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?

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

Q: A 0.25 m wrench is used to tighten a bolt with a required torque of 40 N-m. If the force is applied perpendicular to the wrench, what force is needed?

$$F = \frac{T}{r} = \frac{40\ \text{N-m}}{0.25\ \text{m}} = 160\ \text{N}$$

Q: A seesaw is 2.0 m long and massless. Child A (mass 30 kg) sits at one end. Where should Child B (mass 40 kg) sit to balance the seesaw?

$$r_B = \frac{m_A \times r_A}{m_B} = \frac{30\ \text{kg} \times 2.0\ \text{m}}{40\ \text{kg}} = 1.5\ \text{m}$$

Q: A solid cylinder (mass 10 kg, radius 0.5 m) is rotated about its central axis. What is its moment of inertia?

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} \times 10\ \text{kg} \times (0.5\ \text{m})^2$ = 1.25\ \text{kg-m}^2$$

Q: A thin hoop and a solid sphere both have mass 2 kg and radius 0.3 m. Which has the greater moment of inertia and why?

The thin hoop, because $$I_{\text{hoop}} = m r^2$ and I$$_{\text{sphere}} = \frac{2}{5} m $$r^2$$, so the hoop's mass is farther from the axis.

Q: A solid wheel (mass 5 kg, radius 0.2 m) is accelerated from rest to 10 rad/s in 4 s. What is the angular acceleration?

$$\alpha = \frac{\Delta \omega}{\Delta t} = \frac{10\ \text{rad/s} - 0}{4\ \text{s}} = 2.5\ \text{rad/s}^2$$

Q: A pulley (radius 0.15 m) is pulled with a force of 12 N. What is the torque applied?

$$T = F r = 12\ \text{N} \times 0.15\ \text{m} = 1.8\ \text{N-m}$$

Q: A baton consists of two 0.2 kg masses at each end of a 0.5 m rod (rod mass negligible). What is the moment of inertia about the center?

$$I = 2 \times m r^2$ = 2 \times 0.2\ \text{kg} \times (0.25\ \text{m})^2$ = 0.025\ \text{kg-m}^2$$

Q: A solid wheel (mass 8 kg, radius 0.3 m) is accelerated at $$\alpha = 2\ \text{rad/s}^2. $$What torque is required?

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} \times 8\ \text{kg} \times (0.3\ \text{m})^2$ = 0.36\ \text{kg-m}^2$$; $$T = I \alpha = 0.36\ \text{kg-m}^2 \times 2\ \text{rad/s}^2 = 0.72\ \text{N-m}$$

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

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

Question 2

A door is 1.2 m wide. To open it with the least force, where and in what direction should you push?

Accepted Answer

Push at the edge farthest from the hinges, perpendicular to the door.

Question 3

A 0.25 m wrench is used to tighten a bolt with a required torque of 40 N-m. If the force is applied perpendicular to the wrench, what force is needed?

Accepted Answer

$$F = \frac{T}{r} = \frac{40\ 	ext{N-m}}{0.25\ 	ext{m}} = 160\ 	ext{N}$$

Question 4

A seesaw is 2.0 m long and massless. Child A (mass 30 kg) sits at one end. Where should Child B (mass 40 kg) sit to balance the seesaw?

Accepted Answer

$$r_B = \frac{m_A 	imes r_A}{m_B} = \frac{30\ 	ext{kg} 	imes 2.0\ 	ext{m}}{40\ 	ext{kg}} = 1.5\ 	ext{m}$$

Question 5

A solid cylinder (mass 10 kg, radius 0.5 m) is rotated about its central axis. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} 	imes 10\ 	ext{kg} 	imes (0.5\ 	ext{m})^2$ = 1.25\ 	ext{kg-m}^2$$

Question 6

A thin hoop and a solid sphere both have mass 2 kg and radius 0.3 m. Which has the greater moment of inertia and why?

Accepted Answer

The thin hoop, because $$I_{	ext{hoop}} = m r^2$ and I$$_{	ext{sphere}} = \frac{2}{5} m $$r^2$$, so the hoop's mass is farther from the axis.

Question 7

A solid wheel (mass 5 kg, radius 0.2 m) is accelerated from rest to 10 rad/s in 4 s. What is the angular acceleration?

Accepted Answer

$$\alpha = \frac{\Delta \omega}{\Delta t} = \frac{10\ 	ext{rad/s} - 0}{4\ 	ext{s}} = 2.5\ 	ext{rad/s}^2$$

Question 8

A pulley (radius 0.15 m) is pulled with a force of 12 N. What is the torque applied?

Accepted Answer

$$T = F r = 12\ 	ext{N} 	imes 0.15\ 	ext{m} = 1.8\ 	ext{N-m}$$

Question 9

A baton consists of two 0.2 kg masses at each end of a 0.5 m rod (rod mass negligible). What is the moment of inertia about the center?

Accepted Answer

$$I = 2 	imes m r^2$ = 2 	imes 0.2\ 	ext{kg} 	imes (0.25\ 	ext{m})^2$ = 0.025\ 	ext{kg-m}^2$$

Question 10

A solid wheel (mass 8 kg, radius 0.3 m) is accelerated at $$\alpha = 2\ 	ext{rad/s}^2. $$What torque is required?

Accepted Answer

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} 	imes 8\ 	ext{kg} 	imes (0.3\ 	ext{m})^2$ = 0.36\ 	ext{kg-m}^2$$; $$T = I \alpha = 0.36\ 	ext{kg-m}^2 	imes 2\ 	ext{rad/s}^2 = 0.72\ 	ext{N-m}$$

Question 11

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

Accepted Answer

$$T = F r \sin 	heta = (70\ 	ext{kg} 	imes 9.8\ 	ext{N/kg}) 	imes 0.18\ 	ext{m} 	imes \sin 30^\circ = 61.74\ 	ext{N-m}$$

Question 12

A pulley (radius 0.2 m) is pulled with a force of 16 N. Its rotational speed increases from 0 to 17 rev/min in 5 s. What is the moment of inertia?

Accepted Answer

$$\omega = 17 	imes \frac{2\pi}{60} = 1.78\ 	ext{rad/s}$$; $$\alpha = \frac{1.78}{5} = 0.356\ 	ext{rad/s}^2$$; $$T = F r = 16 	imes 0.2 = 3.2\ 	ext{N-m}$$; $$I = \frac{T}{\alpha} = \frac{3.2}{0.356} = 9.0\ 	ext{kg-m}^2$$

Question 13

A solid wheel (mass 6 kg, diameter 0.5 m) is at rest. You want it to rotate at 10 rev/s in 20 s. What torque must be applied?

Accepted Answer

$$\omega = 10 	imes 2\pi = 62.8\ 	ext{rad/s}$$; $$\alpha = \frac{62.8}{20} = 3.14\ 	ext{rad/s}^2$$; $$r = 0.25\ 	ext{m}$$; $$I = \frac{1}{2} m r^2$ = 0.5 	imes 6 	imes 0.25^2$ = 0.1875\ 	ext{kg-m}^2$$; $$T = I \alpha = 0.1875 	imes 3.14 = 0.59\ 	ext{N-m}$$

Question 14

Two people pull on ropes wrapped around the edge of a large wheel (mass 12 kg, diameter 2.4 m). 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 15

A thin rectangular plate (mass 3 kg, length 2 m, width 1 m) rotates about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{12} m (l^2$ + w^2$) = \frac{1}{12} 	imes 3 	imes (2^2$ + 1^2$) = \frac{1}{12} 	imes 3 	imes 5 = 1.25\ 	ext{kg-m}^2$$

Question 16

A solid sphere (mass 4 kg, radius 0.2 m) rotates about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{2}{5} m r^2$ = \frac{2}{5} 	imes 4 	imes 0.2^2$ = 0.064\ 	ext{kg-m}^2$$

Question 17

A chain is wrapped around a pulley and pulled with a force of 13 N. The pulley's radius is 0.15 m. Its rotational speed increases from 0 to 14 rev/min in 4.5 s. What is the moment of inertia?

Accepted Answer

$$\omega = 14 	imes \frac{2\pi}{60} = 1.47\ 	ext{rad/s}$$; $$\alpha = \frac{1.47}{4.5} = 0.327\ 	ext{rad/s}^2$$; $$T = 13 	imes 0.15 = 1.95\ 	ext{N-m}$$; $$I = \frac{T}{\alpha} = \frac{1.95}{0.327} = 5.96\ 	ext{kg-m}^2$$

Question 18

A ball on a low-friction, tilted surface slides downhill without rotating. On a rough surface, it rolls. Why?

Accepted Answer

On a rough surface, friction provides a torque that causes rotation; on a smooth surface, no torque acts, so the ball slides.

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

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