Q: A mechanic uses a 0.40 m wrench to loosen a bolt. If she applies a force of 80 N at an angle of 30° to the handle, what is the magnitude of the torque produced?

$$T = (80\ \text{N})(0.40\ \text{m}) \sin 30^\circ = 16\ \text{N-m}$$

Q: A door is 0.9 m wide. If you push with a force of 25 N perpendicular to the door at the edge, what is the torque about the hinges?

$$T = (25\ \text{N})(0.9\ \text{m}) = 22.5\ \text{N-m}$$

Q: A seesaw is balanced with a 60 kg child sitting 1.2 m from the pivot on one side. How far from the pivot should a 40 kg child sit on the other side to balance the seesaw?

$$r = \frac{60\ \text{kg} \times 1.2\ \text{m}}{40\ \text{kg}} = 1.8\ \text{m}$$

Q: A solid wheel of mass 10 kg and radius 0.25 m is accelerated from rest to 10 rad/s in 5 s. What torque is required?

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} \times 10 \times (0.25)^2$ = 0.3125\ \text{kg-m}^2$$; $$\alpha = \frac{10}{5} = 2\ \text{rad/s}^2$$; $$T = I \alpha = 0.3125 \times 2 = 0.625\ \text{N-m}$$

Q: A 0.5 kg mass is attached to the end of a 0.3 m rod and rotated about the opposite end. What is the moment of inertia of the mass about the axis?

$$I = m r^2$ = 0.5 \times (0.3)^2$ = 0.045\ \text{kg-m}^2$$

Q: A solid, uniform cylinder of mass 2 kg and radius 0.5 m is rotated about its center. What is its moment of inertia?

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} \times 2 \times (0.5)^2$ = 0.25\ \text{kg-m}^2$$

Q: A pulley of radius 0.15 m is accelerated by a force of 12 N. If its rotational speed increases from 0 to 10 rev/min in 4 s, what is the moment of inertia of the pulley?

$$\omega = 10 \times \frac{2\pi}{60} = 1.05\ \text{rad/s}$$; $$\alpha = \frac{1.05}{4} = 0.2625\ \text{rad/s}^2$$; $$T = F r = 12 \times 0.15 = 1.8\ \text{N-m}$$; $$I = \frac{T}{\alpha} = \frac{1.8}{0.2625} = 6.86\ \text{kg-m}^2$$

Q: A thin hoop and a solid sphere both have mass 1 kg and radius 2 m. Which has the greater moment of inertia, and by how much?

The thin hoop has a greater moment of inertia by $$I_{\text{hoop}} - I_{\text{sphere}} = 4 - 1.6 = 2.4\ \text{kg-m}^2$$

Q: A solid wheel of mass 5 kg and radius 0.2 m is at rest. If a force of 10 N is applied tangentially to the rim, what is the angular acceleration?

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} \times 5 \times (0.2)^2$ = 0.1\ \text{kg-m}^2$$; $$T = F r = 10 \times 0.2 = 2\ \text{N-m}$$; $$\alpha = \frac{T}{I} = \frac{2}{0.1} = 20\ \text{rad/s}^2$$

Question 1

A mechanic uses a 0.40 m wrench to loosen a bolt. If she applies a force of 80 N at an angle of 30° to the handle, what is the magnitude of the torque produced?

Accepted Answer

$$T = (80\ 	ext{N})(0.40\ 	ext{m}) \sin 30^\circ = 16\ 	ext{N-m}$$

Question 2

Which of the following best describes the lever arm in the context of rotational dynamics?

Accepted Answer

The perpendicular distance from the axis of rotation to the line of action of the force.

Question 3

A door is 0.9 m wide. If you push with a force of 25 N perpendicular to the door at the edge, what is the torque about the hinges?

Accepted Answer

$$T = (25\ 	ext{N})(0.9\ 	ext{m}) = 22.5\ 	ext{N-m}$$

Question 4

A seesaw is balanced with a 60 kg child sitting 1.2 m from the pivot on one side. How far from the pivot should a 40 kg child sit on the other side to balance the seesaw?

Accepted Answer

$$r = \frac{60\ 	ext{kg} 	imes 1.2\ 	ext{m}}{40\ 	ext{kg}} = 1.8\ 	ext{m}$$

Question 5

A solid wheel of mass 10 kg and radius 0.25 m is accelerated from rest to 10 rad/s in 5 s. What torque is required?

Accepted Answer

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} 	imes 10 	imes (0.25)^2$ = 0.3125\ 	ext{kg-m}^2$$; $$\alpha = \frac{10}{5} = 2\ 	ext{rad/s}^2$$; $$T = I \alpha = 0.3125 	imes 2 = 0.625\ 	ext{N-m}$$

Question 6

Which of the following objects has the greatest moment of inertia about its central axis, assuming equal mass and radius?

Accepted Answer

A thin hoop

Question 7

A 0.5 kg mass is attached to the end of a 0.3 m rod and rotated about the opposite end. What is the moment of inertia of the mass about the axis?

Accepted Answer

$$I = m r^2$ = 0.5 	imes (0.3)^2$ = 0.045\ 	ext{kg-m}^2$$

Question 8

A solid, uniform cylinder of mass 2 kg and radius 0.5 m is rotated about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} 	imes 2 	imes (0.5)^2$ = 0.25\ 	ext{kg-m}^2$$

Question 9

A pulley of radius 0.15 m is accelerated by a force of 12 N. If its rotational speed increases from 0 to 10 rev/min in 4 s, what is the moment of inertia of the pulley?

Accepted Answer

$$\omega = 10 	imes \frac{2\pi}{60} = 1.05\ 	ext{rad/s}$$; $$\alpha = \frac{1.05}{4} = 0.2625\ 	ext{rad/s}^2$$; $$T = F r = 12 	imes 0.15 = 1.8\ 	ext{N-m}$$; $$I = \frac{T}{\alpha} = \frac{1.8}{0.2625} = 6.86\ 	ext{kg-m}^2$$

Question 10

A thin hoop and a solid sphere both have mass 1 kg and radius 2 m. Which has the greater moment of inertia, and by how much?

Accepted Answer

The thin hoop has a greater moment of inertia by $$I_{	ext{hoop}} - I_{	ext{sphere}} = 4 - 1.6 = 2.4\ 	ext{kg-m}^2$$

Question 11

A solid wheel of mass 5 kg and radius 0.2 m is at rest. If a force of 10 N is applied tangentially to the rim, what is the angular acceleration?

Accepted Answer

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} 	imes 5 	imes (0.2)^2$ = 0.1\ 	ext{kg-m}^2$$; $$T = F r = 10 	imes 0.2 = 2\ 	ext{N-m}$$; $$\alpha = \frac{T}{I} = \frac{2}{0.1} = 20\ 	ext{rad/s}^2$$

Question 12

A bicycle pedal is 0.18 m from the center of the chain ring. If a cyclist of mass 70 kg stands on the pedal at an angle of 30° above the horizontal, what torque is exerted?

Accepted Answer

$$F = 70 	imes 9.8 = 686\ 	ext{N}$$; $$T = F r \sin 30^\circ = 686 	imes 0.18 	imes 0.5 = 61.74\ 	ext{N-m}$$

Question 13

If you want to open a door with the least amount of force, where and how should you push?

Accepted Answer

Push as far from the hinges as possible, at a right angle to the door.

Question 14

A solid wheel of mass 8 kg and radius 0.3 m is accelerated by a force of 20 N applied tangentially. What is the angular acceleration?

Accepted Answer

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} 	imes 8 	imes (0.3)^2$ = 0.36\ 	ext{kg-m}^2$$; $$T = F r = 20 	imes 0.3 = 6\ 	ext{N-m}$$; $$\alpha = \frac{T}{I} = \frac{6}{0.36} = 16.67\ 	ext{rad/s}^2$$

Question 15

Two people pull on ropes wrapped around the edge of a large wheel (diameter 2.4 m, mass 12 kg). One pulls clockwise with 43 N, the other counterclockwise with 67 N. What is the net torque?

Accepted Answer

$$r = 1.2\ 	ext{m}$$; 	ext{Net torque} = (67 - 43) 	imes 1.2 = 28.8\ 	ext{N-m}$$\ 	ext{counterclockwise}$$

Question 16

A disk with moment of inertia $$0.26\ 	ext{kg-m}^2 is $$attached to a smaller disk (diameter $$0.18\ 	ext{m}$$, mass $$2.5\ 	ext{kg}). $$What force is needed on a strap around the smaller disk to give the system an angular acceleration of $$2.57\ 	ext{rad/s}^2$$?

Accepted Answer

$$I_{	ext{small}} = m r^2$ = 2.5 	imes (0.09)^2$ = 0.02025\ 	ext{kg-m}^2$$; $$I_{	ext{total}} = 0.26 + 0.02025 = 0.28025\ 	ext{kg-m}^2$$; $$T = I \alpha = 0.28025 	imes 2.57 = 0.720\ 	ext{N-m}$$; $$F = \frac{T}{r} = \frac{0.720}{0.09} = 8.0\ 	ext{N}$$

Question 17

A ball rolls down a rough incline but slides down a smooth incline. Why does the ball roll on the rough surface?

Accepted Answer

Friction provides a torque that causes angular acceleration, allowing the ball to roll.

Question 18

A long, uniform rod of mass 2 kg and length 1 m is rotated about its end. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{3} m l^2$ = \frac{1}{3} 	imes 2 	imes (1)^2$ = 0.666\ 	ext{kg-m}^2$$

Rotational Dynamics: Torque, Moment of Inertia, and Newton's Second Law for Rotation

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