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

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

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?

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

Q: A seesaw is balanced with a 60 kg child sitting 1.2 m from the pivot. Where should a 40 kg child sit 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?

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

Q: A thin hoop and a solid sphere both have mass $$m$$ and radius $$r. $$Which has the greater moment of inertia about its central axis?

The thin hoop, because $$I_{\text{hoop}} = m r^2$ and I$$_{\text{sphere}} = \frac{2}{5} m $$r^2.$$

Q: A pulley with radius 0.15 m is pulled with a force of 20 N. What is the torque exerted on the pulley?

$$T = F r = 20 \times 0.15 = 3.0\ \text{N-m}$$

Q: A baton consists of two 0.20 kg masses at each end of a 0.50 m rod. What is the moment of inertia about the center of the rod?

$$I = 2 \times m r^2$ = 2 \times 0.20 \times (0.25)^2$ = 0.025\ \text{kg-m}^2$$

Q: A solid cylinder of mass 2 kg and radius 0.3 m is rotated about its central axis. What is its moment of inertia?

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} \times 2 \times (0.3)^2$ = 0.09\ \text{kg-m}^2$$

Q: 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?

$$T = F r \sin \theta = (70 \times 9.8) \times 0.18 \times \sin 30^\circ = 123.48\ \text{N-m}$$

Question 1

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

Accepted Answer

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

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 50 N at the edge, perpendicular to the door, what is the torque about the hinges?

Accepted Answer

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

Question 4

A seesaw is balanced with a 60 kg child sitting 1.2 m from the pivot. Where should a 40 kg child sit 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

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

Question 6

A thin hoop and a solid sphere both have mass $$m$$ and radius $$r. $$Which has the greater moment of inertia about its central axis?

Accepted Answer

The thin hoop, because $$I_{	ext{hoop}} = m r^2$ and I$$_{	ext{sphere}} = \frac{2}{5} m $$r^2.$$

Question 7

A pulley with radius 0.15 m is pulled with a force of 20 N. What is the torque exerted on the pulley?

Accepted Answer

$$T = F r = 20 	imes 0.15 = 3.0\ 	ext{N-m}$$

Question 8

A baton consists of two 0.20 kg masses at each end of a 0.50 m rod. What is the moment of inertia about the center of the rod?

Accepted Answer

$$I = 2 	imes m r^2$ = 2 	imes 0.20 	imes (0.25)^2$ = 0.025\ 	ext{kg-m}^2$$

Question 9

A solid cylinder of mass 2 kg and radius 0.3 m is rotated about its central axis. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} 	imes 2 	imes (0.3)^2$ = 0.09\ 	ext{kg-m}^2$$

Question 10

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

$$T = F r \sin 	heta = (70 	imes 9.8) 	imes 0.18 	imes \sin 30^\circ = 123.48\ 	ext{N-m}$$

Question 11

If a torque of 40 N-m is required to turn a bolt and you can exert a maximum force of 100 N, what is the minimum length of the lever arm needed?

Accepted Answer

$$r = \frac{T}{F} = \frac{40}{100} = 0.40\ 	ext{m}$$

Question 12

A solid wheel (mass 5 kg, radius 0.2 m) is accelerated by a force of 10 N applied at its 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 13

A rope is wrapped around a pulley of radius 0.12 m. If the rope is pulled with a force of 15 N, what is the torque on the pulley?

Accepted Answer

$$T = F r = 15 	imes 0.12 = 1.8\ 	ext{N-m}$$

Question 14

A solid sphere of mass 3 kg and radius 0.2 m is rotated about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{2}{5} m r^2$ = \frac{2}{5} 	imes 3 	imes (0.2)^2$ = 0.048\ 	ext{kg-m}^2$$

Question 15

If two people pull on opposite sides of a wheel (diameter 1.0 m) with forces of 30 N (clockwise) and 50 N (counterclockwise), what is the net torque?

Accepted Answer

$$T_{	ext{net}} = (50 - 30) 	imes 0.5 = 10\ 	ext{N-m}$$

Question 16

A disk with moment of inertia $$0.20\ 	ext{kg-m}^2 is $$subjected to a net torque of $$2.0\ 	ext{N-m}. $$What is its angular acceleration?

Accepted Answer

$$\alpha = \frac{T_{	ext{net}}}{I} = \frac{2.0}{0.20} = 10\ 	ext{rad/s}^2$$

Question 17

A book is rotated about its edge and then about its center. Which orientation requires more torque to achieve the same angular acceleration, and why?

Accepted Answer

Rotating about the edge requires more torque because the moment of inertia is greater when the mass is farther from the axis.

Question 18

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

Accepted Answer

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

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

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