Q: 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?

$$I = \frac{1}{2}MR^2$ = \frac{1}{2} \times 2.0 \times (0.5)^2$ = 0.25\ \mathrm{kg\cdot m^2$}$$

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

$$I = \frac{1}{3}ML^2$ = \frac{1}{3} \times 3.0 \times (2.0)^2$ = 4.0\ \mathrm{kg\cdot m^2$}$$

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

$$\omega = \$$omega_0$$ + \alpha t = 0 + 2.0 \times 5.0 = 10.0\ \mathrm{rad/s}$$

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?

$$\tau = rF = 0.2 \times 10 = 2.0\ \mathrm{N\cdot m}$$

Q: A plank of length $$4.0\ \mathrm{m}$$ and mass $$20\ \mathrm{kg}$$ rests horizontally on two supports placed $$0.5\ \mathrm{m}$$ from each end. What is the force on each support?

$$F = \frac{Mg}{2} = \frac{20 \times 9.8}{2} = 98\ \mathrm{N}$$

Q: A uniform beam of length $$L$$ and mass $$M is $$attached to a wall by a hinge and held horizontal by a cable at its end making an angle $$\theta$$ above the beam. What is the tension $$T in $$the cable?

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

Q: A rod of mass $$2.0\ \mathrm{kg}$$ and length $$1.0\ \mathrm{m} is $$pivoted at one end. What is its angular momentum about the pivot if it rotates at $$\omega = 3.0\ \mathrm{rad/s}$$?

$$L = I\omega = \frac{1}{3}ML^2$ \times \omega = \frac{1}{3} \times 2.0 \times (1.0)^2$ \times 3.0 = 2.0\ \mathrm{kg\cdot m^2$/s}$$

Q: A $$5.0\ \mathrm{kg}$$ mass is attached to a string and swung in a horizontal circle of radius $$2.0\ \mathrm{m} at an $$angular speed of $$4.0\ \mathrm{rad/s}. $$What is its angular momentum?

$$L = mvr = mr^2$\omega = 5.0 \times (2.0)^2$ \times 4.0 = 80\ \mathrm{kg\cdot m^2$/s}$$

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

$$I = \frac{1}{2}MR^2$ = \frac{1}{2} 	imes 2.0 	imes (0.5)^2$ = 0.25\ \mathrm{kg\cdot m^2$}$$

Question 2

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

Accepted Answer

$$I = \frac{1}{3}ML^2$ = \frac{1}{3} 	imes 3.0 	imes (2.0)^2$ = 4.0\ \mathrm{kg\cdot m^2$}$$

Question 3

A wheel starts from rest and 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 = 0 + 2.0 	imes 5.0 = 10.0\ \mathrm{rad/s}$$

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

$$	au = rF = 0.2 	imes 10 = 2.0\ \mathrm{N\cdot m}$$

Question 5

A solid sphere (mass $$M$$, 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 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 base of the ladder

Question 7

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

$$2\$$omega_0$$

Question 8

A plank of length $$4.0\ \mathrm{m}$$ and mass $$20\ \mathrm{kg}$$ rests horizontally on two supports placed $$0.5\ \mathrm{m}$$ from each end. What is the force on each support?

Accepted Answer

$$F = \frac{Mg}{2} = \frac{20 	imes 9.8}{2} = 98\ \mathrm{N}$$

Question 9

A uniform beam of length $$L$$ and mass $$M is $$attached to a wall by a hinge and held horizontal by a cable at its end making an angle $$	heta$$ above the beam. What is the tension $$T in $$the cable?

Accepted Answer

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

Question 10

A disk rotates with constant angular acceleration. If it makes $$20$$ revolutions in $$5$$ seconds starting from rest, what is its angular acceleration $$\alpha$$?

Accepted Answer

$$\alpha = \frac{8\pi}{5^2$} = \frac{8\pi}{25}\ \mathrm{$$rad/s^2$$}$$

Question 11

A rod of mass $$2.0\ \mathrm{kg}$$ and length $$1.0\ \mathrm{m} is $$pivoted at one end. What is its angular momentum about the pivot if it rotates at $$\omega = 3.0\ \mathrm{rad/s}$$?

Accepted Answer

$$L = I\omega = \frac{1}{3}ML^2$ 	imes \omega = \frac{1}{3} 	imes 2.0 	imes (1.0)^2$ 	imes 3.0 = 2.0\ \mathrm{kg\cdot m^2$/s}$$

Question 12

A $$5.0\ \mathrm{kg}$$ mass is attached to a string and swung in a horizontal circle of radius $$2.0\ \mathrm{m} at an $$angular speed of $$4.0\ \mathrm{rad/s}. $$What is its angular momentum?

Accepted Answer

$$L = mvr = mr^2$\omega = 5.0 	imes (2.0)^2$ 	imes 4.0 = 80\ \mathrm{kg\cdot m^2$/s}$$

Question 13

A $$10\ \mathrm{kg}$$ uniform beam $$4.0\ \mathrm{m}$$ long is supported horizontally at one end by a hinge and at the other by a vertical cable. What is the tension in the cable?

Accepted Answer

$$T = \frac{MgL/2}{L} = \frac{Mg}{2} = \frac{10 	imes 9.8}{2} = 49\ \mathrm{N}$$

Question 14

A $$2.0\ \mathrm{kg}$$ mass hangs from a steel wire of length $$1.0\ \mathrm{m}$$ and cross-sectional area $$1.0 	imes 10$$^{-6}$$\ \mathrm{m^2$}. If $$Young's modulus for steel is $$2.0 	imes 10$$^{11}$$\ \mathrm{N/m^2$}$$, what is the elongation $$\Delta L$$?

Accepted Answer

$$\Delta L = \frac{MgL}{A Y} = \frac{2.0 	imes 9.8 	imes 1.0}{1.0 	imes 10$$^{-6}$$ 	imes 2.0 	imes 10$$^{11}$$} = 9.8 	imes 10$$^{-5}$$\ \mathrm{m}$$

Question 15

A uniform beam of length $$L$$ and mass $$M is $$supported at its ends. A person of mass $$m$$ stands at a distance $$d$$ from one end. What is the upward force $$F_1 at $$the nearer support?

Accepted Answer

$$F_1$$ = \frac{M g}{2} + m g \left(1 - \frac{d}{L}\right)$$

Question 16

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

Accepted Answer

The solid disk, because it has a smaller moment of inertia and thus a greater acceleration.

Question 17

A force $$F is $$applied at an angle $$	heta to a $$wrench of length $$L. $$What is the torque about the pivot?

Accepted Answer

$$	au = F L \sin	heta$$

Question 18

A $$0.5\ \mathrm{kg}$$ mass is attached to a light rod of length $$0.8\ \mathrm{m}$$, rotating about one end. What is the rotational kinetic energy if it spins at $$6.0\ \mathrm{rad/s}$$?

Accepted Answer

$$K = \frac{1}{2} I \$$omega^2$$ = \frac{1}{2} 	imes 0.5 	imes (0.8)^2$ 	imes (6.0)^2$ = 5.76\ \mathrm{J}$$

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

Study Guide - Practice Questions

Study Guide - Flashcards