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 magnitude of the torque produced?

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

Q: A door is 0.9 m wide. If you push with a force of 50 N perpendicular to the door at the edge, 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 two children. Child A (mass 40 kg) sits 1.5 m from the pivot. Child B (mass 60 kg) sits on the other side. How far from the pivot should Child B sit to balance the seesaw?

$$r_B = \frac{40\ \text{kg} \times 1.5\ \text{m}}{60\ \text{kg}} = 1.0\ \text{m}$$

Q: A solid wheel of mass 10 kg and radius 0.20 m is accelerated from rest to 10 rad/s in 5 s. What is the required torque?

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} \times 10\ \text{kg} \times (0.20\ \text{m})^2$ = 0.2\ \text{kg-m}^2$$ \\ $$\alpha = \frac{\Delta \omega}{\Delta t} = \frac{10\ \text{rad/s}}{5\ \text{s}} = 2\ \text{rad/s}^2$$ \\ $$T = I \alpha = 0.2 \times 2 = 0.4\ \text{N-m}$$

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 of radius 0.15 m is accelerated from rest to 14 rev/min in 4.5 s by a force of 13.0 N. What is the moment of inertia of the pulley?

$$\omega_f = 14 \times \frac{2\pi}{60} = 1.47\ \text{rad/s}$$ \\ $$\alpha = \frac{1.47}{4.5} = 0.327\ \text{rad/s}^2$$ \\ $$T = F r = 13.0 \times 0.15 = 1.95\ \text{N-m}$$ \\ $$I = \frac{T}{\alpha} = \frac{1.95}{0.327} = 5.96\ \text{kg-m}^2$$

Q: A solid 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 bicycle pedal is 0.18 m from the center of the chain ring. If a rider of mass 65 kg stands on the pedal at an angle of 35° above the horizontal, what torque is exerted?

$$T = (65 \times 9.8) \times 0.18 \times \sin 35^\circ = 65 \times 9.8 \times 0.18 \times 0.574 = 66.1\ \text{N-m}$$

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 magnitude of the torque produced?

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 perpendicular to the door at the edge, 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 two children. Child A (mass 40 kg) sits 1.5 m from the pivot. Child B (mass 60 kg) sits on the other side. How far from the pivot should Child B sit to balance the seesaw?

Accepted Answer

$$r_B = \frac{40\ 	ext{kg} 	imes 1.5\ 	ext{m}}{60\ 	ext{kg}} = 1.0\ 	ext{m}$$

Question 5

A solid wheel of mass 10 kg and radius 0.20 m is accelerated from rest to 10 rad/s in 5 s. What is the required torque?

Accepted Answer

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} 	imes 10\ 	ext{kg} 	imes (0.20\ 	ext{m})^2$ = 0.2\ 	ext{kg-m}^2$$ \ $$\alpha = \frac{\Delta \omega}{\Delta t} = \frac{10\ 	ext{rad/s}}{5\ 	ext{s}} = 2\ 	ext{rad/s}^2$$ \ $$T = I \alpha = 0.2 	imes 2 = 0.4\ 	ext{N-m}$$

Question 6

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

Accepted Answer

Rotating the book about an axis through its edge, parallel to its longer side.

Question 7

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 8

A pulley of radius 0.15 m is accelerated from rest to 14 rev/min in 4.5 s by a force of 13.0 N. What is the moment of inertia of the pulley?

Accepted Answer

$$\omega_f = 14 	imes \frac{2\pi}{60} = 1.47\ 	ext{rad/s}$$ \ $$\alpha = \frac{1.47}{4.5} = 0.327\ 	ext{rad/s}^2$$ \ $$T = F r = 13.0 	imes 0.15 = 1.95\ 	ext{N-m}$$ \ $$I = \frac{T}{\alpha} = \frac{1.95}{0.327} = 5.96\ 	ext{kg-m}^2$$

Question 9

A solid 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 10

A bicycle pedal is 0.18 m from the center of the chain ring. If a rider of mass 65 kg stands on the pedal at an angle of 35° above the horizontal, what torque is exerted?

Accepted Answer

$$T = (65 	imes 9.8) 	imes 0.18 	imes \sin 35^\circ = 65 	imes 9.8 	imes 0.18 	imes 0.574 = 66.1\ 	ext{N-m}$$

Question 11

If you want to maximize the torque when opening a door, which strategy should you use?

Accepted Answer

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

Question 12

A solid wheel (mass 5.2 kg, diameter 0.55 m) is at rest. You need it to rotate at 12 rev/s in 35 s. What torque must be applied?

Accepted Answer

$$\omega_f = 12 	imes 2\pi = 75.4\ 	ext{rad/s}$$ \ $$\alpha = \frac{75.4}{35} = 2.15\ 	ext{rad/s}^2$$ \ $$I = \frac{1}{2} 	imes 5.2 	imes (0.275)^2$ = 0.196\ 	ext{kg-m}^2$$ \ $$T = I \alpha = 0.196 	imes 2.15 = 0.42\ 	ext{N-m}$$

Question 13

A chain is wrapped around a pulley and pulled with a force of 16.0 N. The pulley has a radius of 0.20 m. The rotational speed increases from 0.0 to 17.0 rev/min in 5.00 s. What is the moment of inertia of the pulley?

Accepted Answer

$$\omega_f = 17 	imes \frac{2\pi}{60} = 1.78\ 	ext{rad/s}$$ \ $$\alpha = \frac{1.78}{5.00} = 0.356\ 	ext{rad/s}^2$$ \ $$T = 16.0 	imes 0.20 = 3.2\ 	ext{N-m}$$ \ $$I = \frac{3.2}{0.356} = 9.0\ 	ext{kg-m}^2$$

Question 14

If two people pull on ropes wrapped around the edge of a large wheel (diameter 2.4 m, mass 12 kg), one with 43 N clockwise and the other with 67 N counterclockwise, what is the net torque?

Accepted Answer

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

Question 15

A baton consists of a thin rod with two masses (0.30 kg each) at its ends, length 0.66 m. What is the moment of inertia about the midpoint?

Accepted Answer

$$I = 2 	imes 0.30 	imes (0.33)^2$ = 0.066\ 	ext{kg-m}^2$$

Question 16

According to Newton's second law for rotational motion, what happens to angular acceleration if the net torque is doubled and the moment of inertia remains constant?

Accepted Answer

Angular acceleration doubles.

Question 17

A solid wheel accelerates at $$3.25\ 	ext{rad/s}^2$$ when a force of $$4.5\ 	ext{N}$$ exerts a torque on it. If the wheel is replaced by a wheel with all its mass on the rim ($$I = m r^2$), what force should be exerted to achieve the same angular acceleration?

Accepted Answer

A greater force is needed because the moment of inertia is larger; $$F = \frac{I \alpha}{r}$$, so $$F \u003e 4.5\ 	ext{N}.$$

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

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