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?

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

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 (30 kg) sits at one end. Where should Child B (40 kg) sit to balance the seesaw?

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

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 torque is required?

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

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

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

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 N. What is the moment of inertia of the pulley?

$$\omega_f = 14 \times \frac{2\pi}{60} = 1.47\ \text{rad/s}$$\ $$\alpha = \frac{1.47}{4.5} = 0.327\ \text{rad/s}^2$$\ $$T = F r = 13 \times 0.15 = 1.95\ \text{N-m}$$\ $$I = \frac{T}{\alpha} = \frac{1.95}{0.327} = 5.96\ \text{kg-m}^2$$

Q: A baton consists of a thin rod (negligible mass) with two 0.30 kg masses at each end, 0.66 m apart. What is the moment of inertia about the center?

$$I = 2 \times m r^2$ = 2 \times 0.30 \times (0.33)^2$ = 0.066\ \text{kg-m}^2$$

Q: A solid wheel (mass 5.2 kg, diameter 0.55 m) is at rest and must rotate at 12 rev/s in 35 s. What force must be applied to a strap wrapped around the wheel?

$$r = 0.275\ \text{m}$$\ $$I = \frac{1}{2} m r^2$ = \frac{1}{2} \times 5.2 \times (0.275)^2$ = 0.393\ \text{kg-m}^2$$\ $$\omega = 12 \times 2\pi = 75.4\ \text{rad/s}$$\ $$\alpha = \frac{75.4}{35} = 2.15\ \text{rad/s}^2$$\ $$T = I \alpha = 0.393 \times 2.15 = 0.845\ \text{N-m}$$\ $$F = \frac{T}{r} = \frac{0.845}{0.275} = 3.07\ \text{N}$$

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

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

Question 2

A door is 0.9 m wide. To open it with the least force, where and in what direction should you apply the force?

Accepted Answer

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 (30 kg) sits at one end. Where should Child B (40 kg) sit to balance the seesaw?

Accepted Answer

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

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 torque is required?

Accepted Answer

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

Question 6

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

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

Question 7

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 its mass is farther from the axis: $$I_{hoop} = m r^2$ = 4\ 	ext{kg-m}^2$$, $$I_{sphere} = \frac{2}{5} m r^2$ = 1.6\ 	ext{kg-m}^2$$

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 N. What is the moment of inertia of the pulley?

Accepted Answer

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

Question 9

A baton consists of a thin rod (negligible mass) with two 0.30 kg masses at each end, 0.66 m apart. What is the moment of inertia about the center?

Accepted Answer

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

Question 10

A solid wheel (mass 5.2 kg, diameter 0.55 m) is at rest and must rotate at 12 rev/s in 35 s. What force must be applied to a strap wrapped around the wheel?

Accepted Answer

$$r = 0.275\ 	ext{m}$$\ $$I = \frac{1}{2} m r^2$ = \frac{1}{2} 	imes 5.2 	imes (0.275)^2$ = 0.393\ 	ext{kg-m}^2$$\ $$\omega = 12 	imes 2\pi = 75.4\ 	ext{rad/s}$$\ $$\alpha = \frac{75.4}{35} = 2.15\ 	ext{rad/s}^2$$\ $$T = I \alpha = 0.393 	imes 2.15 = 0.845\ 	ext{N-m}$$\ $$F = \frac{T}{r} = \frac{0.845}{0.275} = 3.07\ 	ext{N}$$

Question 11

A chain is wrapped around a pulley (radius 0.20 m) and pulled with a force of 16 N. The pulley speeds up 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 12

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

$$F = m g = 65 	imes 9.8 = 637\ 	ext{N}$$\ $$T = F r \sin 35^\circ = 637 	imes 0.18 	imes 0.574 = 65.7\ 	ext{N-m}$$

Question 13

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

$$r = 1.2\ 	ext{m}$$\ $$T_{net} = 67 	imes 1.2 - 43 	imes 1.2 = (80.4 - 51.6) = 28.8\ 	ext{N-m}$$\ counterclockwise

Question 14

A solid, uniform cylinder of mass 1 kg and radius 2 m is rotated about its center. What is its moment of inertia?

Accepted Answer

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

Question 15

A long, uniform rod of mass 0.5 kg and length 1 m is rotated about its end. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{3} m l^2$ = \frac{1}{3} 	imes 0.5 	imes 1^2$ = 0.167\ 	ext{kg-m}^2$$

Question 16

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

$$I_{small} = m r^2$ = 2.5 	imes (0.09)^2$ = 0.0203\ 	ext{kg-m}^2$$\ $$I_{total} = 0.26 + 0.0203 = 0.2803\ 	ext{kg-m}^2$$\ $$T = I_{total} \alpha = 0.2803 	imes 2.57 = 0.720\ 	ext{N-m}$$\ $$F = \frac{T}{r} = \frac{0.720}{0.09} = 8.0\ 	ext{N}$$

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 replaced by a wheel with all mass on the rim ($$I = m r^2$), what force should be exerted for the same angular acceleration?

Accepted Answer

A greater force, because the moment of inertia is larger: $$F_{rim} = \frac{I_{rim} \alpha}{r}$$

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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