Q: A force of $$F = 10\ \mathrm{N} is $$applied tangentially to the rim of a wheel of radius $$r = 0.2\ \mathrm{m}. $$What is the torque produced?

$$\tau = rF = 2.0\ \mathrm{N\cdot m}$$

Q: A plank of length $$4.0\ \mathrm{m}$$ and mass $$20\ \mathrm{kg}$$ rests on two supports $$3.0\ \mathrm{m}$$ apart. If a $$60\ \mathrm{kg}$$ person stands $$1.0\ \mathrm{m}$$ from the left end, what is the force on the left support? (Assume $$g = 10\ \mathrm{m/s^2$})$$

$$F_{\text{left}} = 350\ \mathrm{N}$$

Q: A uniform beam of length $$L$$ and mass $$M is $$attached to a wall at one end and held horizontal by a cable at the other end. If the cable makes an angle $$\theta$$ with the beam, what is the tension $$T in $$the cable?

$$T = \frac{MgL}{2L\sin\theta} = \frac{Mg}{2\sin\theta}$$

Question 1

A solid disk of mass $$M = 2.0\ \mathrm{kg}$$ and radius $$R = 0.5\ \mathrm{m}$$ rotates about its central axis. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{2}MR^2$ = 0.25\ \mathrm{kg\cdot m^2$}$$

Question 2

A uniform rod of length $$L = 2.0\ \mathrm{m}$$ and mass $$M = 3.0\ \mathrm{kg}$$ rotates about an axis perpendicular to the rod and passing through one end. What is its moment of inertia?

Accepted Answer

$$I = \frac{1}{3}ML^2$ = 4.0\ \mathrm{kg\cdot m^2$}$$

Question 3

A wheel initially at rest undergoes a constant angular acceleration of $$\alpha = 2.0\ \mathrm{$$rad/s^2$$}. $$What is its angular velocity after $$t = 5.0\ \mathrm{s}$$?

Accepted Answer

$$\omega = \$$omega_0$$ + \alpha t = 10.0\ \mathrm{rad/s}$$

Question 4

A force of $$F = 10\ \mathrm{N} is $$applied tangentially to the rim of a wheel of radius $$r = 0.2\ \mathrm{m}. $$What is the torque produced?

Accepted Answer

$$	au = rF = 2.0\ \mathrm{N\cdot m}$$

Question 5

A uniform solid sphere of mass $$M$$ and radius $$R$$ rolls without slipping down an incline of height $$h. $$What is its speed at the bottom using conservation of energy?

Accepted Answer

$$v = \sqrt{\frac{10gh}{7}}$$

Question 6

A figure skater pulls in her arms to decrease her moment of inertia to half its original value. If her initial angular velocity was $$\$$omega_0$$, what is her final angular velocity?

Accepted Answer

$$\omega_f = 2\$$omega_0$$

Question 7

A uniform ladder of length $$L$$ and mass $$M$$ leans against a frictionless wall, making an angle $$	heta$$ with the ground. Which force provides the horizontal equilibrium?

Accepted Answer

The static friction force at the ground

Question 8

A beam of length $$L is $$supported at both ends and a mass $$m is $$placed at a distance $$x$$ from the left support. Which equation gives the upward force $$F_1 at $$the left support?

Accepted Answer

$$F_1$$ = \frac{(L-x)m g}{L}$$

Question 9

A plank of length $$4.0\ \mathrm{m}$$ and mass $$20\ \mathrm{kg}$$ rests on two supports $$3.0\ \mathrm{m}$$ apart. If a $$60\ \mathrm{kg}$$ person stands $$1.0\ \mathrm{m}$$ from the left end, what is the force on the left support? (Assume $$g = 10\ \mathrm{m/s^2$})$$

Accepted Answer

$$F_{	ext{left}} = 350\ \mathrm{N}$$

Question 10

A uniform beam of length $$L$$ and mass $$M is $$attached to a wall at one end and held horizontal by a cable at the other end. If the cable makes an angle $$	heta$$ with the beam, what is the tension $$T in $$the cable?

Accepted Answer

$$T = \frac{MgL}{2L\sin	heta} = \frac{Mg}{2\sin	heta}$$

Question 11

A $$2.0\ \mathrm{kg}$$ mass is attached to a spring and stretches it by $$0.05\ \mathrm{m}$$ under gravity. What is the spring constant $$k$$?

Accepted Answer

$$k = \frac{mg}{\Delta x} = 392\ \mathrm{N/m}$$

Question 12

A thin hoop of mass $$M$$ and radius $$R$$ rotates about its central axis with angular velocity $$\omega. $$What is its rotational kinetic energy?

Accepted Answer

$$K = \frac{1}{2}I\$$omega^2$$ = \frac{1}{2}MR^2$\$$omega^2$$

Question 13

A disk of mass $$M$$ and radius $$R$$ rotates about an axis through its center. If the axis is shifted parallel to a distance $$d$$, what is the new moment of inertia?

Accepted Answer

$$I = \frac{1}{2}MR^2$ + Md^2$

Question 14

A child sits on a rotating merry-go-round at a distance $$r$$ from the center, moving with angular velocity $$\omega. $$What is the child's angular momentum?

Accepted Answer

$$L = m r^2$ \omega$$

Question 15

A solid cylinder of mass $$M$$ and radius $$R$$ rolls without slipping at speed $$v. $$What is its total kinetic energy?

Accepted Answer

$$K = \frac{1}{2}Mv^2$ + \frac{1}{4}Mv^2$ = \frac{3}{4}Mv^2$

Question 16

A rod of length $$L$$ and mass $$M is $$pivoted at its center. Two masses $$m$$ are attached at each end. What is the total moment of inertia about the center?

Accepted Answer

$$I = \frac{1}{12}ML^2$ + 2m\left(\frac{L}{2}\$$right)^2$$

Question 17

A uniform beam of length $$L$$ and mass $$M is in $$equilibrium, supported horizontally at its ends. If a person of mass $$m$$ stands at a distance $$x$$ from the left end, what is the condition for rotational equilibrium?

Accepted Answer

$$F_{	ext{left}} \cdot 0 + F_{	ext{right}} \cdot L = Mg \cdot \frac{L}{2} + mgx$$

Question 18

A steel wire of length $$L = 2.0\ \mathrm{m}$$ and cross-sectional area $$A = 1.0\ \mathrm{mm^2$}$$ stretches by $$1.0\ \mathrm{mm}$$ under a load. What is the stress in the wire if $$Y = 2.0 	imes 10$$^{11}$$\ \mathrm{N/m^2$}$$?

Accepted Answer

$$	ext{Stress} = Y \cdot \frac{\Delta L}{L} = 1.0 	imes 10^8$\ \mathrm{N/m^2$}$$

Rotational Motion, Dynamics, and Equilibrium: Study Guide (Chapters 9–11)

Study Guide - Practice Questions

Study Guide - Flashcards