Q: Which of the following best describes the torque produced by a force $$\vec{F}$$ applied at a position vector $$\vec{r}$$ relative to point $$O$$?

$$\vec{\tau} = \vec{r} \times \vec{F}$$

Q: A plumber stands on the end of a wrench handle, applying a $$900\ \mathrm{N}$$ force at a point $$0.80\ \mathrm{m}$$ from the center of a pipe fitting. The handle makes an angle of $$19^\circ$$ with the horizontal. What is the magnitude of the torque applied about the center?

$$\tau = (900\ \mathrm{N})(0.80\ \mathrm{m}) \sin 19^\circ = 234\ \mathrm{N\cdot m}$$

Q: A metal bar lies in the $$xy-$$plane with one end at the origin. A force $$\vec{F} = (7.00\ \mathrm{N})\hat{i} + (-3.00\ \mathrm{N})\hat{j} is $$applied at the point $$(3.00\ \mathrm{m}, 4.00\ \mathrm{m}). $$What is the magnitude of the torque about the origin?

$$\tau = r_x F_y - r_y F_x = (3.00)(-3.00) - (4.00)(7.00) = -9.00 - 28.00 = -37.00\ \mathrm{N\cdot m}$$

Q: A wheel rotates without friction about a stationary horizontal axis at its center. A constant tangential force of $$80.0\ \mathrm{N} is $$applied to the rim (radius $$0.120\ \mathrm{m}). $$Starting from rest, the wheel reaches $$12.0\ \mathrm{rev/s}$$ after $$2.00\ \mathrm{s}. $$What is the moment of inertia of the wheel?

$$I = \frac{\tau}{\alpha} = \frac{Fr}{\omega / t} = \frac{(80.0)(0.120)}{(2\pi \times 12.0) / 2.00} = 0.254\ \mathrm{kg\cdot m^2$}$$

Q: A 15.0 kg bucket of water is suspended by a light rope wrapped around a solid uniform cylinder (diameter $$0.300\ \mathrm{m}$$, mass $$12.0\ \mathrm{kg}). $$The bucket falls $$10.0\ \mathrm{m}. $$What is the speed of the bucket when it strikes the water?

$$v = \sqrt{\frac{2mgh}{m + \frac{1}{2}M}} = \sqrt{\frac{2 \times 15.0 \times 9.8 \times 10.0}{15.0 + 6.0}} = 9.1\ \mathrm{m/s}$$

Question 1

A baseball bat of mass $$M$$ has its center of mass at a distance $$d$$ from the end. If the moment of inertia about the center of mass is $$I_{cm}$$, what is the moment of inertia about a parallel axis through the end of the bat using the parallel-axis theorem?

Accepted Answer

$$I_{end} = I_{cm} + Md^2$

Question 2

A part of a mechanical linkage has a mass of $$3.6\ \mathrm{kg}. $$Its moment of inertia about an axis $$0.15\ \mathrm{m}$$ from its center of mass is $$0.132\ \mathrm{kg\cdot m^2$}. $$What is the moment of inertia about the center of mass?

Accepted Answer

$$I_{cm} = 0.132\ \mathrm{kg\cdot m^2$} - (3.6\ \mathrm{kg})(0.15\ \mathrm{m})^2$ = 0.051\ \mathrm{kg\cdot m^2$}$$

Question 3

Which of the following best describes the torque produced by a force $$\vec{F}$$ applied at a position vector $$\vec{r}$$ relative to point $$O$$?

Accepted Answer

$$\vec{	au} = \vec{r} 	imes \vec{F}$$

Question 4

A plumber stands on the end of a wrench handle, applying a $$900\ \mathrm{N}$$ force at a point $$0.80\ \mathrm{m}$$ from the center of a pipe fitting. The handle makes an angle of $$19^\circ$$ with the horizontal. What is the magnitude of the torque applied about the center?

Accepted Answer

$$	au = (900\ \mathrm{N})(0.80\ \mathrm{m}) \sin 19^\circ = 234\ \mathrm{N\cdot m}$$

Question 5

A metal bar lies in the $$xy-$$plane with one end at the origin. A force $$\vec{F} = (7.00\ \mathrm{N})\hat{i} + (-3.00\ \mathrm{N})\hat{j} is $$applied at the point $$(3.00\ \mathrm{m}, 4.00\ \mathrm{m}). $$What is the magnitude of the torque about the origin?

Accepted Answer

$$	au = r_x F_y - r_y F_x = (3.00)(-3.00) - (4.00)(7.00) = -9.00 - 28.00 = -37.00\ \mathrm{N\cdot m}$$

Question 6

Which of the following is the correct rotational analog of Newton’s second law for a rigid body about the $$z-$$axis?

Accepted Answer

$$\sum 	au_z = I \alpha_z$$

Question 7

A wheel rotates without friction about a stationary horizontal axis at its center. A constant tangential force of $$80.0\ \mathrm{N} is $$applied to the rim (radius $$0.120\ \mathrm{m}). $$Starting from rest, the wheel reaches $$12.0\ \mathrm{rev/s}$$ after $$2.00\ \mathrm{s}. $$What is the moment of inertia of the wheel?

Accepted Answer

$$I = \frac{	au}{\alpha} = \frac{Fr}{\omega / t} = \frac{(80.0)(0.120)}{(2\pi 	imes 12.0) / 2.00} = 0.254\ \mathrm{kg\cdot m^2$}$$

Question 8

A 15.0 kg bucket of water is suspended by a light rope wrapped around a solid uniform cylinder (diameter $$0.300\ \mathrm{m}$$, mass $$12.0\ \mathrm{kg}). $$The bucket falls $$10.0\ \mathrm{m}. $$What is the speed of the bucket when it strikes the water?

Accepted Answer

$$v = \sqrt{\frac{2mgh}{m + \frac{1}{2}M}} = \sqrt{\frac{2 	imes 15.0 	imes 9.8 	imes 10.0}{15.0 + 6.0}} = 9.1\ \mathrm{m/s}$$

Question 9

A yo-yo consists of a solid cylinder of mass $$M$$ and radius $$R. If it $$descends a distance $$h$$ while unwinding, what is the speed $$v_{cm} of $$its center of mass using energy considerations?

Accepted Answer

$$v_{cm} = \sqrt{\frac{4gh}{3}}$$

Question 10

Which condition must be satisfied for a wheel to roll without slipping?

Accepted Answer

$$v_{cm} = R\omega$$

Question 11

A sphere rolls down an inclined plane without slipping. Which force is responsible for producing the torque that causes the sphere to rotate?

Accepted Answer

The friction force at the point of contact

Question 12

Which of the following statements about rolling friction is correct?

Accepted Answer

Rolling friction can be ignored if both the rolling object and the surface are perfectly rigid.

Question 13

A rigid body is acted on by several forces. Which of the following statements is true regarding internal forces and the rotational motion of the body?

Accepted Answer

Internal forces do not affect the rotational motion because their torques cancel each other.

Question 14

The total kinetic energy of a rigid body that is both translating and rotating is given by which of the following expressions?

Accepted Answer

$$K = \frac{1}{2}Mv_{cm}^2 + \frac{1}{2}I\$$omega^2$$

Question 15

A drag racer’s rear tires are spinning and producing smoke, indicating that the tires are slipping. Which of the following is true?

Accepted Answer

$$v_{cm} 
eq R\omega$$

Question 16

Which of the following best describes the relationship between the net external force and the acceleration of the center of mass of a rigid body?

Accepted Answer

$$\sum \vec{F}_{ext} = M\vec{a}_{cm}$$

Question 17

A maple seed falls while rotating about its center of mass. Which of the following best explains why it falls more slowly than a non-rotating object of the same mass and shape?

Accepted Answer

Some gravitational potential energy is converted into rotational kinetic energy, reducing the translational speed.

Study Guidance: Rotational Dynamics, Torque, and Moment of Inertia

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