Deck 7: Rotational Motion

Question 1

Five forces act on a rod that is free to pivot at point P, as shown in the figure. Which of these forces is producing a counter-clockwise torque about point P? (There could be more than one correct choice.)

A) force A
B) force B
C) force C
D) force D
E) force E

Answer 1

C

Question 2

Three masses are located in the x-y plane as follows: a mass of 6 kg is at (0 m, 0 m), a mass of 4 kg is at (3 m, 0 m), and a mass of 2 kg is at (0 m, 3 m). Where is the center of mass (or center of gravity) of the system?

A) (1 m, 2 m)
B) (2 m, 1 m)
C) (1 m, 1 m)
D) (1 m, 0.5 m)
E) (0.5 m, 1 m)

Answer 2

D

Question 3

In the figure, four point masses are placed as shown. Assume that all the numbers in the figure are accurate to two significant figures. What are the x and y coordinates of the center of mass (or center of gravity) of this arrangement?

A) (2.2 m, 2.6 m)
B) (2.2 m, 2.7 m)
C) (2.3 m, 2.6 m)
D) (2.3 m, 2.7 m)
E) (2.3 m, 2.8 m)

Answer 3

E

Question 4

A 2.0-m rope is lying on a table. You pick up one end and start raising it vertically. How high above the table is the center of mass (or center of gravity) of the rope when half of the rope has lifted off the table?

A) 0.25 m
B) 0.50 m
C) 0.75 m
D) 1.0 m
E) 1.5 m

Question 5

Three small masses are positioned as follows: 2.0 kg at (0.0 m, 0.0 m), 2.0 kg at (2.0 m, 0.0 m), and 4.0 kg at (2.0 m, 1.0 m). Determine the coordinates of the center of mass (or center of gravity) of this system.

A) (0.50 m, 1.5 m)
B) (1.5 m, 0.50 m)
C) (2.5 m, 1.5 m)
D) (2.5 m, 0.50 m)

Question 6

When a rigid object rotates about a fixed axis, what is true about all the points in the object? (There could be more than one correct choice.)

A) They all have the same angular speed.
B) They all have the same tangential speed.
C) They all have the same angular acceleration.
D) They all have the same tangential acceleration.
E) They all have the same radial acceleration.

Question 7

The figure shows scale drawings of four objects, each of the same mass and uniform thickness, with the mass distributed uniformly. Which one has the greatest moment of inertia when rotated about an axis perpendicular to the plane of the drawing at point P?

A) A
B) B
C) C
D) D
E) The moment of inertia is the same for all of these objects.

Question 8

As shown in the figure, a given force is applied to a rod in several different ways. In which case is the torque about the pivot P due to this force the greatest?

A) 1
B) 2
C) 3
D) 4
E) 5

Question 9

There must be equal amounts of mass on both sides of the center of mass (or center of gravity) of a system.

Question 10

A car is traveling along a freeway at 65 mph. What is the linear speed, relative to the highway, of each of the following points on one of its tires?
(a) the highest point on the tire
(b) the lowest point on a tire
(c) the center of the tire

Question 11

Two forces produce equal torques on a door about the door hinge. The first force is applied at the midpoint of the door; the second force is applied at the doorknob. Both forces are applied perpendicular to the door. Which force has a greater magnitude?

A) the first force (at the midpoint)
B) the second force (at the doorknob)
C) The two forces are equal.

Question 12

Two children, Ahmed and Jacques, ride on a merry-go-round. Ahmed is at a greater distance from the axis of rotation than Jacques. Which of the following are true statements? (There could be more than one correct choice.)

A) Jacques has a greater angular speed than Ahmed.
B) Jacques and Ahmed have the same angular speed.
C) Jacques has a smaller angular speed than Ahmed.
D) Ahmed has a greater tangential speed than Jacques.
E) Jacques and Ahmed have the same tangential speed.

Question 13

The rotating systems shown in the figure differ only in that the two identical movable masses are positioned a distance r from the axis of rotation (left), or a distance r/2 from the axis of rotation (right). If you release the hanging blocks simultaneously from rest,

A) the block at the left lands first.
B) the block at the right lands first.
C) both blocks land at the same time.

Question 14

Three small masses are positioned at the following coordinates: 3.0 kg at (3.0 m, 2.0 m); 4.0 kg at (0.0 m, -1.0 m); and 5.0 kg at (5.0 m, -7.0 m). What are the coordinates of the center of mass (or center of gravity) of this system?

Question 15

The center of mass (or center of gravity) of a two-particle system is at the origin. One particle is located at (3.0 m, 0.0 m) and has a mass of 2.0 kg. The other particle has a mass of 3.0 kg. What is the location of the 3.0-kg particle?

A) (-3.0 m, 0.0 m)
B) (-2.0 m, 0.0 m)
C) (2.0 m, 0.0 m)
D) (3.0 m, 0.0 m)

Question 16

A 30-kg child stands at one end of a floating 20-kg canoe that is 5.0-m long and initially at rest in the water. The child then slowly walks to the other end of the canoe. How far does the canoe move in the water, assuming water friction is negligible?

A) 1.0 m
B) 2.0 m
C) 3.0 m
D) 4.0 m
E) 5.0 m

Question 17

As shown in the figure, a 60-cm length of uniform wire, of mass 60 g, is bent into a right triangle. What are the x and y coordinates of the center of mass (or center of gravity) of this triangle?

A) (8.0 cm, 3.0 cm)
B) (8.0 cm, 5.0 cm)
C) (9.0 cm, 4.0 cm)
D) (10 cm, 3.0 cm)
E) (10 cm, 5.0 cm)

Question 18

A 60.0-kg man stands at one end of a 20.0-kg uniform 10.0-m long board. How far from the man is the center of mass (or center of gravity) of the man-board system?

A) 1.25 m
B) 2.50 m
C) 5.00 m
D) 7.50 m
E) 9.00 m

Question 19

Two equal-magnitude forces are applied to a door at the doorknob. The first force is applied perpendicular to the door, and the second force is applied at 30° to the plane of the door. Which force exerts the greater torque about the door hinge?

A) the first force (applied perpendicular to the door)
B) the second force (applied at an angle)
C) Both forces exert equal non-zero torques.
D) Both forces exert zero torque.

Question 20

Three balls are moving along a straight line having the instantaneous positions shown in the figure. At that instant, find the location and velocity of the center of mass (or center of gravity) of this system.

Question 21

A chicken is running in a circular path with an angular speed of 1.52 rad/s. How long does it take the chicken to complete one revolution?

A) 4.13 s
B) 2.07 s
C) 118 s
D) 4.77 s
E) 8.26 s

Question 22

Through how many degrees does a 33 rpm turntable rotate in

A) 63°
B) 35°
C) 46°
D) 74°

Question 23

A cylinder of radius 8.0 cm rolls 20 cm in 5.0 s without slipping. Through how many degrees does the cylinder turn during this time?

Question 24

A 3.0-kg mass is located at (0.0 m, 8.0 m), and a 1.0-kg mass is located at (12 m, 0.0 m). You want to add a 4.0-kg mass so that the center of mass (or center of gravity) of the three-mass system will be at the origin. What should be the coordinates of the 4.0-kg mass?

A) (-3.0 m, -6.0 m)
B) (-12 m, -8.0 m)
C) (3.0 m, 6.0 m)
D) (-6.0 m, -3.0 m)

Question 25

Consider the sun and its largest planet Jupiter. On the line joining them, how far from the center of the sun is their center mass? Is it within or outside the sun? (Jupiter-sun distance is 778 × 10⁶ km, diameter of the sun is 1.4 × 10⁶ km, the sun is 1000 times as massive as Jupiter)

Question 26

A bicycle wheel has an outside diameter of 66 cm. Through what distance does a point on the rim move as the wheel rotates through an angle of 70°?

Question 27

When a fan is turned off, its angular speed decreases from 10 rad/s to 6.3 rad/s in 5.0 s. What is the magnitude of the average angular acceleration of the fan?

A) 0.86 rad/s²
B) 0.74 rad/s²
C) 0.37 rad/s²
D) 11 rad/s²
E) 1.2 rad/s²

Question 28

When Mary is 3.00 m from the center of a merry-go-round, her tangential speed is a constant 1.88 m/s.
(a) What is her angular speed in rad/s?
(b) What is the magnitude of her linear acceleration?

Question 29

An artificial satellite in a low orbit circles the earth every 98 minutes. What is its angular speed in rad/s?

Question 30

Three masses, 1.0 kg, 2.0 kg, and 3.0 kg, are located at (0.0 m, 0.0 m), (1.0 m, 1.0 m), and (2.0 m, -2.0 m), respectively. What is the location of the center of mass (or center of gravity) of this system?

A) (1.3 m, 0.67 m)
B) (1.3 m, -0.67 m)
C) (-1.3 m, 0.67 m)
D) (-1.3 m, -0.67 m)

Question 31

A string is wound tightly around a fixed pulley having a radius of 5.0 cm. As the string is pulled, the pulley rotates without any slipping of the string. What is the angular speed of the pulley when the string is moving at 5.0 m/s?

A) 100 rad/s
B) 50 rad/s
C) 25 rad/s
D) 20 rad/s
E) 10 rad/s

Question 32

The sun subtends an angle of 0.00928 rad when viewed from the surface of the earth, and its distance from Earth is 1.5 × 10¹¹ m. What is the diameter of the sun?

Question 33

Express the angular speed of an old 33 1/3 rpm LP in rad/s.

Question 34

An electrical motor spins at a constant If the rotor radius is what is the linear acceleration of the edge of the rotor?

A) 5707 m/s²
B) 281.6 m/s²
C) 572,400 m/s²
D) 28.20 m/s²

Question 35

A child is riding a merry-go-round that is turning at 7.18 rpm. If the child is standing 4.65 m from the center of the merry-go-round, how fast is the child moving?

A) 5.64 m/s
B) 3.50 m/s
C) 0.556 m/s
D) 1.75 m/s
E) 1.80 m/s

Question 36

At a certain instant, a compact disc is rotating at 210 rpm. What is its angular speed in rad/s?

A) 11 rad/s
B) 22 rad/s
C) 45 rad/s
D) 69 rad/s
E) 660 rad/s

Question 37

The diameter of the Moon is 3.47 × 10⁶ m, and it subtends an angle of 0.00904 rad when viewed from the surface of Earth. How far is the Moon from Earth?

Question 38

A 1.0-g bead is at (-2.0 cm, 2.0 cm), a 2.0-g bead is at (2.0 cm, -4.0 cm), and a 3.0-g bead is at (2.0 cm, 0.0 cm). What are the coordinates of the center of mass (or center of gravity) of this system of beads?

Question 39

A wheel of diameter 0.70 m rolls on the floor without slipping. A point at the top of the wheel moves with a speed 2.0 m/s relative to the floor.
(a) At what speed is the central axis of the wheel moving relative to the floor?
(b) What is the angular speed of the wheel?

Question 40

What is the angular speed, in rad/s, of a flywheel turning at 813.0 rpm?

A) 85.14 rad/s
B) 13.53 rad/s
C) 63.84 rad/s
D) 95.33 rad/s

Question 41

A bicycle has wheels that are 60 cm in diameter. What is the angular speed of these wheels when it is moving at 4.0 m/s?

A) 1.2 rad/s
B) 4.8 rad/s
C) 0.36 rad/s
D) 13 rad/s
E) 7.6 rad/s

Question 42

When an old LP turntable was revolving at 33 rpm, it was shut off and uniformly slowed down and stopped in 5.5 seconds.
(a) What was the magnitude of its angular acceleration (in rad/s²) as it slowed down?
(b) Through how many revolutions did it turn while stopping?

Question 43

A wheel accelerates from rest to at a uniform rate of Through what angle (in radians) did the wheel turn while accelerating?

A) 30 rad
B) 24 rad
C) 60 rad
D) 38 rad

Question 44

A machinist turns on the power on to a grinding wheel at time t = 0 s. The wheel accelerates uniformly from rest for 10 s and reaches the operating angular speed of $58 \mathrm { rad } / \mathrm { s }$ The wheel is run at that angular velocity for 30 s, and then power is shut off. The wheel slows down uniformly at $1.4 \mathrm { rad } / \mathrm { s } ^ { 2 }$ until the wheel stops. What is the total number of revolutions made by the wheel in this situation?

A) 510
B) 280
C) 320
D) 470
E) 750

Question 45

A bicycle wheel has an initial angular speed of 7.2 rad/s. After turning through of a revolution, the angular speed is reduced to 2.2 rad/s. If the angular acceleration of the wheel was constant during the motion, how long will it take the wheel to make the revolution?

Question 46

An old LP record that is originally rotating at 33.3 rad/s is given a uniform angular acceleration of 2.15 rad/s². Through what angle has the record turned when its angular speed reaches 72.0 rad/s?

A) 83.2 rad
B) 316 rad
C) 697 rad
D) 66.8 rad
E) 948 rad

Question 47

A machinist turns on the power to a grinding wheel at time t = 0 s. The wheel accelerates uniformly from rest for 10 s and reaches the operating angular speed of $38 \mathrm { rad } / \mathrm { s }$ The wheel is run at that angular speed for 30 s and then power is shut off. The wheel slows down uniformly at $2.1 \mathrm { rad } / \mathrm { s } ^ { 2 }$ until the wheel stops. In this situation, what is the angular acceleration of the wheel between $t = 0 \mathrm {~s}$ and $t = 10 \mathrm {~s} ?$

A) 3.8 rad/ $s ^ { 2 }$
B) 4.6 rad/ $s ^ { 2 }$
C) 5.3 rad/ $s ^ { 2 }$
D) 6.1 rad/ $s ^ { 2 }$
E) 6.8 rad/ $s ^ { 2 }$

Question 48

An old 78 rpm record rotates through an angle of 320° as it slows down uniformly from 78.0 rpm to 22.8 rpm. What is the magnitude of the angular acceleration of the record?

A) 2.34 rad/s²
B) 5.46 rad/s²
C) 6.50 rad/s²
D) 8.35 rad/s²
E) 10.9 rad/s²

Question 49

How long does it take for a rotating object to speed up from 15.0 rad/s to 33.3 rad/s if it has a uniform angular acceleration of 3.45 rad/s²?

A) 4.35 s
B) 5.30 s
C) 9.57 s
D) 10.6 s
E) 63.1 s

Question 50

In the figure, point P is on the rim of a wheel of radius 2.0 m. At time t = 0, the wheel is at rest, and P is on the x-axis. The wheel undergoes a uniform counterclockwise angular acceleration of 0.010 rad/s² about the center O.
(a) At time t = 0, what is the tangential acceleration of P?
(b) What is the linear speed of P when it reaches the y-axis?
(c) What is the magnitude of the net linear acceleration of P when it reaches the y-axis?
(d) How long after starting does it take for P to return to its original position on the x-axis?

Question 51

A child is riding a merry-go-round that has an instantaneous angular speed of 1.25 rad/s and an angular acceleration of 0.745 rad/s². The child is standing 4.65 m from the center of the merry-go-round. What is the magnitude of the linear acceleration of the child?

A) 8.05 m/s²
B) 7.27 m/s²
C) 2.58 m/s²
D) 3.46 m/s²
E) 4.10 m/s²

Question 52

A turntable 45 cm in diameter starts from rest and makes its first complete revolution in 3.4 s with constant angular acceleration. If it maintains the same acceleration, how long will it take the turntable to make its second complete revolution?

Question 53

A wheel starts from rest and has a uniform angular acceleration of 4.0 rad/s². After the wheel completes its first revolution, how long does it take for it to make its second complete revolution?

Question 54

A rolling wheel of diameter of 68 cm slows down uniformly from 8.4 m/s to rest over a distance of 115 m. What is the magnitude of its angular acceleration if there was no slipping?

A) 1.8 rad/s²
B) 0.90 rad/s²
C) 5.7 rad/s²
D) 11 rad/s²

Question 55

A wheel rotates through an angle of 13.8 rad as it slows down uniformly from 22.0 rad/s to 13.5 rad/s. What is the magnitude of the angular acceleration of the wheel?

A) 0.616 rad/s²
B) 5.45 rad/s²
C) 111 rad/s²
D) 22.5 rad/s²
E) 10.9 rad/s²

Question 56

A pulley has an initial angular speed of 12.5 rad/s and a constant angular acceleration of 3.41 rad/s². Through what angle does the pulley turn in 5.26 s?

A) 113 rad
B) 22.6 rad
C) 42.6 rad
D) 19.3 rad
E) 160 rad

Question 57

A scooter has wheels with a diameter of 120 mm. What is the angular speed of the wheels when the scooter is moving forward at 6.00 m/s?

A) 47.7 rpm
B) 955 rpm
C) 72.0 rpm
D) 50.0 rpm
E) 100 rpm

Question 58

A machinist turns on the power on to a grinding wheel at time t = 0 s. The wheel accelerates uniformly from rest for 10 s and reaches the operating angular speed of $96 \mathrm { rad } / \mathrm { s }$ The wheel is run at that angular velocity for 40 s and then power is shut off. The wheel slows down uniformly at $1.5 \mathrm { rad } / \mathrm { s } ^ { 2 }$ until the wheel stops. For how long a time after the power is shut off does it take the wheel to stop?

A) 64 s
B) 62 s
C) 66 s
D) 68 s
E) 70 s

Question 59

A bowling ball of mass 7.5 kg and diameter 18 cm rolls without slipping down a 10-m bowling lane with a constant speed 4.3 m/s.
(a) Through what angle does the bowling ball turn as it travels the length of the lane?
(b) What is the angular speed of the bowling ball?
(c) Calculate the maximum radial acceleration that a point on the surface of the bowling ball could have.
(d) Calculate the tangential acceleration of a point on the surface of the bowling ball.

Question 60

A wheel accelerates with a constant angular acceleration of 4.5 rad/s² from an initial angular speed of 1.0 rad/s. (a) Through what angle does the wheel turn in the first 2.0 s, and (b) what is its angular speed at that time?

Question 61

The drive chain in a bicycle is applying a torque of 0.850 N ∙ m to the wheel of the bicycle. The wheel has a moment of inertia of 0.100 kg ∙ m². What is the angular acceleration of the wheel?

Question 62

A triatomic molecule is oriented as follows along the x-axis: mass m is at the origin, mass 2m is at x = a, and, mass 3m is at x = 2a. What is the moment of inertia of this molecule about the y-axis?

A) 2ma²
B) 3ma²
C) 12ma²
D) 14ma²

Question 63

A centrifuge in a medical laboratory rotates at a rotational speed of 3600 rev/min. When switched off, it makes 50 complete turns at a constant angular acceleration before coming to rest.
(a) What was the magnitude of the angular acceleration of the centrifuge as it slowed down?
(d) How long did it take for the centrifuge to come to rest after being turned off?

Question 64

A 95-N force exerted at the end of a 0.32-m long torque wrench produces a torque of 15 N ∙ m. What is the angle (less than 90°) between the wrench handle and the direction of the applied force?

A) 30°
B) 24°
C) 36°
D) 42°

Question 65

A force of $16.88 \mathrm {~N}$ is applied tangentially to a wheel of radius 0.340 m and causes an angular acceleration of 1.20 rad/s². What is the moment of inertia of the wheel?

A) 4.78 kg ? m²
B) 3.59 kg ? m²
C) 5.98 kg ? m²
D) 7.17 kg ? m²

Question 66

A majorette fastens two batons together at their centers to form an X shape. Each baton consists of an extremely light 1.2-m bar with small 0.25-kg balls at each end. What is the moment of inertia of this baton about an axis through the center of the X?

Question 67

A Ferris wheel rotating at 20 rad/s slows down with a constant angular acceleration of magnitude 5.0 rad/s². How many revolutions does it make while slowing down before coming to rest?

A) 40
B) 20
C) 6.4
D) 3.2

Question 68

A man in a gym is holding an 8.0-kg weight at arm's length, a distance of 0.55 m from his shoulder joint. What is the torque about his shoulder joint due to the weight if his arm is horizontal?

A) 0.24 N ∙ m
B) 4.4 N ∙ m
C) 43 N ∙ m
D) 15 N ∙ m
E) 0 N ∙ m

Question 69

The drive chain in a bicycle is applying a torque of 0.850 N ∙ m to the wheel of the bicycle. You can treat the wheel as a thin uniform hoop (or ring) with a mass of 0.750 kg and a radius of 33.0 cm. What is the angular acceleration of the wheel?

A) 10.4 rad/s²
B) 20.8 rad/s²
C) 5.20 rad/s²
D) 3.43 rad/s²
E) 1.06 rad/s²

Question 70

A torque of 12 N ? m is applied to a solid, uniform disk of radius 0.50 m. If the disk accelerates at $1.6 \mathrm { rad } / \mathrm { s } ^ { 2 }$ what is the mass of the disk?

A) 60 kg
B) 45 kg
C) 30 kg
D) 15 kg

Question 71

A potter's wheel has the shape of a solid uniform disk of mass and radius 0.65 m. It spins about an axis perpendicular to the disk at its center. A small 2.1 kg lump of very dense clay is dropped onto the wheel at a distance 0.41 m from the axis. What is the moment of inertia of the system about the axis of spin?

A) 1.8 kg ∙ m²
B) 1.5 kg ∙ m²
C) 0.40 kg ∙ m²
D) 2.5 kg ∙ m²

Question 72

Two uniform solid spheres have the same mass, but one has twice the radius of the other. The ratio of the larger sphere's moment of inertia about a central axis to that of the smaller sphere is

A) 4/5.
B) 8/5.
C) 1/2.
D) 2.
E) 4.

Question 73

A force of 17 N is applied to the end of a 0.63-m long torque wrench at an angle 45° from a line joining the pivot point to the handle. What is the magnitude of the torque about the pivot point produced by this force?

A) 7.6 N ∙ m
B) 10.7 N ∙ m
C) 12.0 N ∙ m
D) 9.7 N ∙ m

Question 74

In the figure, a weightlifter's barbell consists of two identical small but dense spherical weights, each of mass 50 kg. These weights are connected by a thin 0.96-m rod with a mass of 24 kg. Find the moment of inertia of the barbell through the axis perpendicular to the rod at its center, assuming the two weights are small enough to be treated as point masses.

Question 75

A mechanic is examining the wheel of a bicycle to adjust the brake. With the bicycle off the ground, he manually rotates the wheel until it reaches an angular speed of 12.0 rad/s and then allows it to coast to a stop. If the wheel has a moment of inertia of 0.100 kg ∙ m², and the wheel slows uniformly to a stop in 160 s, what is the magnitude of the retarding torque?

A) 1.00 N ∙ m
B) 0.00750 N ∙ m
C) 0.0787 N ∙ m
D) 1.33 N ∙ m
E) 1.67 N ∙ m

Question 76

A man in a gym is holding an 8.0-kg weight at arm's length, a distance of 0.55 m from his shoulder joint. What is the torque about his shoulder joint due to the weight if his arm is held at 30° below the horizontal?

A) 22 N ∙ m
B) 2.2 N ∙ m
C) 4.4 N ∙ m
D) 13 N ∙ m
E) 37 N ∙ m

Question 77

The L-shaped object shown in the figure consists of three small masses connected by thin uniform rods, each rod of mass 3.00 kg. Assume that the masses shown are accurate to three significant figures. What is the moment of inertia of this object (a) about the x-axis, and (b) about the y-axis?

Question 78

The lug nuts on a car wheel require tightening to a torque of 90 N ∙ m. If a 30-cm long wrench is used, what is the magnitude of the minimum force required using the wrench?

A) 300 N
B) 150 N
C) 30 N
D) 15 N

Question 79

The L-shaped object shown in the figure consists of three small masses connected by extremely light rods. Assume that the masses shown are accurate to three significant figures. What is the moment of inertia of this object (a) about the x-axis, and (b) about the y-axis?

Question 80

The figure shows a person's foot. In that figure, the Achilles tendon exerts a force of magnitude F = 720 N. What is the magnitude of the torque that this force produces about the ankle joint?

A) 12 N ∙ m
B) 16 N ∙ m
C) 21 N ∙ m
D) 26 N ∙ m
E) 36 N ∙ m