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

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

Q: A seesaw is balanced with two children sitting at different distances from the pivot. If child A (mass 40 kg) sits 1.2 m from the pivot and child B (mass 50 kg) sits on the other side, how far from the pivot should child B sit to balance the seesaw?

$$r_B = \frac{m_A r_A}{m_B} = \frac{40\ \text{kg} \times 1.2\ \text{m}}{50\ \text{kg}} = 0.96\ \text{m}$$

Q: A thin hoop of mass 2 kg and radius 1 m is rotated about its central axis. What is its moment of inertia?

$$I = mr^2$ = 2 \times 1^2$ = 2\ \text{kg-m}^2$$

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

$$\omega_f = 10 \times \frac{2\pi}{60} = 1.047\ \text{rad/s}$$; $$\alpha = \frac{1.047}{4} = 0.262\ \text{rad/s}^2$$; $$T = Fr = 12 \times 0.15 = 1.8\ \text{N-m}$$; $$I = \frac{T}{\alpha} = \frac{1.8}{0.262} = 6.87\ \text{kg-m}^2$$

Q: A solid cylinder (mass 3 kg, radius 0.5 m) is spun up from rest to 6 rad/s in 4 s. What force must be applied tangentially at the rim?

$$I = \frac{1}{2}mr^2$ = \frac{1}{2} \times 3 \times 0.5^2$ = 0.375\ \text{kg-m}^2$$; $$\alpha = \frac{6}{4} = 1.5\ \text{rad/s}^2$$; $$T = I\alpha = 0.375 \times 1.5 = 0.5625\ \text{N-m}$$; $$F = \frac{T}{r} = \frac{0.5625}{0.5} = 1.125\ \text{N}$$

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

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

Accepted Answer

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

Question 4

A seesaw is balanced with two children sitting at different distances from the pivot. If child A (mass 40 kg) sits 1.2 m from the pivot and child B (mass 50 kg) sits on the other side, how far from the pivot should child B sit to balance the seesaw?

Accepted Answer

$$r_B = \frac{m_A r_A}{m_B} = \frac{40\ 	ext{kg} 	imes 1.2\ 	ext{m}}{50\ 	ext{kg}} = 0.96\ 	ext{m}$$

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 is the required torque?

Accepted Answer

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

Question 6

Which scenario will result in the greatest moment of inertia for a book?

Accepted Answer

Rotating the book about an axis along its edge, perpendicular to its length.

Question 7

A thin hoop of mass 2 kg and radius 1 m is rotated about its central axis. What is its moment of inertia?

Accepted Answer

$$I = mr^2$ = 2 	imes 1^2$ = 2\ 	ext{kg-m}^2$$

Question 8

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

Accepted Answer

The thin hoop, because all its mass is farther from the axis: $$I_{hoop} = mr^2$ = 4\ 	ext{kg-m}^2$$; $$I_{sphere} = \frac{2}{5}mr^2$ = 0.8\ 	ext{kg-m}^2.$$

Question 9

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

Accepted Answer

$$\omega_f = 10 	imes \frac{2\pi}{60} = 1.047\ 	ext{rad/s}$$; $$\alpha = \frac{1.047}{4} = 0.262\ 	ext{rad/s}^2$$; $$T = Fr = 12 	imes 0.15 = 1.8\ 	ext{N-m}$$; $$I = \frac{T}{\alpha} = \frac{1.8}{0.262} = 6.87\ 	ext{kg-m}^2$$

Question 10

A solid cylinder (mass 3 kg, radius 0.5 m) is spun up from rest to 6 rad/s in 4 s. What force must be applied tangentially at the rim?

Accepted Answer

$$I = \frac{1}{2}mr^2$ = \frac{1}{2} 	imes 3 	imes 0.5^2$ = 0.375\ 	ext{kg-m}^2$$; $$\alpha = \frac{6}{4} = 1.5\ 	ext{rad/s}^2$$; $$T = I\alpha = 0.375 	imes 1.5 = 0.5625\ 	ext{N-m}$$; $$F = \frac{T}{r} = \frac{0.5625}{0.5} = 1.125\ 	ext{N}$$

Question 11

A bicycle wheel (radius 0.38 m) is accelerated by pulling on the tire with a force of 20 N. If the same acceleration is needed by pulling on the gear (radius 0.14 m), what force is required?

Accepted Answer

$$F_{gear} = F_{tire} 	imes \frac{r_{tire}}{r_{gear}} = 20 	imes \frac{0.38}{0.14} = 54.3\ 	ext{N}$$

Question 12

A baton consists of two masses (0.4 kg each) at the ends of a 0.8 m rod. What is the moment of inertia about the center?

Accepted Answer

$$I = 2 	imes m r^2$ = 2 	imes 0.4 	imes (0.4)^2$ = 0.128\ 	ext{kg-m}^2$$

Question 13

If a torque of 50 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

$$r = \frac{T}{F} = \frac{50}{100} = 0.5\ 	ext{m}$$

Question 14

A solid wheel (mass 5 kg, radius 0.2 m) is spun up from rest to 20 rad/s in 10 s. What is the angular acceleration?

Accepted Answer

$$\alpha = \frac{\Delta \omega}{\Delta t} = \frac{20}{10} = 2\ 	ext{rad/s}^2$$

Question 15

A rope is wrapped around a pulley (radius 0.2 m) and pulled with a force of 10 N. If the net torque is 2 N-m, what is the moment of inertia if the angular acceleration is 4 rad/s²?

Accepted Answer

$$I = \frac{T_{net}}{\alpha} = \frac{2}{4} = 0.5\ 	ext{kg-m}^2$$

Question 16

Two people pull on ropes wrapped around a 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

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

Question 17

A ball rolls down a rough incline but slides down a smooth one. 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 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 to give the system an angular acceleration of $$2.57\ 	ext{rad/s}^2 by $$pulling on the smaller disk?

Accepted Answer

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

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

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