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?

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

Q: A 65 kg cyclist stands on a pedal that is 0.18 m from the center of the chain ring, making an angle of 35° above the horizontal. What torque does the cyclist exert?

$$T = Fr \sin \theta = (65\ \text{kg} \times 9.8\ \text{N/kg})(0.18\ \text{m}) \sin 35^\circ = 67.2\ \text{N-m}$$

Q: If a torque of 55.0 N-m is required to turn a bolt and the maximum force you can exert is 135 N, what is the minimum length of the lever arm needed?

$$r = \frac{T}{F} = \frac{55.0\ \text{N-m}}{135\ \text{N}} = 0.407\ \text{m}$$

Q: A solid wheel of mass 15 kg and diameter 0.44 m starts at rest. What torque is needed to reach an angular velocity of $$8.0\ \text{rev/s} in 15 s$$?

$$T = I \alpha = (\frac{1}{2} m r^2$) \alpha = (0.36\ \text{kg-m}^2)(3.4\ \text{rad/s}^2) = 1.2\ \text{N-m}$$

Q: A thin hoop of mass 1.0 kg and radius 2.0 m rotates about its central diameter. What is its moment of inertia?

$$I = m r^2$ = (1.0\ \text{kg})(2.0\ \text{m})^2$ = 4.0\ \text{kg-m}^2$$

Q: A seesaw is 1.75 m long. Kariann (56 kg) and Aysha (43 kg) want to balance it. Where should the pivot be placed from Aysha's position?

$$r_A = \frac{F_K \times 1.75\ \text{m}}{F_K + F_A} = \frac{(56 \times 9.8) \times 1.75}{(56 \times 9.8) + (43 \times 9.8)} = 0.99\ \text{m}$$

Q: A solid, uniform cylinder of mass 1.0 kg and radius 2.0 m rotates about its center. What is its moment of inertia?

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} \times 1.0\ \text{kg} \times (2.0\ \text{m})^2$ = 2.0\ \text{kg-m}^2$$

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

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

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 doorknob, perpendicular to the door

Question 3

A 65 kg cyclist stands on a pedal that is 0.18 m from the center of the chain ring, making an angle of 35° above the horizontal. What torque does the cyclist exert?

Accepted Answer

$$T = Fr \sin 	heta = (65\ 	ext{kg} 	imes 9.8\ 	ext{N/kg})(0.18\ 	ext{m}) \sin 35^\circ = 67.2\ 	ext{N-m}$$

Question 4

If a torque of 55.0 N-m is required to turn a bolt and the maximum force you can exert is 135 N, what is the minimum length of the lever arm needed?

Accepted Answer

$$r = \frac{T}{F} = \frac{55.0\ 	ext{N-m}}{135\ 	ext{N}} = 0.407\ 	ext{m}$$

Question 5

Two children of equal mass sit 0.3 m from the center of a seesaw. If they move to 0.6 m from the center, by what factor does the moment of inertia increase?

Accepted Answer

The moment of inertia increases by a factor of 4

Question 6

A solid wheel of mass 15 kg and diameter 0.44 m starts at rest. What torque is needed to reach an angular velocity of $$8.0\ 	ext{rev/s} in 15 s$$?

Accepted Answer

$$T = I \alpha = (\frac{1}{2} m r^2$) \alpha = (0.36\ 	ext{kg-m}^2)(3.4\ 	ext{rad/s}^2) = 1.2\ 	ext{N-m}$$

Question 7

A thin hoop of mass 1.0 kg and radius 2.0 m rotates about its central diameter. What is its moment of inertia?

Accepted Answer

$$I = m r^2$ = (1.0\ 	ext{kg})(2.0\ 	ext{m})^2$ = 4.0\ 	ext{kg-m}^2$$

Question 8

A seesaw is 1.75 m long. Kariann (56 kg) and Aysha (43 kg) want to balance it. Where should the pivot be placed from Aysha's position?

Accepted Answer

$$r_A = \frac{F_K 	imes 1.75\ 	ext{m}}{F_K + F_A} = \frac{(56 	imes 9.8) 	imes 1.75}{(56 	imes 9.8) + (43 	imes 9.8)} = 0.99\ 	ext{m}$$

Question 9

A solid, uniform cylinder of mass 1.0 kg and radius 2.0 m rotates about its center. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{2} m r^2$ = \frac{1}{2} 	imes 1.0\ 	ext{kg} 	imes (2.0\ 	ext{m})^2$ = 2.0\ 	ext{kg-m}^2$$

Question 10

A pulley of radius 0.20 m is pulled with a force of 16.0 N. Its 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

$$I = \frac{T_{net}}{\alpha} = \frac{F r}{\alpha}$$, where $$\alpha = \frac{\Delta \omega}{\Delta t}$$; $$\omega = 17.0 	imes \frac{2\pi}{60} = 1.78\ 	ext{rad/s}$$; $$\alpha = \frac{1.78}{5.00} = 0.356\ 	ext{rad/s}^2$$; $$I = \frac{16.0 	imes 0.20}{0.356} = 9.0\ 	ext{kg-m}^2$$

Question 11

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

Accepted Answer

The moment of inertia depends on both the mass and its distribution relative to the axis of rotation.

Question 12

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; $$F = \frac{I \alpha}{r}$$

Question 13

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 of the pulley?

Accepted Answer

$$\omega = 14.0 	imes \frac{2\pi}{60} = 1.47\ 	ext{rad/s}$$; $$\alpha = \frac{1.47}{4.50} = 0.327\ 	ext{rad/s}^2$$; $$I = \frac{13.0 	imes 0.150}{0.327} = 5.97\ 	ext{kg-m}^2$$

Question 14

A baton is modeled as a thin rod with two masses of $$0.30\ 	ext{kg} at $$each end, length $$0.66\ 	ext{m}. $$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 15

If you apply a force perpendicular to a wrench, how does the required force compare to applying the same torque at an angle?

Accepted Answer

The required force is less when applied perpendicular, because $$\sin 	heta is $$maximized at $$90^\circ$$

Question 16

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

Accepted Answer

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

Question 17

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 18

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 its mass is farther from the axis

Question 19

A disk with moment of inertia $$0.26\ 	ext{kg-m}^2 is $$attached to a smaller disk (diameter $$0.180\ 	ext{m}$$, 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

$$I_{small} = m r^2$ = 2.5 	imes (0.090)^2$ = 0.020\ 	ext{kg-m}^2$$; $$I_{total} = 0.26 + 0.020 = 0.28\ 	ext{kg-m}^2$$; $$T = I_{total} \alpha = 0.28 	imes 2.57 = 0.72\ 	ext{N-m}$$; $$F = \frac{T}{r} = \frac{0.72}{0.090} = 8.0\ 	ext{N}$$

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

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