Deck 10: Rotation

Question 1

The angular speed of the second hand of a watch is:

A) ( $\pi$ /1800) rad/s
B) ( $\pi$ /60) rad/s
C) ( $\pi$ /30) rad/s
D) (2 $\pi$ ) rad/s
E) (60) rad/s

Answer 1

( $\pi$ /30) rad/s

Question 2

One revolution per minute is about:

A) 0.0524 rad/s
B) 0.105 rad/s
C) 0.95 rad/s
D) 1.57 rad/s
E) 6.28 rad/s

Answer 2

0.105 rad/s

Question 3

If the angular velocity vector of a spinning body points out of the page then, when viewed from above the page, the body is spinning:

A) clockwise about an axis that is perpendicular to the page
B) counterclockwise about an axis that is perpendicular to the page
C) about an axis that is parallel to the page
D) about an axis that is changing orientation
E) about an axis that is getting longer

Answer 3

counterclockwise about an axis that is perpendicular to the page

Question 4

An object rotates from θ₁ to θ₂ through an angle that is less than 2π radians. Which of the following represents its angular displacement?

A) θ₁
B) θ₂
C) θ₁ - θ₂
D) θ₂ - θ₁
E) θ₁ + θ₂

Question 5

A wheel initially has an angular velocity of -36 rad/s but after 6.0 s its angular velocity is -24 rad/s. If its angular acceleration is constant the value is:

A) 2.0 rad/s²
B) -2.0 rad/s²
C) 3.0 rad/s²
D) -3.0 rad/s²
E) -6.0 rad/s²

Question 6

The angular velocity of a rotating wheel increases 2 rev/s every minute. The angular acceleration of this wheel is:

A) 4 $\pi$ ² rad/s²
B) 2 $\pi$ rad/s²
C) 1/30 rad/s²
D) 2 $\pi$ /30 rad/s²
E) 4 $\pi$ rad/s²

Question 7

A wheel initially has an angular velocity of 36 rad/s but after 6.0s its angular velocity is 24 rad/s. If its angular acceleration is constant the value is:

A) 2.0 rad/s²
B) -2.0 rad/s²
C) 3.0 rad/s²
D) -3.0 rad/s²
E) 6.0 rad/s²

Question 8

If a wheel turning at a constant rate completes 100 revolutions in 10 s its angular speed is:

A) 0.31 rad/s
B) 0.63 rad/s
C) 10 rad/s
D) 31 rad/s
E) 63 rad/s

Question 9

If a wheel is turning at 3.0 rad/s, the time it takes to complete one revolution is about:

A) 0.33 s
B) 0.67 s
C) 1.0 s
D) 1.3 s
E) 2.1 s

Question 10

A wheel initially has an angular velocity of 18 rad/s. It has a constant angular acceleration of 2.0 rad/s² and is slowing at first. What time elapses before its angular velocity is18 rad/s in the direction opposite to its initial angular velocity?

A) 3.0 s
B) 6.0 s
C) 9.0 s
D) 18 s
E) 36 s

Question 11

The angular speed of the minute hand of a watch is:

A) (60/ $\pi$ ) rad/s
B) (1800/ $\pi$ ) rad/s
C) ( $\pi$ ) rad/s
D) ( $\pi$ /1800) rad/s
E) ( $\pi$ /60) rad/s

Question 12

A flywheel rotating at 12 rev/s is brought to rest in 6 s. The magnitude of the average angular acceleration of the wheel during this process is:

A) 1/ $\pi$ rad/s²
B) 2 rad/s²
C) 4 rad/s²
D) 4 $\pi$ rad/s²
E) 72 rad/s²

Question 13

A phonograph turntable, initially rotating at 0.75 rev/s, slows down and stops in 30 s. The magnitude of its average angular acceleration for this process is:

A) 1.5 rad/s²
B) 1.5 $\pi$ rad/s²
C) $\pi$ /40 rad/s²
D) $\pi$ /20 rad/s²
E) 0.75 rad/s²

Question 14

A radian is about:

A) 25 $\degree$
B) 37 $\degree$
C) 45 $\degree$
D) 57 $\degree$
E) 90 $\degree$

Question 15

The angular velocity vector of a spinning body points out of the page. If the angular acceleration vector points into the page then:

A) the body is slowing down
B) the body is speeding up
C) the body is starting to turn in the opposite direction
D) the axis of rotation is changing orientation
E) none of the above

Question 16

Ten seconds after an electric fan is turned on, the fan rotates at 300 rev/min. Its average angular acceleration is:

A) 3.14 rad/s²
B) 30 rad/s²
C) 30 rev/s²
D) 50 rev/min²
E) 1800 rev/s²

Question 17

One revolution is the same as:

A) 1 rad
B) 57 rad
C) $\pi$ /2 rad
D) $\pi$ rad
E) 2 $\pi$ rad

Question 18

One-dimensional linear position is measured along a line, from a point designated x = 0. One-dimensional angular position:

A) is measured along a line, from a point designated θ = 0.
B) is measured along the axis of rotation.
C) is the angle that an internal reference line makes with a fixed external reference line.
D) is measured relative to the positive y axis.
E) is meaningless, as rotations take place in two dimensions.

Question 19

A child, riding on a large merry-go-round, travels a distance of 3000 m in a circle of diameter 40 m. The total angle through which she revolves is:

A) 50 rad
B) 75 rad
C) 150 rad
D) 314 rad
E) none of these

Question 20

If a wheel turns with constant angular speed then:

A) each point on its rim moves with constant velocity
B) each point on its rim moves with constant acceleration
C) the wheel turns through equal angles in equal times
D) the angle through which the wheel turns in each second increases as time goes on
E) the angle through which the wheel turns in each second decreases as time goes on

Question 21

The coordinate of an object is given as a function of time by θ = 7t - 3t², where θ is in radians and t is in seconds. Its angular velocity at t = 3 s is:

A) −11 rad/s
B) −3.7 rad/s
C) 1.0 rad/s
D) 3.7 rad/s
E) 11 rad/s

Question 22

A wheel starts from rest and has an angular acceleration that is given by $\alpha$ (t) = (6.0 rad/s⁴)t^2. After it has turned through 10 rev its angular velocity is:

A) 63 rad/s
B) 75 rad/s
C) 89 rad/s
D) 130 rad/s
E) 210 rad/s

Question 23

This graph shows the angular velocity of a turntable as a function of time. What is its angular acceleration at t = 3.5 s?

A) −10 rad/s²
B) −5 rad/s²
C) 0 rad/s²
D) 5 rad/s²
E) 10 rad/s²

Question 24

Instantaneous angular speed is:

A) total angular displacement divided by time
B) the integral of the displacement over time
C) the rate at which the angular acceleration is changing
D) the magnitude of the instantaneous angular velocity
E) a vector directed along the axis of rotation

Question 25

This graph shows the angular position of an object as a function of time. What is its average angular velocity between t = 5 s and t = 9 s?

A) 3 rad/s
B) −3 rad/s
C) 12 rad/s
D) −12 rad/s
E) Need additional information.

Question 26

The coordinate of an object is given as a function of time by θ = 7t - 3t², where θ is in radians and t is in seconds. Its average velocity over the interval from t = 0 to t = 2 s is:

A) 5 rad/s
B) -5 rad/s
C) 11 rad/s
D) -11 rad/s
E) 1 rad/s

Question 27

A wheel is spinning at 27 rad/s but is slowing with an angular acceleration that has a magnitude given by (3.0 rad/s⁴)t^2. It stops in a time of:

A) 1.7 s
B) 2.6 s
C) 3.0 s
D) 4.4 s
E) 9.0 s

Question 28

A wheel starts from rest and has an angular acceleration of 4.0 rad/s². When it has made 10 rev its angular velocity is:

A) 8.9 rad/s
B) 16 rad/s
C) 22 rad/s
D) 32 rad/s
E) 250 rad/s

Question 29

A flywheel is initially rotating at 20 rad/s and has a constant angular acceleration. After 9.0 s it has rotated through 450 rad. Its angular acceleration is:

A) 3.3 rad/s
B) 4.4 rad/s
C) 6.7 rad/s
D) 11 rad/s
E) 48 rad/s

Question 30

This graph shows the angular position of an object as a function of time. What is its instantaneous angular velocity at t = 1.5 s?

A) −6 rad/s
B) 6 rad/s
C) 9 rad/s
D) 12 rad/s
E) Need additional information.

Question 31

The angular velocity of a rotating turntable is given in rad/s by ω(t) = 4.5 + 0.64t - 2.7t². What is its angular acceleration at t = 2.0 s?

A) −10 rad/s²
B) -5.0 rad/s²
C) −5.4 rad/s²
D) 2.4 rad/s²
E) 3.1 rad/s²

Question 32

The fan shown has been turned on and is slowing as it rotates clockwise. The direction of the acceleration of the point X on the fan tip could be:

A)
B)
C)
D) $\leftarrow$
E) $\rightarrow$

Question 33

A wheel rotates with a constant angular acceleration of $\pi$ rad/s². During a certain time interval its angular displacement is $\pi$ rad. At the end of the interval its angular velocity is 2 $\pi$ rad/s. Its angular velocity at the beginning of the interval is:

A) 0 rad/s
B) 1 rad/s
C) $\pi$ rad/s
D) rad/s
E) 2 $\pi$ rad/s

Question 34

The angular velocity of a rotating turntable is given in rad/s by ω(t) = 4.5 + 0.64t - 2.7t². What is its average angular acceleration between t = 1.0 s and t = 3.0 s?

A) 0.64 rad/s²
B) −5.4 rad/s²
C) −7.7 rad/s²
D) −10 rad/s²
E) −27 rad/s²

Question 35

A wheel initially has an angular velocity of 18 rad/s but it is slowing at a rate of 2.0 rad/s². By the time it stops it will have turned through:

A) 81 rad
B) 160 rad
C) 245 rad
D) 330 rad
E) 410 rad

Question 36

A wheel starts from rest and has an angular acceleration that is given by $\alpha$ (t) = 6 rad/s⁴)t^2. The angle through which it turns in time t is given by:

A) [(1/8)t⁴] rad/s⁴
B) [(1/4)t⁴] rad/s⁴
C) [(1/2)t⁴] rad/s⁴
D) (t⁴) rad/s⁴
E) 12 rad

Question 37

A wheel starts from rest and has an angular acceleration that is given by $\alpha$ (t) = (6.0 rad/s⁴)t². The time it takes to make 10 rev is:

A) 1.3 s
B) 2.1 s
C) 2.8 s
D) 3.3 s
E) 4.0 s

Question 38

A wheel starts from rest and has an angular acceleration of 4.0 rad/s². The time it takes to make 10 revolutions is:

A) 0.50 s
B) 0.71 s
C) 2.2 s
D) 2.8 s
E) 5.6 s

Question 39

An object rotates from θ₁ to θ₂ through an angle that is less than π radians. Which of the following results in a positive angular displacement?

A) θ₁ = 45°, θ₂= −45°
B) θ₁ = 45°, θ₂= 15°
C) θ₁ = 45°, θ₂= −40°
D) θ₁ = 135°, θ₂= −135°
E) θ₁ = −135°, θ₂= 135°

Question 40

This graph shows the angular velocity of a turntable as a function of time. What is its average angular acceleration between t = 2 s and t = 4 s?

A) −10 rad/s²
B) −5 rad/s²
C) 0 rad/s²
D) 5 rad/s²
E) 10 rad/s²

Question 41

Three identical balls are tied by light strings to the same rod and rotate around it, as shown below. Rank the balls according to their rotational inertia, least to greatest.

A) 1, 2, 3
B) 3, 2, 1
C) 3, then 1 and 2 tie
D) 1, 3, 2
E) All are the same

Question 42

A flywheel of diameter 1.2 m has a constant angular acceleration of 5.0 rad/s². The tangential acceleration of a point on its rim is:

A) 5.0 rad/s²
B) 3.0 m/s²
C) 5.0 m/s²
D) 6.0 m/s²
E) 12 m/s²

Question 43

Three balls, with masses of 3M, 2M, and M, are fastened to a massless rod of length L as shown. The rotational inertia about the left end of the rod is:

A) ML²/2
B) ML²
C) 3ML²/2
D) 6ML²
E) 3ML²

Question 44

Four identical particles, each with mass m, are arranged in the x, y plane as shown. They are connected by light sticks to form a rigid body. If m = 2.0 kg and a = 1.0 m, the rotational inertia of this array about the y-axis is:

A) 4.0 kg∙m²
B) 12 kg∙m²
C) 9.6 kg∙m²
D) 4.8 kg∙m²
E) none of these

Question 45

For a wheel spinning on an axis through its center, the ratio of the radial acceleration of a point on the rim to the radial acceleration of a point halfway between the center and the rim is:

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

Question 46

The rotational inertia of a thin cylindrical shell of mass M, radius R, and length L about its central axis (X - X') is:

A) MR²/2
B) ML²/2
C) ML²
D) MR²
E) none of these

Question 47

A pulley with a radius of 3.0 cm and a rotational inertia of 4.5 * 10^-3 kg∙m² is suspended from the ceiling. A rope passes over it with a 2.0-kg block attached to one end and a 4.0-kg block attached to the other. The rope does not slip on the pulley. At any instant after the blocks start moving the object with the greatest kinetic energy is:

A) the heavier block
B) the lighter block
C) the pulley
D) either block (the two blocks have the same kinetic energy)
E) none (all three objects have the same kinetic energy)

Question 48

For a wheel spinning with constant angular acceleration on an axis through its center, the ratio of the speed of a point on the rim to the speed of a point halfway between the center and the rim is:

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

Question 49

The figure shows a cylinder of radius 0.7 m rotating about its axis at 10 rad/s. The speed of the point P is:

A) 7.0 m/s
B) 14 $\pi$ rad/s
C) 7 $\pi$ rad/s
D) 0.70 m/s
E) none of these

Question 50

The rotational inertia of a wheel about its axle does not depend upon its:

A) diameter
B) mass
C) distribution of mass
D) speed of rotation
E) material composition

Question 51

A car travels north at constant velocity. It goes over a piece of mud which sticks to the tire. The initial acceleration of the mud, as it leaves the ground, is:

A) vertically upward
B) horizontally to the north
C) horizontally to the south
D) zero
E) upward and forward at 45 $\degree$ to the horizontal

Question 52

Two wheels are identical but wheel B is spinning with twice the angular speed of wheel A. The ratio of the magnitude of the radial acceleration of a point on the rim of B to the magnitude of the radial acceleration of a point on the rim of A is:

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

Question 53

The magnitude of the acceleration of a point on a spinning wheel is increased by a factor of 4 if:

A) the magnitudes of the angular velocity and the angular acceleration are each multiplied by a factor of 4
B) the magnitude of the angular velocity is multiplied by a factor of 4 and the angular acceleration is not changed
C) the magnitudes of the angular velocity and the angular acceleration are each multiplied by a factor of 2
D) the magnitude of the angular velocity is multiplied by a factor of 2 and the angular acceleration is not changed
E) the magnitude of the angular velocity is multiplied by a factor of 2 and the magnitude of the angular acceleration is multiplied by a factor of 4

Question 54

A wheel starts from rest and spins with a constant angular acceleration. As time goes on the acceleration vector for a point on the rim:

A) decreases in magnitude and becomes more nearly tangent to the rim
B) decreases in magnitude and becomes more nearly radial
C) increases in magnitude and becomes more nearly tangent to the rim
D) increases in magnitude and becomes more nearly radial
E) increases in magnitude but retains the same angle with the tangent to the rim

Question 55

A wheel of diameter 3.0 cm has a 4.0 m cord wrapped around its periphery. Starting from rest, the wheel is given a constant angular acceleration of 2 rad/s². The cord will unwind in:

A) 0.82 s
B) 2.0 s
C) 12 s
D) 16 s
E) 130 s

Question 56

String is wrapped around the periphery of a 5.0-cm radius cylinder, free to rotate on its axis. The string is pulled straight out at a constant rate of 10 cm/s and does not slip on the cylinder. As each small segment of string leaves the cylinder, the segment's acceleration changes by:

A) 0 m/s²
B) 0.010 m/s²
C) 0.020 m/s²
D) 0.10 m/s²
E) 0.20 m/s²

Question 57

A particle moves in a circular path of radius 0.10 m with a constant angular speed of 5 rev/s. The acceleration of the particle is:

A) 0.10 $\pi$ m/s²
B) 0.50 m/s²
C) 500 $\pi$ m/s²
D) 2.5 m/s²
E) 10 $\pi$ ² m/s²

Question 58

Wrapping paper is being unwrapped from a 5.0-cm radius tube, free to rotate on its axis. If it is pulled at the constant rate of 10 cm/s and does not slip on the tube, the angular velocity of the tube is:

A) 2.0 rad/s
B) 5.0 rad/s
C) 10 rad/s
D) 25 rad/s
E) 50 rad/s

Question 59

For a wheel spinning on an axis through its center, the ratio of the tangential acceleration of a point on the rim to the tangential acceleration of a point halfway between the center and the rim is:

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

Question 60

A pulley with a radius of 3.0 cm and a rotational inertia of 4.5 * 10^-3 kg∙m² is suspended from the ceiling. A rope passes over it with a 2.0-kg block attached to one end and a 4.0-kg block attached to the other. The rope does not slip on the pulley. When the velocity of the heavier block is 2.0 m/s the total kinetic energy of the pulley and blocks is:

A) 2.0 J
B) 12 J
C) 14 J
D) 22 J
E) 28 J

Question 61

A solid uniform sphere of radius R and mass M has a rotational inertia about a diameter that is given by (2/5)MR². A light string of length 2.5 R is attached to the surface and used to suspend the sphere from the ceiling. Its rotational inertia about the point of attachment at the ceiling is:

A) (2/5)MR²
B) 9MR²
C) 16MR²
D) 47/5MR²
E) (82/5)MR²

Question 62

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Question 63

The rotational inertia of a solid uniform sphere about a diameter is (2/5)MR², where M is its mass and R is its radius. If the sphere is pivoted about an axis that is tangent to its surface, its rotational inertia is:

A) MR²
B) (2/5)MR²
C) (3/5)MR²
D) (5/2)MR²
E) (7/5)MR²

Question 64

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Question 65

When a thin uniform stick of mass M and length L is pivoted about its midpoint, its rotational inertia is ML²/12. When pivoted about a parallel axis through one end, its rotational inertia is:

A) ML²/12
B) ML²/6
C) ML²/3
D) 7ML²/12
E) 13ML²/12

Question 66

A rod is pivoted about its center. A 5-N force is applied 4 m from the pivot and another 5-N force is applied 2 m from the pivot, as shown. The magnitude of the total torque about the pivot is:

A) 0 N.m
B) 5.0 N.m
C) 8.7 N.m
D) 15 N.m
E) 26 N.m

Question 67

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Question 68

Consider four objects, each having the same mass and the same radius:
1. a solid sphere
2. a hollow sphere
3. a flat disk in the x,y plane
4. a hoop in the x,y plane
The order of increasing rotational inertia about an axis through the center of mass and parallel to the z axis is:

A) 1, 2, 3, 4
B) 4, 3, 2, 1
C) 1, 3, 2, 4
D) 4, 2, 3, 1
E) 3, 1, 2, 4

Question 69

a hoop in the x,y plane The order of increasing rotational inertia about an axis through the center of mass and parallel to the z axis is:

A) 1, 2, 3, 4
B) 4, 3, 2, 1
C) 1, 3, 2, 4
D) 4, 2, 3, 1
E) 3, 1, 2, 4

Question 70

The torque exerted on an object can be written as . Here, :

A) is the radius of the object.
B) is a vector pointing from the axis of rotation to the point where the force is applied.
C) is always perpendicular to .
D) is a vector pointing from the point where the force is applied to the axis of rotation.
E) points along the axis of rotation.

Question 71

Two uniform circular disks having the same mass and the same thickness are made from different materials. The disk with the smaller rotational inertia is:

A) the one made from the more dense material
B) the one made from the less dense material
C) neither - both rotational inertias are the same
D) the disk with the larger angular velocity
E) the disk with the larger torque

Question 72

To increase the rotational inertia of a solid disk about its axis without changing its mass:

A) drill holes near the rim and put the material near the axis
B) drill holes near the axis and put the material near the rim
C) drill holes at points on a circle near the rim and put the material at points between the holes
D) drill holes at points on a circle near the axis and put the material at points between the holes
E) do none of the above (the rotational inertia cannot be changed without changing the mass)

Question 73

A thin rod of length L has a density that increases along its length, ρ = ρ₀x. What is the rotational inertia of the rod around its less dense end?

A) ML²/12
B) ML²/6
C) ML²/3
D) ML²/2
E) ML²

Question 74

A meter stick on a horizontal frictionless table top is pivoted at the 80-cm mark. A horizontal force is applied perpendicularly to the end of the stick at 0 cm, as shown. A second horizontal force (not shown) is applied at the 100-cm end of the stick. If the stick does not rotate:

A)
B)
C)
D)
E)

Question 75

A disk is free to rotate on a fixed axis. A force of given magnitude F, in the plane of the disk, is to be applied. Of the following alternatives the greatest angular acceleration is obtained if the force is:

A) applied tangentially halfway between the axis and the rim
B) applied tangentially at the rim
C) applied radially halfway between the axis and the rim
D) applied radially at the rim
E) applied at the rim but neither radially nor tangentially

Question 76

A force with a given magnitude is to be applied to a wheel. The torque can be maximized by:

A) applying the force near the axle, radially outward from the axle
B) applying the force near the rim, radially outward from the axle
C) applying the force near the axle, parallel to a tangent to the wheel
D) applying the force at the rim, tangent to the rim
E) applying the force at the rim, at 45 $\degree$ to the tangent

Question 77

The figure shows forces acting on a meter stick, which is constrained to rotate around the axis indicated by the dot $\bullet$ Which force(s) create a positive torque around that axis?

A)
B)
C)
D)
E)

Question 78

The meter stick shown below rotates about an axis through the point marked $\bullet$ , 20 cm from one end. Five forces act on the stick: one at each end, one at the pivot point, and two 40 cm from one end, as shown. The magnitudes of the forces are all the same. Rank the forces according to the magnitudes of the torques they produce about the pivot point, least to greatest.

A)
B)
C)
D)
E)

Question 79

A uniform solid cylinder made of lead has the same mass and the same length as a uniform solid cylinder made of wood. The rotational inertia of the lead cylinder compared to the wooden one is:

A) greater
B) less
C) same
D) unknown unless the radii are given
E) unknown unless both the masses and the radii are given

Question 80

A force is applied to a billiard ball. In order to calculate the torque created by the force, you also need to know:

A) the mass of the ball
B) the rotational inertia of the ball
C) the kinetic energy of the ball
D) the angular speed of the ball
E) the location and orientation of the axis of rotation of the ball