Q: A mechanic uses a wrench to loosen a bolt. The wrench is 0.30 m long, and the mechanic applies a force of 120 N at an angle of 45° to the handle. What is the magnitude of the torque produced?

$$T = Fr\sin\theta = (120\,\text{N})(0.30\,\text{m})\sin 45^\circ = 25.5\,\text{N-m}$$

Q: A door is 0.90 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?

$$T = Fr = (50\,\text{N})(0.90\,\text{m}) = 45\,\text{N-m}$$

Q: A seesaw is 2.0 m long and massless. Child A (40 kg) sits at one end. Where should Child B (60 kg) sit to balance the seesaw?

$$r_B = \frac{m_A r_A}{m_B} = \frac{40\,\text{kg} \times 2.0\,\text{m}}{60\,\text{kg}} = 1.33\,\text{m}$$

Q: A solid cylinder (mass $$5.0\,\text{kg}$$, radius $$0.20\,\text{m}) is $$accelerated from rest to $$10\,\text{rad/s} in$$ $$4.0\,\text{s}. $$What is the required torque?

$$I = \frac{1}{2}mr^2$ = \frac{1}{2}(5.0)(0.20)^2$ = 0.10\,\text{kg}\cdot\text{m}^2$$; $$\alpha = \frac{\Delta\omega}{\Delta t} = \frac{10}{4} = 2.5\,\text{rad/s}^2$$; $$T = I\alpha = 0.10 \times 2.5 = 0.25\,\text{N-m}$$

Q: A thin hoop (mass $$2.0\,\text{kg}$$, radius $$1.0\,\text{m}) $$and a solid sphere (same mass and radius) are spun about their central axes. Which has the greater moment of inertia?

The thin hoop, because $$I_{\text{hoop}} = mr^2$ = 2.0 \times 1.0^2$ = 2.0\,\text{kg}\cdot\text{m}^2$$; $$I_{\text{sphere}} = \frac{2}{5}mr^2$ = 0.8\,\text{kg}\cdot\text{m}^2$$

Q: A pulley (radius $$0.15\,\text{m}) is $$pulled with a force of $$10.0\,\text{N}$$, causing its angular speed to increase from $$0.0 to$$ $$12.0\,\text{rev/min} in$$ $$6.0\,\text{s}. $$What is the moment of inertia of the pulley?

$$\omega_f = 12.0 \times \frac{2\pi}{60} = 1.26\,\text{rad/s}$$; $$\alpha = \frac{1.26}{6.0} = 0.21\,\text{rad/s}^2$$; $$T = Fr = 10.0 \times 0.15 = 1.5\,\text{N-m}$$; $$I = \frac{T}{\alpha} = \frac{1.5}{0.21} = 7.14\,\text{kg}\cdot\text{m}^2$$

Q: A bicycle pedal is $$0.18\,\text{m}$$ from the center of the chain ring. If a rider of mass $$70\,\text{kg}$$ stands on the pedal at an angle of $$30^\circ$$ above the horizontal, what torque is exerted?

$$F = mg = 70 \times 9.8 = 686\,\text{N}$$; $$T = Fr\sin\theta = 686 \times 0.18 \times \sin 30^\circ = 61.7\,\text{N-m}$$

Q: A solid wheel (mass $$8.0\,\text{kg}$$, diameter $$0.50\,\text{m}) is at $$rest. You want it to rotate at $$10\,\text{rev/s} in$$ $$20\,\text{s}. $$What force must you exert on a strap wrapped around the wheel's edge?

$$r = 0.25\,\text{m}$$; $$I = \frac{1}{2}mr^2$ = \frac{1}{2} \times 8.0 \times 0.25^2$ = 0.25\,\text{kg}\cdot\text{m}^2$$; $$\omega = 10 \times 2\pi = 62.8\,\text{rad/s}$$; $$\alpha = \frac{62.8}{20} = 3.14\,\text{rad/s}^2$$; $$T = I\alpha = 0.25 \times 3.14 = 0.785\,\text{N-m}$$; $$F = \frac{T}{r} = \frac{0.785}{0.25} = 3.14\,\text{N}$$

Q: A long, uniform rod (mass $$1.0\,\text{kg}$$, length $$2.0\,\text{m}) is $$rotated about its center. What is its moment of inertia?

$$I = \frac{1}{12}ml^2$ = \frac{1}{12} \times 1.0 \times 2.0^2$ = 0.33\,\text{kg}\cdot\text{m}^2$$

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

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

Question 1

A mechanic uses a wrench to loosen a bolt. The wrench is 0.30 m long, and the mechanic applies a force of 120 N at an angle of 45° to the handle. What is the magnitude of the torque produced?

Accepted Answer

$$T = Fr\sin	heta = (120\,	ext{N})(0.30\,	ext{m})\sin 45^\circ = 25.5\,	ext{N-m}$$

Question 2

A door is 0.90 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

$$T = Fr = (50\,	ext{N})(0.90\,	ext{m}) = 45\,	ext{N-m}$$

Question 3

A seesaw is 2.0 m long and massless. Child A (40 kg) sits at one end. Where should Child B (60 kg) sit to balance the seesaw?

Accepted Answer

$$r_B = \frac{m_A r_A}{m_B} = \frac{40\,	ext{kg} 	imes 2.0\,	ext{m}}{60\,	ext{kg}} = 1.33\,	ext{m}$$

Question 4

A solid cylinder (mass $$5.0\,	ext{kg}$$, radius $$0.20\,	ext{m}) is $$accelerated from rest to $$10\,	ext{rad/s} in$$ $$4.0\,	ext{s}. $$What is the required torque?

Accepted Answer

$$I = \frac{1}{2}mr^2$ = \frac{1}{2}(5.0)(0.20)^2$ = 0.10\,	ext{kg}\cdot	ext{m}^2$$; $$\alpha = \frac{\Delta\omega}{\Delta t} = \frac{10}{4} = 2.5\,	ext{rad/s}^2$$; $$T = I\alpha = 0.10 	imes 2.5 = 0.25\,	ext{N-m}$$

Question 5

A thin hoop (mass $$2.0\,	ext{kg}$$, radius $$1.0\,	ext{m}) $$and a solid sphere (same mass and radius) are spun about their central axes. Which has the greater moment of inertia?

Accepted Answer

The thin hoop, because $$I_{	ext{hoop}} = mr^2$ = 2.0 	imes 1.0^2$ = 2.0\,	ext{kg}\cdot	ext{m}^2$$; $$I_{	ext{sphere}} = \frac{2}{5}mr^2$ = 0.8\,	ext{kg}\cdot	ext{m}^2$$

Question 6

A pulley (radius $$0.15\,	ext{m}) is $$pulled with a force of $$10.0\,	ext{N}$$, causing its angular speed to increase from $$0.0 to$$ $$12.0\,	ext{rev/min} in$$ $$6.0\,	ext{s}. $$What is the moment of inertia of the pulley?

Accepted Answer

$$\omega_f = 12.0 	imes \frac{2\pi}{60} = 1.26\,	ext{rad/s}$$; $$\alpha = \frac{1.26}{6.0} = 0.21\,	ext{rad/s}^2$$; $$T = Fr = 10.0 	imes 0.15 = 1.5\,	ext{N-m}$$; $$I = \frac{T}{\alpha} = \frac{1.5}{0.21} = 7.14\,	ext{kg}\cdot	ext{m}^2$$

Question 7

A bicycle pedal is $$0.18\,	ext{m}$$ from the center of the chain ring. If a rider of mass $$70\,	ext{kg}$$ stands on the pedal at an angle of $$30^\circ$$ above the horizontal, what torque is exerted?

Accepted Answer

$$F = mg = 70 	imes 9.8 = 686\,	ext{N}$$; $$T = Fr\sin	heta = 686 	imes 0.18 	imes \sin 30^\circ = 61.7\,	ext{N-m}$$

Question 8

A solid wheel (mass $$8.0\,	ext{kg}$$, diameter $$0.50\,	ext{m}) is at $$rest. You want it to rotate at $$10\,	ext{rev/s} in$$ $$20\,	ext{s}. $$What force must you exert on a strap wrapped around the wheel's edge?

Accepted Answer

$$r = 0.25\,	ext{m}$$; $$I = \frac{1}{2}mr^2$ = \frac{1}{2} 	imes 8.0 	imes 0.25^2$ = 0.25\,	ext{kg}\cdot	ext{m}^2$$; $$\omega = 10 	imes 2\pi = 62.8\,	ext{rad/s}$$; $$\alpha = \frac{62.8}{20} = 3.14\,	ext{rad/s}^2$$; $$T = I\alpha = 0.25 	imes 3.14 = 0.785\,	ext{N-m}$$; $$F = \frac{T}{r} = \frac{0.785}{0.25} = 3.14\,	ext{N}$$

Question 9

A long, uniform rod (mass $$1.0\,	ext{kg}$$, length $$2.0\,	ext{m}) is $$rotated about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{12}ml^2$ = \frac{1}{12} 	imes 1.0 	imes 2.0^2$ = 0.33\,	ext{kg}\cdot	ext{m}^2$$

Question 10

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

Accepted Answer

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

Question 11

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

Accepted Answer

$$\omega_f = 17.0 	imes \frac{2\pi}{60} = 1.78\,	ext{rad/s}$$; $$\alpha = \frac{1.78}{5.00} = 0.356\,	ext{rad/s}^2$$; $$T = Fr = 16.0 	imes 0.20 = 3.2\,	ext{N-m}$$; $$I = \frac{T}{\alpha} = \frac{3.2}{0.356} = 9.0\,	ext{kg}\cdot	ext{m}^2$$

Question 12

A baton is modeled as a thin rod with two masses ($$0.30\,	ext{kg}$$ each) at its ends, length $$0.66\,	ext{m}. $$What is the moment of inertia about its center?

Accepted Answer

$$r = 0.33\,	ext{m}$$; $$I = 2 	imes m r^2$ = 2 	imes 0.30 	imes 0.33^2$ = 0.066\,	ext{kg}\cdot	ext{m}^2$$

Question 13

If you apply a force at an angle to a door, how does the lever arm change compared to applying the force perpendicular?

Accepted Answer

The lever arm decreases; $$L = r\sin	heta is $$less than $$r$$ unless $$	heta = 90^\circ$$

Question 14

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 hoop (same mass and radius), what force is needed for the same acceleration?

Accepted Answer

The hoop has greater moment of inertia ($$I = mr^2$), so F$$_{	ext{hoop}} \u003e 4.5\,	ext{N}$$

Question 15

A rope is wrapped around a pulley and pulled with a force of $$13.0\,	ext{N}. $$The pulley's radius is $$0.150\,	ext{m}. $$Its rotational speed increases from $$0.0 to$$ $$14.0\,	ext{rev/min} in$$ $$4.50\,	ext{s}. $$What is the moment of inertia?

Accepted Answer

$$\omega_f = 14.0 	imes \frac{2\pi}{60} = 1.47\,	ext{rad/s}$$; $$\alpha = \frac{1.47}{4.50} = 0.327\,	ext{rad/s}^2$$; $$T = Fr = 13.0 	imes 0.150 = 1.95\,	ext{N-m}$$; $$I = \frac{T}{\alpha} = \frac{1.95}{0.327} = 5.96\,	ext{kg}\cdot	ext{m}^2$$

Question 16

A thin rectangular plate (mass $$2.0\,	ext{kg}$$, length $$3.0\,	ext{m}$$, width $$1.0\,	ext{m}) is $$rotated about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{12}m(l^2$ + w^2$) = \frac{1}{12} 	imes 2.0 	imes (3.0^2$ + 1.0^2$) = \frac{2.0}{12} 	imes 10 = 1.67\,	ext{kg}\cdot	ext{m}^2$$

Question 17

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

Accepted Answer

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

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

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