Deck 8: Torque and Angular Momentum

Question 1

A $2.00 \mathrm{~kg}$ mass is located at $(4.00 \mathrm{~m}, 0.00 \mathrm{~m}, 0.00 \mathrm{~m})$ and a $4.00 \mathrm{~kg}$ mass is located at $(0.00 \mathrm{~m}, 3.00 \mathrm{~m}$ , $0.00 \mathrm{~m}$ ). The rotational inertia of this system of masses about the Z-axis, perpendicular to the X-Y plane, is

A) $68 \mathrm{~kg} \mathrm{~m}^{2}$ .
B) $55 \mathrm{~kg} \mathrm{~m}^{2}$ .
C) $62 \mathrm{~kg} \mathrm{~m}^{2}$ .
D) $50 \mathrm{~kg} \mathrm{~m}^{2}$ .
E) $58 \mathrm{~kg} \mathrm{~m}^{2}$ .

Answer 1

$68 \mathrm{~kg} \mathrm{~m}^{2}$ .

Question 2

A $2.00 \mathrm{~kg}$ mass is located at $(4.00 \mathrm{~m}, 0.00 \mathrm{~m}, 0.00 \mathrm{~m})$ and a $4.00 \mathrm{~kg}$ mass is located at $(0.00 \mathrm{~m}, 3.00 \mathrm{~m}$ , $0.00 \mathrm{~m}$ ). The rotational inertia of this system of masses about the $\mathrm{X}$ -axis, perpendicular to the Z-Y plane, is

A) $28 \mathrm{~kg} \mathrm{~m}^{2}$ .
B) $36 \mathrm{~kg} \mathrm{~m}^{2}$ .
C) $33 \mathrm{~kg} \mathrm{~m}^{2}$ .
D) $23 \mathrm{~kg} \mathrm{~m}^{2}$ .
E) $41 \mathrm{~kg} \mathrm{~m}^{2}$ .

Answer 2

$36 \mathrm{~kg} \mathrm{~m}^{2}$ .

Question 3

What is the rotational inertia of a solid iron disk of mass $41.0 \mathrm{~kg}$ with a thickness of $5.00 \mathrm{~cm}$ and radius of $30.0 \mathrm{~cm}$ , about an axis perpendicular to the disk and passing through its center?

A) $0.980 \mathrm{~kg} \mathrm{~m}^{2}$
B) $0.761 \mathrm{~kg} \mathrm{~m}^{2}$
C) $2.29 \mathrm{~kg} \mathrm{~m}^{2}$
D) $1.85 \mathrm{~kg} \mathrm{~m}^{2}$

Answer 3

$1.85 \mathrm{~kg} \mathrm{~m}^{2}$

Question 4

A $4.00 \mathrm{~kg}$ mass is located at $(2.00 \mathrm{~m}, 2.00 \mathrm{~m}, 0.00 \mathrm{~m})$ and a $3.00 \mathrm{~kg}$ mass is located at $(-1.0 \mathrm{~m}, 3.00$ $\mathrm{m}, 0.00 \mathrm{~m}$ ). The rotational inertia of this system of masses about the X-axis, perpendicular to the Z-Y plane, is

A) $24 \mathrm{~kg} \mathrm{~m}^{2}$ .
B) $56 \mathrm{~kg} \mathrm{~m}^{2}$ .
C) $62 \mathrm{~kg} \mathrm{~m}^{2}$ .
D) $43 \mathrm{~kg} \mathrm{~m}^{2}$ .
E) $36 \mathrm{~kg} \mathrm{~m}^{2}$ .

Question 5

A $4.00 \mathrm{~kg}$ mass is located at $(2.00 \mathrm{~m}, 2.00 \mathrm{~m}, 0.00 \mathrm{~m})$ and a $3.00 \mathrm{~kg}$ mass is located at $(-1 \mathrm{~m}, 3.00$ $\mathrm{m}, 0.00 \mathrm{~m}$ ). The rotational inertia of this system of masses about the Y-axis, perpendicular to the X-Z plane, is

A) $32 \mathrm{~kg} \mathrm{~m}^{2}$ .
B) $29 \mathrm{~kg} \mathrm{~m}^{2}$ .
C) $40 \mathrm{~kg} \mathrm{~m}^{2}$ .
D) $24 \mathrm{~kg} \mathrm{~m}^{2}$ .
E) $19 \mathrm{~kg} \mathrm{~m}^{2}$ .

Question 6

A $6.00 \mathrm{~kg}$ mass is located at $(2.00 \mathrm{~m}, 2.00 \mathrm{~m}, 2.00 \mathrm{~m})$ and a $5.00 \mathrm{~kg}$ mass is located at $(-1.0 \mathrm{~m}, 3.00 \mathrm{~m}$ , $-2.00 \mathrm{~m}$ ). The rotational inertia of this system of masses about the Z-axis, perpendicular to the $\mathrm{X}-\mathrm{Y}$ plane, is

A) $98 \mathrm{~kg} \mathrm{~m}^{2}$ .
B) $79 \mathrm{~kg} \mathrm{~m}^{2}$ .
C) $60 \mathrm{~kg} \mathrm{~m}^{2}$ .
D) $112 \mathrm{~kg} \mathrm{~m}^{2}$ .
E) $85 \mathrm{~kg} \mathrm{~m}^{2}$ .

Question 7

A $6.00 \mathrm{~kg}$ mass is located at $(2.00 \mathrm{~m}, 2.00 \mathrm{~m}, 2.00 \mathrm{~m})$ and a $5.00 \mathrm{~kg}$ mass is located at $(-1.0 \mathrm{~m}, 3.00 \mathrm{~m}$ , $-2.00 \mathrm{~m}$ ). The rotational inertia of this system of masses about the X-axis, perpendicular to the Z-Y plane, is

A) $167 \mathrm{~kg} \mathrm{~m}^{2}$ .
B) $281 \mathrm{~kg} \mathrm{~m}^{2}$ .
C) $113 \mathrm{~kg} \mathrm{~m}^{2}$ .
D) $69 \mathrm{~kg} \mathrm{~m}^{2}$ .
E) $85 \mathrm{~kg} \mathrm{~m}^{2}$ .

Question 8

A $6.00 \mathrm{~kg}$ mass is located at $(2.00 \mathrm{~m}, 2.00 \mathrm{~m}, 2.00 \mathrm{~m})$ and a $5.00 \mathrm{~kg}$ mass is located at $(-1.0 \mathrm{~m}, 3.00 \mathrm{~m}$ , $-2.00 \mathrm{~m}$ ). The rotational inertia of this system of masses about the Y-axis, perpendicular to the Z-X plane, is

A) $55 \mathrm{~kg} \mathrm{~m}^{2}$ .
B) $73 \mathrm{~kg} \mathrm{~m}^{2}$ .
C) $60 \mathrm{~kg} \mathrm{~m}^{2}$ .
D) $66 \mathrm{~kg} \mathrm{~m}^{2}$ .
E) $48 \mathrm{~kg} \mathrm{~m}^{2}$ .

Question 9

The rotational inertia of a thin rod about one end is $1 / 3 \mathrm{ML}^{2}$ . What is the rotational inertia of the same rod about a point located $0.300 \mathrm{~L}$ from the end?

A) $0.300 \mathrm{ML}^{2}$
B) $0.205 \mathrm{ML}^{2}$
C) $0.240 \mathrm{ML}^{2}$
D) $0.198 \mathrm{ML}^{2}$
E) $0.123 \mathrm{ML}^{2}$

Question 10

The rotational inertia of a thin rod about one end is $1 / 3 \mathrm{ML}^{2}$ . What is the rotational inertia of the same rod about a point located $0.40 \mathrm{~L}$ from the end?

A) $0.093 \mathrm{ML}^{2}$
B) $0.493 \mathrm{ML}^{2}$
C) $0.073 \mathrm{ML}^{2}$
D) $0.243 \mathrm{ML}^{2}$
E) $0.056 \mathrm{ML}^{2}$

Question 11

A $30.0 \mathrm{~cm}$ wrench is used to generate a torque at a bolt. A force of $40 \mathrm{~N}$ is applied perpendicularly at the end of the wrench. The torque generated at the bolt is

A) $9.0 \mathrm{~N} \cdot \mathrm{m}$ .
B) $7.0 \mathrm{~N} \cdot \mathrm{m}$ .
C) $12 \mathrm{~N} \cdot \mathrm{m}$ .
D) $20 \mathrm{~N} \cdot \mathrm{m}$ .
E) $5.0 \mathrm{~N} \cdot \mathrm{m}$ .

Question 12

A $30.0 \mathrm{~cm}$ wrench is used to generate a torque at a bolt. A force of $50.0 \mathrm{~N}$ is applied at the end of the wrench at an angle of 70.0 degrees. The torque generated at the bolt is

A) $10.4 \mathrm{~N} \cdot \mathrm{m}$ .
B) $14.1 \mathrm{~N} \cdot \mathrm{m}$ .
C) $19.7 \mathrm{~N} \cdot \mathrm{m}$ .
D) $26.2 \mathrm{~N} \cdot \mathrm{m}$ .
E) $21.5 \mathrm{~N} \cdot \mathrm{m}$ .

Question 13

A torque of $20.0 \mathrm{~N} \cdot \mathrm{m}$ is applied to a bolt. The bolt rotates through an angle of 180 degrees. The work done in turning the bolt is

A) $49.9 \mathrm{~J}$ .
B) $62.8 \mathrm{~J}$ .
C) $72.5 \mathrm{~J}$ .
D) $51.9 \mathrm{~J}$ .
E) $58.4 \mathrm{~J}$

Question 14

A torque of $15.0 \mathrm{~N} \cdot \mathrm{m}$ is applied to a bolt. The bolt rotates through an angle of 360 degrees. The work done in turning the bolt is

A) $96.7 \mathrm{~J}$ .
B) $91.3 \mathrm{~J}$ .
C) $98.1 \mathrm{~J}$ .
D) $89.9 \mathrm{~J}$ .
E) $94.2 \mathrm{~J}$ .

Question 15

A $2.00 \mathrm{~kg}$ mass is located at $(4.00 \mathrm{~m}, 0.00 \mathrm{~m}, 0.00 \mathrm{~m})$ and a $4.00 \mathrm{~kg}$ mass is located at $(0.00 \mathrm{~m}, 3.00 \mathrm{~m}$ , $0.00 \mathrm{~m}$ ). The center of gravity of the system of masses is

A) $(1.33 \mathrm{~m}, 2.00 \mathrm{~m}, 0)$ .
B) $(1.33 \mathrm{~m}, 1.00 \mathrm{~m}, 0)$ .
C) $(2.00 \mathrm{~m}, 1.33 \mathrm{~m}, 0)$ .
D) $(1.50 \mathrm{~m}, 1.33 \mathrm{~m}, 0)$ .
E) $(1.33 \mathrm{~m}, 1.50 \mathrm{~m}, 0)$ .

Question 16

A $5.00 \mathrm{~kg}$ mass is located at $(2.00 \mathrm{~m}, 0.00 \mathrm{~m}, 0.00 \mathrm{~m})$ and a $3.00 \mathrm{~kg}$ mass is located at $(0.00 \mathrm{~m}, 4.00 \mathrm{~m}$ , $0.00 \mathrm{~m}$ ). The center of gravity of the system of masses is

A) $(1.00 \mathrm{~m}, 1.00 \mathrm{~m}, 0)$ .
B) $(1.50 \mathrm{~m}, 1.25 \mathrm{~m}, 0)$ .
C) $(1.25 \mathrm{~m}, 1.50 \mathrm{~m}, 0)$ .
D) $(1.25 \mathrm{~m}, 1.25 \mathrm{~m}, 0)$ .
E) $(1.50 \mathrm{~m}, 1.50 \mathrm{~m}, 0)$ .

Question 17

A $5.00 \mathrm{~kg}$ mass is located at ( $2.00 \mathrm{~m}, 0.00 \mathrm{~m}, 3.00 \mathrm{~m})$ and a $2.00 \mathrm{~kg}$ mass is located at $(0.00 \mathrm{~m}, 4.00 \mathrm{~m}$ , $-2.00 \mathrm{~m}$ ). The center of gravity of the system of masses is

A) $(7 / 7 \mathrm{~m}, 10 / 7 \mathrm{~m}, 11 / 7 \mathrm{~m})$ .
B) $(11 / 7 \mathrm{~m}, 7 / 7 \mathrm{~m}, 8 / 7 \mathrm{~m})$ .
C) $(8 / 7 \mathrm{~m}, 7 / 7 \mathrm{~m}, 10 / 7 \mathrm{~m})$ .
D) $(10 / 7 \mathrm{~m}, 7 / 7 \mathrm{~m}, 8 / 7 \mathrm{~m})$ .
E) $(10 / 7 \mathrm{~m}, 8 / 7 \mathrm{~m}, 11 / 7 \mathrm{~m})$ .

Question 18

A $5.00 \mathrm{~kg}$ mass is located at $(1.0 \mathrm{~m}, 0.00 \mathrm{~m}, 3.00 \mathrm{~m})$ , a $2.00 \mathrm{~kg}$ mass is located at $(0.00 \mathrm{~m}, 3.00 \mathrm{~m},-2.00$ $\mathrm{m})$ , and a $3.00 \mathrm{~kg}$ mass is located at $(-1.0 \mathrm{~m},-2.00 \mathrm{~m}, 0.00 \mathrm{~m})$ . The center of gravity of the system of masses is

A) $(10 / 10 \mathrm{~m}, 2 / 10 \mathrm{~m}, 3 / 10 \mathrm{~m})$ .
B) $(1 / 10 \mathrm{~m}, 10 / 10 \mathrm{~m}, 1 / 10 \mathrm{~m})$ .
C) $(3 / 10 \mathrm{~m}, 2 / 10 \mathrm{~m}, 10 / 10 \mathrm{~m})$ .
D) $(2 / 10 \mathrm{~m}, 0 \mathrm{~m}, 11 / 10 \mathrm{~m})$ .
E) $(2 / 10 \mathrm{~m}, 10 / 10 \mathrm{~m}, 0 \mathrm{~m})$ .

Question 19

A $6.00 \mathrm{~kg}$ mass is located at $(1.0 \mathrm{~m},-2.00 \mathrm{~m}, 3.00 \mathrm{~m})$ , a $5.00 \mathrm{~kg}$ mass is located at $(1.0 \mathrm{~m}, 3.00 \mathrm{~m},-2.00$ $\mathrm{m})$ , and a $4.00 \mathrm{~kg}$ mass is located at $(-1.0 \mathrm{~m},-2.00 \mathrm{~m}, 2.00 \mathrm{~m})$ . The center of gravity of the system of masses is

A) $(5 / 15 \mathrm{~m},-17 / 15 \mathrm{~m}, 5 / 15 \mathrm{~m})$ .
B) $(12 / 15 \mathrm{~m},-5 / 15 \mathrm{~m}, 16 / 15 \mathrm{~m})$ .
C) $(7 / 15 \mathrm{~m},-5 / 15 \mathrm{~m}, 16 / 15 \mathrm{~m})$ .
D) $(16 / 15 \mathrm{~m},-1 / 15 \mathrm{~m}, 17 / 15 \mathrm{~m})$ .
E) $(5 / 15 \mathrm{~m},-1 / 15 \mathrm{~m}, 17 / 15 \mathrm{~m})$ .

Question 20

A $1.500 \mathrm{~m}$ long uniform beam of mass $30.00 \mathrm{~kg}$ is supported by a wire as shown in the figure. The beam makes an angle of 10.00 degrees with the horizontal and the wire makes an angle of 30.00 degrees with the beam. A $50.00 \mathrm{~kg}$ mass, $\mathrm{m}$ , is attached to the end of the beam. What is the tension in the wire?

A) $1,855 \mathrm{~N}$
B) $1,255 \mathrm{~N}$
C) $1,435 \mathrm{~N}$
D) $2,034 \mathrm{~N}$
E) $1,035 \mathrm{~N}$

Question 21

A $1.500 \mathrm{~m}$ long uniform beam of mass $30.00 \mathrm{~kg}$ is supported by a wire as shown in the figure. The beam makes an angle of 10.00 degrees with the horizontal and the wire makes an angle of 30.00 degrees with the beam. A $50.00 \mathrm{~kg}$ mass, $\mathrm{m}$ , is attached to the end of the beam. What are the vertical and horizontal components of the force of the wall on the beam at the hinge?

A) $\mathrm{H}=1,458 \mathrm{~N}, \mathrm{~V}=454.0 \mathrm{~N}$
B) $\mathrm{H}=750 \mathrm{~N}, \mathrm{~V}=297.3 \mathrm{~N}$
C) $\mathrm{H}=1,300 \mathrm{~N}, \mathrm{~V}=403.4 \mathrm{~N}$
D) $\mathrm{H}=979 \mathrm{~N}, \mathrm{~V}=324.5 \mathrm{~N}$
E) $\mathrm{H}=1,179 \mathrm{~N}, \mathrm{~V}=354.9 \mathrm{~N}$

Question 22

A $75.0 \mathrm{~kg}$ ladder that is $3.00 \mathrm{~m}$ long is placed against a wall at an angle theta. The center of gravity of the ladder is at a point $1.20 \mathrm{~m}$ from the base of the ladder. The coefficient of static friction at the base of the ladder is 0.800 . There is no friction between the wall and the ladder. What is the vertical force of the ground on the ladder?

A) $735 \mathrm{~N}$
B) $832 \mathrm{~N}$
C) $625 \mathrm{~N}$
D) $900 \mathrm{~N}$
E) $640 \mathrm{~N}$

Question 23

A $75.0 \mathrm{~kg}$ ladder that is $3.00 \mathrm{~m}$ long is placed against a wall at an angle theta. The center of gravity of the ladder is at a point $1.20 \mathrm{~m}$ from the base of the ladder. The coefficient of static friction at the base of the ladder is 0.800 . There is no friction between the wall and the ladder. What is the minimum angle the ladder must make with the horizontal for the ladder not to slip and fall?

A) 40.55 degrees
B) 30.34 degrees
C) 26.57 degrees
D) 36.35 degrees
E) 46.52 degrees

Question 24

A $75.0 \mathrm{~kg}$ ladder that is $3.00 \mathrm{~m}$ long is placed against a wall at an angle theta. The center of gravity of the ladder is at a point $1.2 \mathrm{~m}$ from the base of the ladder. The coefficient of static friction at the base of the ladder is 0.400 . There is no friction between the wall and the ladder. What is the minimum angle the ladder must make with the horizontal for the ladder not to slip and fall?

A) 35 degrees
B) 53 degrees
C) 45 degrees
D) 60 degrees
E) 65 degrees

Question 25

A $10 \mathrm{~kg}$ object has a moment of inertia of $1.25 \mathrm{~kg} \mathrm{~m}^{2}$ . If a torque of $2.5 \mathrm{~N} \cdot \mathrm{m}$ is applied to the object, the angular acceleration is

A) $2.0 \mathrm{rad} / \mathrm{s}^{2}$ .
B) $4.0 \mathrm{rad} / \mathrm{s}^{2}$ .
C) $8.0 \mathrm{rad} / \mathrm{s}^{2}$ .
D) $10 \mathrm{rad} / \mathrm{s}^{2}$ .
E) $6.0 \mathrm{rad} / \mathrm{s}^{2}$ .

Question 26

A $8.0 \mathrm{~kg}$ object has a moment of inertia of $1.00 \mathrm{~kg} \mathrm{~m}^{2}$ . What torque is needed to give the object an angular acceleration of $1.5 \mathrm{rad} / \mathrm{s} 2$ ?

A) $1.0 \mathrm{~N} \cdot \mathrm{m}$
B) $2.5 \mathrm{~N} \cdot \mathrm{m}$
C) $2.0 \mathrm{~N} \cdot \mathrm{m}$
D) $3.0 \mathrm{~N} \cdot \mathrm{m}$
E) $1.5 \mathrm{~N} \cdot \mathrm{m}$

Question 27

A $10 \mathrm{~kg}$ sphere with a $25.0 \mathrm{~cm}$ radius has a moment of inertia of $2 / 5 \mathrm{MR} 2$ . If a torque of $2.0 \mathrm{~N} \cdot \mathrm{m}$ is applied to the object, the angular acceleration is

A) $2.0 \mathrm{rad} / \mathrm{s}^{2}$ .
B) $4.0 \mathrm{rad} / \mathrm{s}^{2}$ .
C) $8.0 \mathrm{rad} / \mathrm{s}^{2}$ .
D) $1.0 \mathrm{rad} / \mathrm{s}^{2}$ .
E) $6.0 \mathrm{rad} / \mathrm{s}^{2}$ .

Question 28

An $8.00 \mathrm{~kg}$ object has a moment of inertia of $1.50 \mathrm{~kg} \mathrm{~m}^{2}$ . If a torque of $2.00 \mathrm{~N} \cdot \mathrm{m}$ is applied to the object, the angular acceleration is

A) $1.33 \mathrm{rad} / \mathrm{s}^{2}$ .
B) $1.00 \mathrm{rad} / \mathrm{s}^{2}$ .
C) $2.01 \mathrm{rad} / \mathrm{s}^{2}$ .
D) $0.750 \mathrm{rad} / \mathrm{s}^{2}$ .
E) $2.67 \mathrm{rad} / \mathrm{s}^{2}$ .

Question 29

A $5.00 \mathrm{~kg}$ object has a moment of inertia of $1.20 \mathrm{~kg} \mathrm{~m}^{2}$ . What torque is needed to give the object an angular acceleration of $2.0 \mathrm{rad} / \mathrm{s}^{2}$ ?

A) $3.0 \mathrm{~N} \cdot \mathrm{m}$
B) $2.6 \mathrm{~N} \cdot \mathrm{m}$
C) $2.4 \mathrm{~N} \cdot \mathrm{m}$
D) $3.2 \mathrm{~N} \cdot \mathrm{m}$
E) $2.8 \mathrm{~N} \cdot \mathrm{m}$

Question 30

A $10 \mathrm{~kg}$ solid cylinder with a $50.0 \mathrm{~cm}$ radius has a moment of inertia of $1 / 2 \mathrm{MR}^{2}$ . If a torque of $2.0 \mathrm{~N} \bullet \mathrm{m}$ is applied to the object, the angular acceleration is

A) $1.8 \mathrm{rad} / \mathrm{s}^{2}$ .
B) $1.6 \mathrm{rad} / \mathrm{s}^{2}$ .
C) $2.3 \mathrm{rad} / \mathrm{s}^{2}$ .
D) $1.0 \mathrm{rad} / \mathrm{s}^{2}$ .
E) $2.1 \mathrm{rad} / \mathrm{s}^{2}$ .

Question 31

A torque of $2.00 \mathrm{~N} \cdot \mathrm{m}$ is applied to a $10.0 \mathrm{~kg}$ object to give it an angular acceleration. If the angular acceleration is $1.75 \mathrm{rad} / \mathrm{s}^{2}$ , then the moment of inertia is

A) $1.35 \mathrm{~kg} \mathrm{~m}^{2}$ .
B) $1.05 \mathrm{~kg} \mathrm{~m}^{2}$ .
C) $1.20 \mathrm{~kg} \mathrm{~m}^{2}$ .
D) $1.14 \mathrm{~kg} \mathrm{~m}^{2}$ .
E) $1.95 \mathrm{~kg} \mathrm{~m}^{2}$ .

Question 32

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ sliding on a frictionless horizontal surface as shown in the figure. There is no slippage between the string and the pulley. The pulley has a radius of $25.0 \mathrm{~cm}$ and a moment of inertia of $1 / 2 \mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $1.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $2.00 \mathrm{~kg}$ , and M is $4.00 \mathrm{~kg}$ , then what is the downward acceleration of $m_{1}$ ?

A) $2.72 \mathrm{~m} / \mathrm{s}^{2}$
B) $1.55 \mathrm{~m} / \mathrm{s}^{2}$
C) $1.96 \mathrm{~m} / \mathrm{s}^{2}$
D) $2.06 \mathrm{~m} / \mathrm{s}^{2}$
E) $2.33 \mathrm{~m} / \mathrm{s}^{2}$

Question 33

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ sliding on a frictionless horizontal surface as shown in the figure. There is no slippage between the string and the pulley. The pulley has a radius of $25.0 \mathrm{~cm}$ and a moment of inertia of $1 / 2 \mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $1.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $2.00 \mathrm{~kg}$ , and M is $4.00 \mathrm{~kg}$ , then what is the tension in the string attached to $m_{1}$ ?

A) $6.83 \mathrm{~N}$
B) $8.02 \mathrm{~N}$
C) $8.33 \mathrm{~N}$
D) $7.03 \mathrm{~N}$
E) $7.84 \mathrm{~N}$

Question 34

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ sliding on a frictionless horizontal surface as shown in the figure. There is no slippage between the string and the pulley. The pulley has a radius of $25.0 \mathrm{~cm}$ and a moment of inertia of $1 / 2 \mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $1.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $2.00 \mathrm{~kg}$ , and M is $4.00 \mathrm{~kg}$ , then what is the tension in the string attached to $\mathrm{m}_{2}$ ?

A) $2.98 \mathrm{~m} / \mathrm{s}^{2}$
B) $3.23 \mathrm{~m} / \mathrm{s}^{2}$
C) $3.02 \mathrm{~m} / \mathrm{s}^{2}$
D) $3.92 \mathrm{~m} / \mathrm{s}^{2}$
E) $3.65 \mathrm{~m} / \mathrm{s}^{2}$

Question 35

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ sliding on a frictionless horizontal surface as shown in the figure. There is no slippage between the string and the pulley. The pulley has a radius of $25 \mathrm{~cm}$ and a moment of inertia of $1 / 2 \mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $4.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $2.00 \mathrm{~kg}$ , and $\mathrm{M}$ is $4.00 \mathrm{~kg}$ , then what is the downward acceleration of $\mathrm{m}_{1}$ ?

A) $4.9 \mathrm{~m} / \mathrm{s}^{2}$
B) $4.5 \mathrm{~m} / \mathrm{s}^{2}$
C) $3.7 \mathrm{~m} / \mathrm{s}^{2}$
D) $4.1 \mathrm{~m} / \mathrm{s}^{2}$
E) $3.9 \mathrm{~m} / \mathrm{s}^{2}$

Question 36

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ sliding on a frictionless horizontal surface as shown in the figure. There is no slippage between the string and the pulley. The pulley has a radius of $25.0 \mathrm{~cm}$ and a moment of inertia of $1 / 2 \mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $4.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $4.00 \mathrm{~kg}$ , and $\mathrm{M}$ is $4.00 \mathrm{~kg}$ , then what is the downward acceleration of $m_{1}$ ?

A) $3.92 \mathrm{~m} / \mathrm{s}^{2}$
B) $3.04 \mathrm{~m} / \mathrm{s}^{2}$
C) $2.96 \mathrm{~m} / \mathrm{s}^{2}$
D) $4.42 \mathrm{~m} / \mathrm{s}^{2}$
E) $3.42 \mathrm{~m} / \mathrm{s}^{2}$

Question 37

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ sliding on a frictionless horizontal surface as shown in the figure. There is no slippage between the string and the pulley. The pulley has a radius of $25.0 \mathrm{~cm}$ and a moment of inertia of $1 / 2 \mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $4.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $4.00 \mathrm{~kg}$ , and M is $4.00 \mathrm{~kg}$ , then what is the tension in the string attached to $\mathrm{m}_{1}$ ?

A) $32.7 \mathrm{~N}$
B) $31.0 \mathrm{~N}$
C) $29.0 \mathrm{~N}$
D) $23.5 \mathrm{~N}$
E) $35.6 \mathrm{~N}$

Question 38

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ sliding on a frictionless horizontal surface as shown in the figure. There is no slippage between the string and the pulley. The pulley has a radius of $25.0 \mathrm{~cm}$ and a moment of inertia of $1 / 2 \mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $4.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $4.00 \mathrm{~kg}$ , and M is $4.00 \mathrm{~kg}$ , then what is the tension in the string attached to $\mathrm{m}_{2}$ ?

A) $12.6 \mathrm{~N}$
B) $15.7 \mathrm{~N}$
C) $19.8 \mathrm{~N}$
D) $10.4 \mathrm{~N}$
E) $17.6 \mathrm{~N}$

Question 39

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ as shown in the figure. Both masses move vertically and there is no slippage between the string and the pulley. The pulley has a radius of $30.0 \mathrm{~cm}$ and a moment of inertia of $\mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $4.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $3.00 \mathrm{~kg}$ and $\mathrm{M}$ is $6.00 \mathrm{~kg}$ , then what is the magnitude of the acceleration of the masses?

A) $0.695 \mathrm{~m} / \mathrm{s}^{2}$
B) $0.754 \mathrm{~m} / \mathrm{s}^{2}$
C) $0.805 \mathrm{~m} / \mathrm{s}^{2}$
D) $0.703 \mathrm{~m} / \mathrm{s}^{2}$
E) $0.731 \mathrm{~m} / \mathrm{s}^{2}$

Question 40

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ as shown in the figure. Both masses move vertically and there is no slippage between the string and the pulley. The pulley has a radius of $30.0 \mathrm{~cm}$ and a moment of inertia of $\mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $4.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $3.00 \mathrm{~kg}$ and $\mathrm{M}$ is $6.00 \mathrm{~kg}$ , then what is the tension in the string that is attached to $m_{1}$ ?

A) $58.2 \mathrm{~N}$
B) $74.5 \mathrm{~N}$
C) $36.2 \mathrm{~N}$
D) $44.6 \mathrm{~N}$
E) $60.6 \mathrm{~N}$

Question 41

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ as shown in the figure. Both masses move vertically and there is no slippage between the string and the pulley. The pulley has a radius of $30.0 \mathrm{~cm}$ and a moment of inertia of $\mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $4.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $3.00 \mathrm{~kg}$ and $\mathrm{M}$ is $6.00 \mathrm{~kg}$ , then what is the tension in the string that is attached to $\mathrm{m}_{2}$ ?

A) $41.3 \mathrm{~N}$
B) $20.7 \mathrm{~N}$
C) $31.7 \mathrm{~N}$
D) $25.5 \mathrm{~N}$
E) $35.2 \mathrm{~N}$

Question 42

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ as shown in the figure. Both masses move vertically and there is no slippage between the string and the pulley. The pulley has a radius of $20.0 \mathrm{~cm}$ and a moment of inertia of $\frac{1}{2} \mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $3.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $6.00 \mathrm{~kg}$ and $\mathrm{M}$ is $4.00 \mathrm{~kg}$ , then what is the magnitude of the acceleration of the masses?

A) $2.67 \mathrm{~m} / \mathrm{s}^{2}$
B) $4.05 \mathrm{~m} / \mathrm{s}^{2}$
C) $5.05 \mathrm{~m} / \mathrm{s}^{2}$
D) $3.44 \mathrm{~m} / \mathrm{s}^{2}$
E) $4.75 \mathrm{~m} / \mathrm{s}^{2}$

Question 43

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ as shown in the figure. Both masses move vertically and there is no slippage between the string and the pulley. The pulley has a radius of $20.0 \mathrm{~cm}$ and a moment of inertia of $1 / 2 \mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $3.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $6.00 \mathrm{~kg}$ and $\mathrm{M}$ is $4.00 \mathrm{~kg}$ , then what is the tension in the string that is attached to mass $\mathrm{m}_{1}$ ?

A) $27.4 \mathrm{~N}$
B) $37.4 \mathrm{~N}$
C) $30.2 \mathrm{~N}$
D) $43.5 \mathrm{~N}$
E) $20.8 \mathrm{~N}$

Question 44

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ as shown in the figure. Both masses move vertically and there is no slippage between the string and the pulley. The pulley has a radius of $20.0 \mathrm{~cm}$ and a moment of inertia of $1 \frac{1}{2} \mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $3.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $6.00 \mathrm{~kg}$ and $\mathrm{M}$ is $4.00 \mathrm{~kg}$ , then what is the tension in the string that is attached to mass $\mathrm{m}_{2}$ ?

A) $63.4 \mathrm{~N}$
B) $42.8 \mathrm{~N}$
C) $33.6 \mathrm{~N}$
D) $53.6 \mathrm{~N}$
E) $75.5 \mathrm{~N}$

Question 45

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ sliding on a frictionless incline as shown in the figure. There is no slippage between the string and the pulley. The pulley has a radius of $25.0 \mathrm{~cm}$ and a moment of inertia of $1 / 2 \mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $2.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $1.00 \mathrm{~kg}, \mathrm{M}$ is $4.00 \mathrm{~kg}$ , and the angle is 60.0 degrees, then what is the acceleration of $m_{1}$ ?

A) $3.27 \mathrm{~m} / \mathrm{s}^{2}$ down
B) $2.94 \mathrm{~m} / \mathrm{s}^{2}$ down
C) $1.64 \mathrm{~m} / \mathrm{s}^{2}$ down
D) $3.98 \mathrm{~m} / \mathrm{s}^{2}$ down
E) $3.15 \mathrm{~m} / \mathrm{s}^{2}$ down

Question 46

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ sliding on a frictionless incline as shown in the figure. There is no slippage between the string and the pulley. The pulley has a radius of $25.0 \mathrm{~cm}$ and a moment of inertia of $1 \frac{1}{2} \mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $1.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $2.00 \mathrm{~kg}, \mathrm{M}$ is $4.00 \mathrm{~kg}$ , and the angle is 60.0 degrees, then what is the acceleration of $m_{1}$ ?

A) $0.00 \mathrm{~m} / \mathrm{s}^{2}$
B) $2.20 \mathrm{~m} / \mathrm{s}^{2}$
C) $1.20 \mathrm{~m} / \mathrm{s}^{2}$
D) $1.80 \mathrm{~m} / \mathrm{s}^{2}$
E) $2.80 \mathrm{~m} / \mathrm{s}^{2}$

Question 47

A mass $m_{1}$ is connected by a light string that passes over a pulley of mass $M$ to a mass $m_{2}$ sliding on a frictionless incline as shown in the figure. There is no slippage between the string and the pulley. The pulley has a radius of $30.0 \mathrm{~cm}$ and a moment of inertia of $\mathrm{MR}^{2}$ . If $\mathrm{m}_{1}$ is $4.00 \mathrm{~kg}, \mathrm{~m}_{2}$ is $4.00 \mathrm{~kg}, \mathrm{M}$ is $4.00 \mathrm{~kg}$ , and the angle is 70.0 degrees, then what is the acceleration of $m_{1}$ ?

A) $2.15 \mathrm{~m} / \mathrm{s}^{2}$ down
B) $3.10 \mathrm{~m} / \mathrm{s}^{2}$ down
C) $2.43 \mathrm{~m} / \mathrm{s}^{2}$ down
D) $2.89 \mathrm{~m} / \mathrm{s}^{2}$ down
E) $1.49 \mathrm{~m} / \mathrm{s}^{2}$ down

Question 48

A $4.00 \mathrm{~kg}$ solid sphere ( $\left.\mathrm{I}=2 / 5 \mathrm{MR}^{2}\right)$ is spinning with an angular velocity of $23.0 \mathrm{rad} / \mathrm{s}$ . The diameter of the sphere is $20.0 \mathrm{~cm}$ . The rotational kinetic energy of the spinning sphere is

A) $4.23 \mathrm{~J}$ .
B) $3.02 \mathrm{~J}$ .
C) $3.52 \mathrm{~J}$ .
D) $3.75 \mathrm{~J}$ .
E) $4.02 \mathrm{~J}$

Question 49

A $20.0 \mathrm{~kg}$ hollow cylinder $\left(\mathrm{I}=\mathrm{MR}^{2}\right)$ has a diameter of $50.0 \mathrm{~cm}$ . The cylinder is rolling down a hill with a velocity of $5.00 \mathrm{~m} / \mathrm{s}$ . The rotational kinetic energy of the rolling cylinder is

A) $225 \mathrm{~J}$ .
B) $250 \mathrm{~J}$ .
C) $150 \mathrm{~J}$ .
D) $175 \mathrm{~J}$ .
E) $200 \mathrm{~J}$ .

Question 50

A $4.00 \mathrm{~kg}$ hollow sphere $\left(\mathrm{I}=2 / 3 \mathrm{MR}^{2}\right)$ is spinning with an angular velocity of $10.0 \mathrm{rad} / \mathrm{s}$ . The diameter of the sphere is $20.0 \mathrm{~cm}$ . The rotational kinetic energy of the spinning sphere is

A) $1.50 \mathrm{~J}$ .
B) $0.90 \mathrm{~J}$ .
C) $1.75 \mathrm{~J}$ .
D) $1.33 \mathrm{~J}$ .
E) $0.75 \mathrm{~J}$ .

Question 51

A $4.00 \mathrm{~kg}$ hollow sphere of radius $5.00 \mathrm{~cm}$ starts from rest and rolls without slipping down a 30.0 degree incline. The acceleration of the center of mass of the hollow sphere is

A) $2.50 \mathrm{~m} / \mathrm{s}^{2}$ .
B) $2.00 \mathrm{~m} / \mathrm{s} 2$ .
C) $2.22 \mathrm{~m} / \mathrm{s}^{2}$ .
D) $2.94 \mathrm{~m} / \mathrm{s}^{2}$ .
E) $2.64 \mathrm{~m} / \mathrm{s}^{2}$ .

Question 52

A $4.00 \mathrm{~kg}$ hollow cylinder of radius $5.00 \mathrm{~cm}$ starts from rest and rolls without slipping down a 30.0 degree incline. The acceleration of the center of mass of the cylinder is

A) $2.98 \mathrm{~m} / \mathrm{s}^{2}$ .
B) $3.98 \mathrm{~m} / \mathrm{s}^{2}$ .
C) $4.05 \mathrm{~m} / \mathrm{s}^{2}$
D) $2.45 \mathrm{~m} / \mathrm{s}^{2}$ .
E) $3.35 \mathrm{~m} / \mathrm{s}^{2}$ .

Question 53

A $4.00 \mathrm{~kg}$ hollow cylinder of radius $5.00 \mathrm{~cm}$ starts from rest and rolls without slipping down a 30.0 degree incline. If the length of the incline is $50.0 \mathrm{~cm}$ , then the velocity of the center of mass of the cylinder at the bottom of the incline is

A) $3.02 \mathrm{~m} / \mathrm{s}$ .
B) $1.35 \mathrm{~m} / \mathrm{s}$ .
C) $2.55 \mathrm{~m} / \mathrm{s}$ .
D) $1.82 \mathrm{~m} / \mathrm{s}$ .
E) $1.57 \mathrm{~m} / \mathrm{s}$

Question 54

A $4.00 \mathrm{~kg}$ solid sphere of radius $5.00 \mathrm{~cm}$ starts from rest and rolls without slipping down a 30.0 degree incline. The acceleration of the center of mass of the solid sphere is

A) $3.50 \mathrm{~m} / \mathrm{s}^{2}$ .
B) $3.00 \mathrm{~m} / \mathrm{s}^{2}$ .
C) $2.00 \mathrm{~m} / \mathrm{s}^{2}$ .
D) $2.50 \mathrm{~m} / \mathrm{s}^{2}$ .
E) $1.50 \mathrm{~m} / \mathrm{s}^{2}$ .

Question 55

A $2.00 \mathrm{~kg}$ solid sphere $\left(\mathrm{I}=2 / 5 \mathrm{MR}^{2}\right)$ with a diameter of $50.0 \mathrm{~cm}$ is rotating at an angular velocity of 5.0 $\mathrm{rad} / \mathrm{s}$ . The angular momentum of the rotating sphere is

A) $0.48 \mathrm{~kg} \mathrm{~m}^{2} / \mathrm{s}$ .
B) $0.25 \mathrm{~kg} \mathrm{~m}^{2} / \mathrm{s}$ .
C) $0.55 \mathrm{~kg} \mathrm{~m}^{2} / \mathrm{s}$ .
D) $0.37 \mathrm{~kg} \mathrm{~m}^{2} / \mathrm{s}$ .
E) $0.20 \mathrm{~kg} \mathrm{~m}^{2} / \mathrm{s}$ .

Question 56

A grinding wheel has a mass of $250 \mathrm{~kg}$ and moment of inertia of $500 \mathrm{~kg} \mathrm{~m}^{2}$ . A torque of $100 \mathrm{~N} \cdot \mathrm{m}$ is applied to the grinding wheel. If the wheel starts from rest, what is the angular momentum of the wheel after 5.0 seconds?

A) $650 \mathrm{~kg} \mathrm{~m}^{2} / \mathrm{s}$
B) $500 \mathrm{~kg} \mathrm{~m}^{2} / \mathrm{s}$
C) $250 \mathrm{~kg} \mathrm{~m}^{2} / \mathrm{s}$
D) $450 \mathrm{~kg} \mathrm{~m}^{2} / \mathrm{s}$
E) $300 \mathrm{~kg} \mathrm{~m}^{2} / \mathrm{s}$

Question 57

A 1.5m long dowel (a cylindrical rod) is pivoted about the end so that it is a pendulum of sorts - it can freely swing in a vertical plane. The dowel has a diameter of $2.0 \mathrm{~cm}$ , and its density is $3.2 \mathrm{~g} / \mathrm{cm} 3$ . When it is positioned in its swing such that its angle with the vertical is 22.5 degrees, what is the torque on the rod about the pivot due to its weight?

A) $1.7 \mathrm{~N} \cdot \mathrm{m}$
B) $1.0 \mathrm{~N} \cdot \mathrm{m}$
C) $4.2 \mathrm{~N} \cdot \mathrm{m}$
D) $4.4 \mathrm{~N} \cdot \mathrm{m}$
E) $1.1 \mathrm{~N} \cdot \mathrm{m}$
F) $4.1 \mathrm{~N} \cdot \mathrm{m}$

Question 58

A $1.5 \mathrm{~m}$ long, $0.75 \mathrm{~kg}$ dowel is pivoted about the end so that it is a pendulum of sorts - it can freely swing in a vertical plane. At the other end of the dowel is a brass weight having mass $1.5 \mathrm{~kg}$ . (The center of the brass weight is $1.5 \mathrm{~m}$ from the pivot point.) When the rod is positioned in its swing such that its angle with the vertical is 22.5 degrees, what is the net torque on the rod about the pivot?

A) $28 \mathrm{~N} \cdot \mathrm{m}$
B) $11 \mathrm{~N} \cdot \mathrm{m}$
C) $25 \mathrm{~N} \cdot \mathrm{m}$
D) $10.5 \mathrm{~N} \cdot \mathrm{m}$

Question 59

A $55 \mathrm{~kg}$ girl swings on a swing, whose seat is attached to the pivot by $2.5 \mathrm{~m}$ long rigid rods (considered to be massless in this problem). As she swings, she rises to a maximum height such that the angle of the rods with respect to the vertical is 32 degrees. What is the maximum torque on the rods due to her weight, as she moves during one cycle of her swinging from the bottom of her swing path to the highest point?

A) $2400 \mathrm{~N} \cdot \mathrm{m}$
B) $710 \mathrm{~N} \cdot \mathrm{m}$
C) $2000 \mathrm{~N} \cdot \mathrm{m}$
D) $970 \mathrm{~N} \cdot \mathrm{m}$

Question 60

A $0.50 \mathrm{~kg}$ solid disk spins at $250 \mathrm{rpm}$ . A torque of $12.5 \mathrm{~N} \cdot \mathrm{m}$ is applied for $0.150 \mathrm{~s}$ to bring it to rest. What is the disk's radius?

A) $0.29 \mathrm{~m}$
B) $0.38 \mathrm{~m}$
C) $0.14 \mathrm{~m}$
D) $0.54 \mathrm{~m}$

Question 61

A person rides a bicycle along a straight road. From the point of view of the rider, what's the direction of the angular momentum vector of the front wheel?

A) left
B) up
C) down
D) backward
E) forward
F) right