Q: A mechanic uses a 0.40 m wrench to loosen a bolt. If he applies a force of 120 N at an angle of 30° to the handle, what is the magnitude of the torque produced?

$$T = Fr \sin \theta = (120\ \text{N})(0.40\ \text{m})\sin 30^\circ = 24\ \text{N-m}$$

Q: A door is 0.9 m wide. If you push with a force of 50 N at the edge, perpendicular to the door, what is the torque about the hinges?

$$T = Fr = (50\ \text{N})(0.9\ \text{m}) = 45\ \text{N-m}$$

Q: A 0.25 m wrench is used to tighten a bolt requiring a 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 (30 kg) sits at one end. Where should Child B (40 kg) sit to balance the seesaw?

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

Q: A solid cylinder (mass 10 kg, radius 0.5 m) rotates about its center. 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 (mass 2 kg, radius 0.3 m) rotates about its central axis. What is its moment of inertia?

$$I = m r^2$ = 2\ \text{kg} \times (0.3\ \text{m})^2$ = 0.18\ \text{kg-m}^2$$

Q: A solid sphere and a thin hoop both have mass 1 kg and radius 0.2 m. Which has the greater moment of inertia and why?

The thin hoop, because $$I_{\text{hoop}} = m r^2$ = 0.04\ \text{kg-m}^2 is $$greater than $$I_{\text{sphere}} = \frac{2}{5} m r^2$ = 0.016\ \text{kg-m}^2$$

Q: A solid wheel (mass 8 kg, radius 0.3 m) starts from rest. What torque is needed to reach an angular velocity of 10 rad/s in 20 s?

$$\alpha = \frac{\Delta \omega}{\Delta t} = \frac{10\ \text{rad/s}}{20\ \text{s}} = 0.5\ \text{rad/s}^2$$ $$I = \frac{1}{2} m r^2$ = \frac{1}{2} \times 8 \times 0.09 = 0.36\ \text{kg-m}^2$$ $$T = I \alpha = 0.36 \times 0.5 = 0.18\ \text{N-m}$$

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

$$\omega_f = 12 \times \frac{2\pi}{60} = 1.26\ \text{rad/s}$$ $$\alpha = \frac{1.26}{4} = 0.315\ \text{rad/s}^2$$ $$T = F r = 12 \times 0.15 = 1.8\ \text{N-m}$$ $$I = \frac{T}{\alpha} = \frac{1.8}{0.315} = 5.7\ \text{kg-m}^2$$

Q: A bicycle wheel (radius 0.38 m) is accelerated by pulling on a string wrapped around the tire. If a force of 15 N is applied to the chain (radius 0.14 m), what force is needed on the string to produce the same torque?

$$T_{\text{chain}} = F_{\text{chain}} r_{\text{chain}} = 15 \times 0.14 = 2.1\ \text{N-m}$$ $$F_{\text{string}} = \frac{T_{\text{chain}}}{r_{\text{tire}}} = \frac{2.1}{0.38} = 5.53\ \text{N}$$

Question 1

A mechanic uses a 0.40 m wrench to loosen a bolt. If he applies a force of 120 N at an angle of 30° to the handle, what is the magnitude of the torque produced?

Accepted Answer

$$T = Fr \sin 	heta = (120\ 	ext{N})(0.40\ 	ext{m})\sin 30^\circ = 24\ 	ext{N-m}$$

Question 2

A door is 0.9 m wide. If you push with a force of 50 N at the edge, perpendicular to the door, what is the torque about the hinges?

Accepted Answer

$$T = Fr = (50\ 	ext{N})(0.9\ 	ext{m}) = 45\ 	ext{N-m}$$

Question 3

A 0.25 m wrench is used to tighten a bolt requiring a 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 (30 kg) sits at one end. Where should Child B (40 kg) sit to balance the seesaw?

Accepted Answer

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

Question 5

A solid cylinder (mass 10 kg, radius 0.5 m) rotates about its center. 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 (mass 2 kg, radius 0.3 m) rotates about its central axis. What is its moment of inertia?

Accepted Answer

$$I = m r^2$ = 2\ 	ext{kg} 	imes (0.3\ 	ext{m})^2$ = 0.18\ 	ext{kg-m}^2$$

Question 7

A solid sphere and a thin hoop both have mass 1 kg and radius 0.2 m. Which has the greater moment of inertia and why?

Accepted Answer

The thin hoop, because $$I_{	ext{hoop}} = m r^2$ = 0.04\ 	ext{kg-m}^2 is $$greater than $$I_{	ext{sphere}} = \frac{2}{5} m r^2$ = 0.016\ 	ext{kg-m}^2$$

Question 8

A solid wheel (mass 8 kg, radius 0.3 m) starts from rest. What torque is needed to reach an angular velocity of 10 rad/s in 20 s?

Accepted Answer

$$\alpha = \frac{\Delta \omega}{\Delta t} = \frac{10\ 	ext{rad/s}}{20\ 	ext{s}} = 0.5\ 	ext{rad/s}^2$$ 
$$I = \frac{1}{2} m r^2$ = \frac{1}{2} 	imes 8 	imes 0.09 = 0.36\ 	ext{kg-m}^2$$ 
$$T = I \alpha = 0.36 	imes 0.5 = 0.18\ 	ext{N-m}$$

Question 9

A pulley (radius 0.15 m) is pulled with a force of 12 N. If its rotational speed increases from 0 to 12 rev/min in 4 s, what is its moment of inertia?

Accepted Answer

$$\omega_f = 12 	imes \frac{2\pi}{60} = 1.26\ 	ext{rad/s}$$ 
$$\alpha = \frac{1.26}{4} = 0.315\ 	ext{rad/s}^2$$ 
$$T = F r = 12 	imes 0.15 = 1.8\ 	ext{N-m}$$ 
$$I = \frac{T}{\alpha} = \frac{1.8}{0.315} = 5.7\ 	ext{kg-m}^2$$

Question 10

A bicycle wheel (radius 0.38 m) is accelerated by pulling on a string wrapped around the tire. If a force of 15 N is applied to the chain (radius 0.14 m), what force is needed on the string to produce the same torque?

Accepted Answer

$$T_{	ext{chain}} = F_{	ext{chain}} r_{	ext{chain}} = 15 	imes 0.14 = 2.1\ 	ext{N-m}$$ 
$$F_{	ext{string}} = \frac{T_{	ext{chain}}}{r_{	ext{tire}}} = \frac{2.1}{0.38} = 5.53\ 	ext{N}$$

Question 11

A baton consists of a thin rod (negligible mass) with two 0.25 kg masses at each end, separated by 0.60 m. What is the moment of inertia about the center?

Accepted Answer

$$I = 2 	imes m r^2$ = 2 	imes 0.25 	imes (0.30)^2$ = 0.045\ 	ext{kg-m}^2$$

Question 12

A solid wheel (mass 5.2 kg, diameter 0.55 m) is at rest. What torque is needed to reach 12 rev/s in 35 s?

Accepted Answer

$$\omega_f = 12 	imes 2\pi = 75.4\ 	ext{rad/s}$$ 
$$\alpha = \frac{75.4}{35} = 2.15\ 	ext{rad/s}^2$$ 
$$r = 0.275\ 	ext{m}$$ 
$$I = \frac{1}{2} m r^2$ = 0.5 	imes 5.2 	imes (0.275)^2$ = 0.393\ 	ext{kg-m}^2$$ 
$$T = I \alpha = 0.393 	imes 2.15 = 0.845\ 	ext{N-m}$$

Question 13

A rope is wrapped around a pulley (radius 0.20 m) and pulled with a force of 16 N. The pulley's rotational speed increases from 0 to 17 rev/min in 5 s. What is the moment of inertia?

Accepted Answer

$$\omega_f = 17 	imes \frac{2\pi}{60} = 1.78\ 	ext{rad/s}$$ 
$$\alpha = \frac{1.78}{5} = 0.356\ 	ext{rad/s}^2$$ 
$$T = F r = 16 	imes 0.20 = 3.2\ 	ext{N-m}$$ 
$$I = \frac{T}{\alpha} = \frac{3.2}{0.356} = 9.0\ 	ext{kg-m}^2$$

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

$$r = 1.2\ 	ext{m}$$ 
$$T_{	ext{net}} = (67 - 43) 	imes 1.2 = 28.8\ 	ext{N-m}$$

Question 15

A long, uniform rod (mass 3 kg, length 2 m) rotates about its center. What is its moment of inertia?

Accepted Answer

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

Question 16

A solid wheel (mass 15 kg, radius 0.22 m) is accelerated from rest to 8.0 rev/s in 15 s. What force must be applied to a strap wrapped around the wheel to achieve this?

Accepted Answer

$$\omega_f = 8.0 	imes 2\pi = 16\pi \approx 50.3\ 	ext{rad/s}$$ 
$$\alpha = \frac{50.3}{15} = 3.35\ 	ext{rad/s}^2$$ 
$$I = \frac{1}{2} m r^2$ = 0.5 	imes 15 	imes (0.22)^2$ = 0.363\ 	ext{kg-m}^2$$ 
$$T = I \alpha = 0.363 	imes 3.35 = 1.22\ 	ext{N-m}$$ 
$$F = \frac{T}{r} = \frac{1.22}{0.22} = 5.55\ 	ext{N}$$

Question 17

A disk (moment of inertia $$0.26\ 	ext{kg-m}^2) is $$attached to a smaller disk (diameter $$0.18\ 	ext{m}$$, mass $$2.5\ 	ext{kg}) on $$the same axle. 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

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

Question 18

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

Accepted Answer

On a rough surface, friction provides a torque that causes angular acceleration, making the ball roll.

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

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