Q: A wheel accelerates from rest at a constant angular acceleration $$\alpha = 2.0\ \mathrm{$$rad/s^2$$}. $$How many radians does it rotate in the first $$4.0\ \mathrm{s}$$?

$$\theta = \frac{1}{2}\alpha t^2$ = 16\ \mathrm{rad}$$

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

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

Q: A $$3.0\ \mathrm{kg}$$ mass is suspended from a wire with a cross-sectional area of $$1.0\ \mathrm{mm}^2$$ and Young's modulus $$Y = 2.0 \times 10$$^{11}$$\ \mathrm{N/m^2$}. $$What is the stress in the wire?

$$\sigma = \frac{F}{A} = 2.94 \times 10^7$\ \mathrm{N/m^2$}$$

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 accelerates from rest at a constant angular acceleration $$\alpha = 2.0\ \mathrm{$$rad/s^2$$}. $$How many radians does it rotate in the first $$4.0\ \mathrm{s}$$?

Accepted Answer

$$	heta = \frac{1}{2}\alpha t^2$ = 16\ \mathrm{rad}$$

Question 4

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

Accepted Answer

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

Question 5

A uniform solid sphere (mass $$M$$, radius $$R) $$rolls without slipping down an incline of height $$h. $$What is its speed at the bottom, assuming it started from rest?

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 speed was $$\$$omega_0$$, what is her final angular speed?

Accepted Answer

$$2\$$omega_0$$

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 friction force at the ground

Question 8

A $$2.0\ \mathrm{m}$$ long, $$10\ \mathrm{kg}$$ uniform beam is supported at one end by a hinge and at the other by a cable making a $$30^\circ$$ angle with the beam. What is the tension in the cable if the beam is horizontal?

Accepted Answer

$$T = 98\ \mathrm{N}$$

Question 9

A $$5.0\ \mathrm{kg}$$ mass is attached to a spring and stretches it by $$0.10\ \mathrm{m}. $$What is the spring constant $$k$$?

Accepted Answer

$$k = 490\ \mathrm{N/m}$$

Question 10

A plank of length $$4.0\ \mathrm{m}$$ and mass $$20\ \mathrm{kg}$$ rests on two supports $$1.0\ \mathrm{m}$$ from each end. What is the force on each support?

Accepted Answer

$$F = 98\ \mathrm{N}$$

Question 11

A disk rotates with constant angular acceleration. If its initial angular velocity is $$\$$omega_0$$ = 2\ \mathrm{rad/s}$$ and after $$3\ \mathrm{s}$$ its angular velocity is $$\omega = 8\ \mathrm{rad/s}$$, what is its angular acceleration?

Accepted Answer

$$\alpha = 2\ \mathrm{$$rad/s^2$$}$$

Question 12

A $$0.5\ \mathrm{kg}$$ mass is attached to a string and swung in a horizontal circle of radius $$0.8\ \mathrm{m} at$$ $$4\ \mathrm{rad/s}. $$What is its angular momentum?

Accepted Answer

$$L = mr^2$\omega = 1.28\ \mathrm{kg\cdot m^2$/s}$$

Question 13

A uniform beam of length $$L$$ and mass $$M is $$attached to a wall by a hinge and held horizontal by a cable at angle $$	heta$$ above the beam. What is the torque due to the beam's weight about the hinge?

Accepted Answer

$$	au = Mg\frac{L}{2}$$

Question 14

A hoop and a solid disk, each of mass $$M$$ and radius $$R$$, roll without slipping down an incline from the same height. Which reaches the bottom first and why?

Accepted Answer

The disk, because it has a smaller moment of inertia

Question 15

A $$3.0\ \mathrm{kg}$$ mass is suspended from a wire with a cross-sectional area of $$1.0\ \mathrm{mm}^2$$ and Young's modulus $$Y = 2.0 	imes 10$$^{11}$$\ \mathrm{N/m^2$}. $$What is the stress in the wire?

Accepted Answer

$$\sigma = \frac{F}{A} = 2.94 	imes 10^7$\ \mathrm{N/m^2$}$$

Question 16

A $$2.0\ \mathrm{kg}$$ mass is attached to a spring and oscillates horizontally. If the spring constant is $$k = 200\ \mathrm{N/m}$$, what is the period of oscillation?

Accepted Answer

$$T = 0.63\ \mathrm{s}$$

Question 17

A uniform plank of length $$6.0\ \mathrm{m}$$ and mass $$30\ \mathrm{kg} is $$supported at its ends. A $$60\ \mathrm{kg}$$ person stands $$1.0\ \mathrm{m}$$ from the left end. What is the force on the right support?

Accepted Answer

$$F = 490\ \mathrm{N}$$

Question 18

A $$0.5\ \mathrm{kg}$$ ball moving at $$4.0\ \mathrm{m/s}$$ strikes the end of a $$2.0\ \mathrm{m}$$ long, $$1.0\ \mathrm{kg}$$ rod lying on a frictionless table and sticks to it. What is the angular velocity of the system after the collision?

Accepted Answer

$$\omega = 1.0\ \mathrm{rad/s}$$

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

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