Q: A mechanic uses a 0.40 m wrench to loosen a bolt. If she applies a force of 50 N at an angle of 30° to the handle, what is the magnitude of the torque produced?

$$T = Fr \sin \theta = (50\ \text{N})(0.40\ \text{m})\sin 30^\circ = 10\ \text{N-m}$$

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

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

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?

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

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

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} \times 2\ \text{kg} \times (0.5\ \text{m})^2$ = 0.25\ \text{kg-m}^2$$

Q: A thin hoop (mass 1 kg, radius 2 m) rotates about its central axis. What is its moment of inertia?

$$I = m r^2$ = 1\ \text{kg} \times (2\ \text{m})^2$ = 4\ \text{kg-m}^2$$

Q: A solid sphere (mass 3 kg, radius 0.4 m) rotates about its center. What is its moment of inertia?

$$I = \frac{2}{5} m r^2$ = \frac{2}{5} \times 3\ \text{kg} \times (0.4\ \text{m})^2$ = 0.192\ \text{kg-m}^2$$

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

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

Q: A disk with moment of inertia $$0.26\ \text{kg-m}^2 is $$attached to a smaller disk (diameter $$0.18\ \text{m}$$, mass $$2.5\ \text{kg}). $$What force is needed on a strap around the smaller disk to give the system an angular acceleration of $$2.57\ \text{rad/s}^2$$?

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

Question 1

A mechanic uses a 0.40 m wrench to loosen a bolt. If she applies a force of 50 N at an angle of 30° to the handle, what is the magnitude of the torque produced?

Accepted Answer

$$T = Fr \sin 	heta = (50\ 	ext{N})(0.40\ 	ext{m})\sin 30^\circ = 10\ 	ext{N-m}$$

Question 2

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

Accepted Answer

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

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

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

Question 5

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

Accepted Answer

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} 	imes 2\ 	ext{kg} 	imes (0.5\ 	ext{m})^2$ = 0.25\ 	ext{kg-m}^2$$

Question 6

A thin hoop (mass 1 kg, radius 2 m) rotates about its central axis. What is its moment of inertia?

Accepted Answer

$$I = m r^2$ = 1\ 	ext{kg} 	imes (2\ 	ext{m})^2$ = 4\ 	ext{kg-m}^2$$

Question 7

A solid sphere (mass 3 kg, radius 0.4 m) rotates about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{2}{5} m r^2$ = \frac{2}{5} 	imes 3\ 	ext{kg} 	imes (0.4\ 	ext{m})^2$ = 0.192\ 	ext{kg-m}^2$$

Question 8

A pulley (radius 0.15 m) is pulled with a force of 13 N. Its rotational speed increases from 0 to 14 rev/min in 4.5 s. What is the moment of inertia of the pulley?

Accepted Answer

$$I = \frac{T_{net}}{\alpha} = \frac{Fr}{\alpha}$$, where $$\alpha = \frac{\Delta \omega}{\Delta t}$$, $$\Delta \omega = 14 	imes \frac{2\pi}{60} = 1.47\ 	ext{rad/s}$$, $$\alpha = \frac{1.47}{4.5} = 0.327\ 	ext{rad/s}^2$$, $$T = 13 	imes 0.15 = 1.95\ 	ext{N-m}$$, $$I = \frac{1.95}{0.327} = 5.96\ 	ext{kg-m}^2$$

Question 9

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

Accepted Answer

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

Question 10

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 on a strap around the smaller disk to give the system an angular acceleration of $$2.57\ 	ext{rad/s}^2$$?

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

A thin rod (length 0.66 m) with two masses (0.30 kg each) at its ends is rotated about its center. What is its moment of inertia?

Accepted Answer

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

Question 13

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

Accepted Answer

$$\omega = 2\pi 	imes 8 = 16\pi = 50.3\ 	ext{rad/s}$$, $$\alpha = \frac{50.3}{15} = 3.35\ 	ext{rad/s}^2$$, $$I = 0.5 	imes 15 	imes (0.22)^2$ = 0.363\ 	ext{kg-m}^2$$, $$T = I \alpha = 0.363 	imes 3.35 = 1.22\ 	ext{N-m}$$, $$F = \frac{T}{r} = \frac{1.22}{0.22} = 5.55\ 	ext{N}$$

Question 14

Which of the following statements about moment of inertia is correct?

Accepted Answer

Moment of inertia depends on both the mass and the distribution of mass relative to the axis of rotation.

Question 15

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 the ball to rotate, allowing it to roll.

Question 16

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

$$T_{net} = (67 - 43) 	imes 1.2 = 24 	imes 1.2 = 28.8\ 	ext{N-m} ($$counterclockwise)

Question 17

A hollow sphere and a solid sphere have equal mass and radius. Which has the greater moment of inertia?

Accepted Answer

The hollow sphere, because all its mass is distributed farther from the axis.

Question 18

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

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

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

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