and if a frictional torque of 4.0 N m acts on the system, in what interval of time would the flywheel come to rest?

A) 3.5 min
B) 42 min
C) 14 min
D) 21 min

Question 14

A hoop with mass, M, and radius, R, rolls along a level surface without slipping with a linear speed, v. What is the ratio of rotational to linear kinetic energy? (For a hoop, .)

A) 1/4
B) 1/2
C) 1/1
D) 2/5

Question 15

A bowling ball has a mass of 7.0 kg, a moment of inertia of 2.8 *10^-2 , and a radius of 0.10 m. If it rolls down the lane without slipping at a linear speed of 3.0 m/s, what is its total kinetic energy?

A) 44 J
B) 32 J
C) 11 J
D) 78 J

Question 16

A cylinder with its mass concentrated toward the center has a moment of inertia of 0.2MR². If this cylinder is rolling without slipping along a level surface with a linear speed v, what is the ratio of its rotational kinetic energy to its linear kinetic energy?

A) 1/l0
B) 1/5
C) 1/2
D) 1/1

Question 17

A box slides down a frictionless incline, and a hoop rolls down another incline. Both inclines have the same height, and both the box and the hoop have the same mass. If both objects start from rest, upon reaching the bottom of the incline which one will have the greater kinetic energy and which one will have the greater speed?

A) The box will have both the greater kinetic energy and the greater speed.
B) The hoop will have both the greater kinetic energy and the greater speed.
C) Both will have the same kinetic energy, but the hoop will have the greater speed.
D) Both will have the same kinetic energy, but the box will have the greater speed.

Question 18

The Earth's density increases with depth. If instead the Earth had uniform density, but the same angular velocity and mass, how would its rotational kinetic energy compare to the actual value?

A) It would be the same as the actual value.
B) It would be less than the actual value.
C) It would be greater than the actual value.
D) More information is needed to find the answer.

Question 19

A hoop, solid sphere, and solid cylinder each have the same mass and same radius.When rolling down the same incline, which has the greatest torque from gravity on it?

A) the hoop
B) the sphere
C) the cylinder
D) They all have the same torque.

Question 20

A rotating flywheel can be used as a method to store energy. If it is required that such a device be able to store up to a maximum of 2.00 * 10⁶ J when rotating at 400 rad/s, what moment of inertia is required?

A) 50 kg
M²
B) 25 kg
M²
C) 12.5 kg
M²
D) 6.3 kg
M²

Question 21

A figure skater on ice with arms extended, spins at a rate of 3.5 rev/s. After he draws his arms in, he spins at 6.0 rev/s. By what factor does the skater's kinetic energy change when he draws his arms in?

A) 2.4
B) 1.7
C) 0.58
D) 0.12

Question 22

A solid sphere, a solid cylinder, and a hoop each have the same mass and radius. If they are spinning at the same angular velocity, which one has the greatest angular momentum?

A) the sphere
B) the cylinder
C) the hoop
D) They all have the same angular momentum.

Question 23

The quantity "angular momentum" (in terms of the fundamental quantities of mass, length, and time) is equivalent to:

A) MLT^-2.
B) ML²T^-1.
C) ML²T^-3.
D) ML³T.

Question 24

A solid sphere of mass 2.5 kg and radius 0.12 m starts from rest at the top of a ramp inclined 15 $\degree$ and rolls to the bottom. The upper end of the ramp is 1.2 m higher than the lower end. What is the linear speed of the sphere when it reaches the bottom of the ramp? (Note: I = 0.4MR² for a solid sphere, and g = 9.8 m/s².)

A) 4.1 m/s
B) 4.7 m/s
C) 3.4 m/s
D) 2.4 m/s

Question 25

A 40-kg boy is standing on the edge of a stationary 30-kg circular platform that is free to rotate. The boy tries to walk around the platform in a counterclockwise direction. As he does:

A) the platform doesn't rotate.
B) the platform rotates in a clockwise direction just fast enough so that the boy remains stationary relative to the ground.
C) the platform rotates in a clockwise direction while the boy goes around in a counterclockwise direction relative to the ground.
D) both go around with equal angular velocities but in opposite directions.

Question 26

A solid disk of radius R rolls down an incline in time T. The center of the disk is removed up to a radius of R/2. The remaining portion of the disk with its center gone is again rolled down the same incline. The time it takes is:

A) T.
B) more than T.
C) less than T.
D) requires more information than given in the problem to figure out.

Question 27

A uniform solid sphere rolls down an incline of height 3 m after starting from rest. In order to calculate its speed at the bottom of the incline, one needs to know:

A) the mass of the sphere.
B) the radius of the sphere.
C) the mass and the radius of the sphere.
D) no more than is given in the problem.

Question 28

A tetherball is attached to a pole with a 2.0-m rope. It is circling at 0.20 rev/s. As the rope wraps around the pole, it shortens. How long is the rope when the ball is moving at 6.0 m/s?

A) 1.8 m
B) 0.83 m
C) 1.2 m
D) 1.0 m

Question 29

A hoop (I = MR²) is rolling along the ground at 10 m/s. It comes to a hill and starts going up. Assuming no losses to friction, how far does it travel vertically up the hill before it stops?

A) 10 m
B) 3.7 m
C) 5.1 m
D) 7.7 m

Question 30

An ice skater spins at 3.5 rev/s with his arms are extended. He draws his arms in and spins at 8.3 rev/s. By what factor does his moment of inertia change in the process?

A) 2.4
B) 0.58
C) 0.42
D) 0.12

Question 31

A solid sphere and a solid cylinder both have the same radius and mass. If the sphere is spun about an axis through it center with angular velocity _sphere, what angular velocity must the cylinder have to the same rotational kinetic energy as the sphere? Round your answer to 2 significant figures.

A) _cylinder = _sphere
B) _cylinder = 0.80 _sphere
C) _cylinder = 0.89 _sphere
D) _cylinder = 0.64 _sphere

Question 32

A turntable has a moment of inertia of 3.0 *10^-2 kg·m² and spins freely on a frictionless bearing at 25 rev/min. A 0.60-kg ball of putty is dropped vertically on the turntable and sticks at a point 0.10 m from the center. By what factor does the kinetic energy of the system change after the putty is dropped onto the turntable?

A) 0.91
B) 1.0
C) 0.83
D) 1.5

Question 33

A meter stick is hinged at its lower end and allowed to fall from a vertical position. If its moment of inertia is ML²/3, with what angular speed does it hit the table?

A) 5.42 rad/s
B) 2.71 rad/s
C) 1.22 rad/s
D) 7.67 rad/s

Question 34

A solid cylinder of mass 3.0 kg and radius 0.2 m starts from rest at the top of a ramp, inclined 15 $\degree$ , and rolls to the bottom without slipping. (For a cylinder I = 0.5MR².) The upper end of the ramp is 1.5 m higher than the lower end. Find the linear speed of the cylinder when it reaches the bottom of the ramp. (g = 9.8 m/s²)

A) 4.7 m/s
B) 4.4 m/s
C) 4.0 m/s
D) 2.4 m/s

Question 35

An astronaut is on a 100-m lifeline outside a spaceship, circling the ship with an angular speed of 0.100 rad/s. How far inward can she be pulled before the centripetal acceleration reaches 4g = 39.2 m/s²?

A) 50.0 m
B) 70.6 m
C) 72.7 m
D) 89.9 m

Question 36

A turntable has a moment of inertia of 3.00 *10 ^-2 kg·m² and spins freely on a frictionless bearing at 25.0 rev/min. A 0.600-kg ball of putty is dropped vertically on the turntable and sticks at a point 0.100 m from the center. By what factor does the angular momentum of the system change after the putty is dropped onto the turntable?

A) 1.22
B) 1.00 (no change)
C) 0.820
D) 1.20

Question 37

A figure skater with arms initially extended starts spinning on the ice at 3 rad/s. She then pulls her arms in close to her body. Which of the following results?

A) a smaller rotational rate
B) a greater rotational rate
C) a greater angular momentum
D) a smaller angular momentum

Question 38

An object of radius R and moment of inertia I rolls down an incline of height H after starting from rest. Its total kinetic energy at the bottom of the incline:

A) is gR/I.
B) is I/gH.
C) is 0.5 Ig/H.
D) cannot be found from the given information alone.

Question 39

A turntable has a moment of inertia of 3.00 *10^-2 kg·m² and spins freely on a frictionless bearing at 25.0 rev/min. A 0.600-kg ball of putty is dropped vertically onto the turntable and sticks at a point 0.100 m from the center. What is the new rate of rotation of the system?

A) 20.8 rev/min
B) 22.7 rev/min
C) 33.3 rev/min
D) 27.2 rev/min

Question 40

A ventilation fan with a moment of inertia of 0.034 kg·m² has a net torque of 0.11 N·m applied to it. If it starts from rest, what angular momentum will it have 9.0 s later?

A) 0.88 kg·m²/s
B) 0.99 kg·m²/s
C) 2.0 kg·m²/s
D) 3.25 kg·m²/s

Question 41

A 2.0-kg mass in the xy plane is moving in the negative x direction at 3.0 m/s along the line y = 5.0 m. As the mass passes across the y axis (i.e., passing the point (x, y = 0, 5.0 m)), what is its angular momentum with respect to the z axis?

A) "30 SI units"
B) "150 SI units"
C) "F-30 SI units"
D) "-150 SI units"

Question 42

Perihelion, the near point to the Sun for the Earth in its orbit, occurs in January. In which of the following months is the orbital angular momentum of the Earth the greatest?

A) only in January
B) only in July
C) in both April and October, but in no other months
D) None of the above answers is correct.

Question 43

The Earth's gravity exerts no torque on a satellite orbiting the Earth in an elliptical orbit. Compare the motion of the satellite at the point nearest the Earth (perigee) to the motion at the point farthest from the Earth (apogee). At these two points:

A) the tangential velocities are the same.
B) the angular velocities are the same.
C) the angular momenta are the same.
D) the kinetic energies are the same.

Question 44

An astronomical unit (AU) is the average distance from the Earth to the Sun. An comet in orbit around the Sun has a perihelion (near point to the Sun) of 0.50 AU and an aphelion (far point from the Sun) of 5.0 AU. If the comet's orbital speed at aphelion is v, what is its speed at perihelion?

A) 11v
B) 10v
C) 5.5v
D) 4.5v

Question 45

A toy gyroscope (a.k.a. a top) consists of a solid disk with a thin rod projecting perpendicular to the center of the disk. The rod, which serves as the rotational axis for the gyroscope, projects on both sides of the disk. The gyroscope is placed with one end of the axis on the surface of the table, and spun so that the rotation of the disk viewed from above is counterclockwise. Alas, the axis of rotation is not quite perpendicular table surface and so the gyroscope precesses. Viewing from above, with the end of the axis in contact with the table staying in the same spot, in what direction does the other end of the axis of the gyroscope precess?

A) The precession is clockwise.
B) The precession is counterclockwise.
C) In the northern hemisphere the precession is clockwise, but in the southern hemisphere it is counterclockwise.
D) In the northern hemisphere the precession is counterclockwise, but in the southern hemisphere it is clockwise.

Question 46

An object with mass m and moment of inertia I is spinning with an angular momentum L. Its kinetic energy is:

A) 0.5I²/L.
B) 0.5L²/I.
C) 0.5L²/m.
D) 0.5I²/m.

Question 47

An object of mass m and moment of inertia I has rotational kinetic energy KE_R. Its angular momentum is:

A) 0.5 I/m.
B) (2 IKE_R)^1/2.
C) (2 mKE_R)^1/2.
D) not given.

Question 48

A hoop, a solid sphere, and a solid cylinder each have the same mass, radius, and initial speed and are going up the same inclined plane. Which one goes the farthest?

A) the hoop
B) the sphere
C) the cylinder
D) All go the same distance.

Question 49

The Earth's gravity exerts no torque on a satellite orbiting the Earth in an elliptical orbit. Compare the motion at the point nearest the Earth (perigee) to the motion at the point farthest from the Earth (apogee). At the point closest to the Earth:

A) the angular speed will be greatest although the linear speed will be the same.
B) the speed will be greatest although the angular speed will be the same.
C) the kinetic energy and angular momentum will both be greater.
D) None of the above.

Question 50

A Frisbee is thrown horizontally by a student and travels in a straight line. Viewed from above, the Frisbee is seen to be in clockwise rotation. From the point of view of the student, in which direction is the angular momentum of the Frisbee while in flight?

A) The Frisbee has no angular momentum since it is moving along a straight line away from the student.
B) The direction of the angular momentum is along the line away from the student.
C) The direction of the angular momentum is toward the right since this was a right-handed throw.
D) The direction of the angular momentum is downward, perpendicular to the path of the Frisbee.

Question 51

Consider the use of the terms "rotation" and "revolution". In physics:

A) the words are used interchangeably.
B) the words are used interchangeably but "rotation" is the preferred word.
C) the words have different meaning.
D) "rotation" is the correct word, and "revolution" should not be used.

Question 52

A 6.0 kg mass in the xy plane is moving in the negative x direction at 3.0 m/s along the line y = 4.0 m. As the mass passes the point (x, y = 10 m, 4.0 m), what is its angular momentum with respect to the z axis?

A) "30 SI units"
B) "72 SI units"
C) "-60 SI units"
D) "-180 SI units"