Q: A runner moves along a circular path of radius $$20\ \mathrm{m}$$ and covers an arc length of $$31.4\ \mathrm{m}. $$What is the angular displacement $$\theta (in $$radians) of the runner?

$$\theta = \dfrac{31.4}{20} = 1.57\ \mathrm{rad}$$

Q: A disk rotates from rest with a constant angular acceleration $$\alpha = 2\ \mathrm{$$rad/s^2$$}. $$What is its angular velocity $$\omega$$ after $$5\ \mathrm{s}$$?

$$\omega = 2 \times 5 = 10\ \mathrm{rad/s}$$

Q: A $$2\ \mathrm{kg}$$ mass is attached to a string and swung in a horizontal circle of radius $$1.5\ \mathrm{m} at a $$constant speed of $$4\ \mathrm{m/s}. $$What is the magnitude of the required centripetal force?

$$F_c = \dfrac{2 \times 4^2$}{1.5} = 21.3\ \mathrm{N}$$

Q: A solid disk of mass $$3\ \mathrm{kg}$$ and radius $$0.2\ \mathrm{m}$$ rotates about its center. What is its moment of inertia $$I$$?

$$I = \dfrac{1}{2} \times 3 \times 0.2^2$ = 0.12\ \mathrm{kg\cdot m^2$}$$

Q: A force of $$10\ \mathrm{N} is $$applied perpendicular to a wrench at a distance of $$0.3\ \mathrm{m}$$ from the bolt. What is the torque $$\tau$$ produced?

$$\tau = 10 \times 0.3 = 3\ \mathrm{N\cdot m}$$

Q: A uniform rod of length $$2\ \mathrm{m}$$ and mass $$4\ \mathrm{kg} is $$pivoted at one end. What is its moment of inertia about the pivot?

$$I = \dfrac{1}{3} \times 4 \times 2^2$ = \dfrac{16}{3}\ \mathrm{kg\cdot m^2$}$$

Question 1

A car travels around a circular track of radius $$50\ \mathrm{m}$$, completing one full lap in $$80\ \mathrm{s}. $$What is the magnitude of the car's average angular speed $$\omega$$ during this lap?

Accepted Answer

$$\omega = \dfrac{2\pi}{80}\ \mathrm{rad/s}$$

Question 2

A runner moves along a circular path of radius $$20\ \mathrm{m}$$ and covers an arc length of $$31.4\ \mathrm{m}. $$What is the angular displacement $$	heta (in $$radians) of the runner?

Accepted Answer

$$	heta = \dfrac{31.4}{20} = 1.57\ \mathrm{rad}$$

Question 3

A disk rotates from rest with a constant angular acceleration $$\alpha = 2\ \mathrm{$$rad/s^2$$}. $$What is its angular velocity $$\omega$$ after $$5\ \mathrm{s}$$?

Accepted Answer

$$\omega = 2 	imes 5 = 10\ \mathrm{rad/s}$$

Question 4

A $$2\ \mathrm{kg}$$ mass is attached to a string and swung in a horizontal circle of radius $$1.5\ \mathrm{m} at a $$constant speed of $$4\ \mathrm{m/s}. $$What is the magnitude of the required centripetal force?

Accepted Answer

$$F_c = \dfrac{2 	imes 4^2$}{1.5} = 21.3\ \mathrm{N}$$

Question 5

A solid disk of mass $$3\ \mathrm{kg}$$ and radius $$0.2\ \mathrm{m}$$ rotates about its center. What is its moment of inertia $$I$$?

Accepted Answer

$$I = \dfrac{1}{2} 	imes 3 	imes 0.2^2$ = 0.12\ \mathrm{kg\cdot m^2$}$$

Question 6

A force of $$10\ \mathrm{N} is $$applied perpendicular to a wrench at a distance of $$0.3\ \mathrm{m}$$ from the bolt. What is the torque $$	au$$ produced?

Accepted Answer

$$	au = 10 	imes 0.3 = 3\ \mathrm{N\cdot m}$$

Question 7

A uniform rod of length $$2\ \mathrm{m}$$ and mass $$4\ \mathrm{kg} is $$pivoted at one end. What is its moment of inertia about the pivot?

Accepted Answer

$$I = \dfrac{1}{3} 	imes 4 	imes 2^2$ = \dfrac{16}{3}\ \mathrm{kg\cdot m^2$}$$

Question 8

A wheel of radius $$0.5\ \mathrm{m}$$ rolls without slipping at a linear speed of $$2\ \mathrm{m/s}. $$What is its angular speed $$\omega$$?

Accepted Answer

$$\omega = \dfrac{2}{0.5} = 4\ \mathrm{rad/s}$$

Question 9

A solid sphere and a thin hoop, both of mass $$m$$ and radius $$R$$, are released from rest at the top of an incline. Which reaches the bottom first and why?

Accepted Answer

The solid sphere, because it has a smaller moment of inertia relative to its mass and radius.

Question 10

A ladder of mass $$10\ \mathrm{kg}$$ and length $$4\ \mathrm{m} is $$balanced horizontally on a pivot at its center. Where is its center of gravity located?

Accepted Answer

At the midpoint, $$2\ \mathrm{m}$$ from either end.

Question 11

A force of $$15\ \mathrm{N} is $$applied at an angle of $$30^\circ to a $$lever arm $$0.4\ \mathrm{m}$$ from the pivot. What is the torque produced?

Accepted Answer

$$	au = 15 	imes 0.4 	imes \sin 30^\circ = 3\ \mathrm{N\cdot m}$$

Question 12

A bucket of mass $$5\ \mathrm{kg} is $$attached to a rope wound around a solid cylinder of mass $$2\ \mathrm{kg}$$ and radius $$0.1\ \mathrm{m}. $$When released, what is the acceleration $$a of $$the bucket? (Assume $$g = 9.8\ \mathrm{m/s^2$})$$

Accepted Answer

$$a = \dfrac{5 	imes 9.8}{5 + 1} = 8.17\ \mathrm{m/s^2$}$$

Question 13

A uniform solid cylinder and a hollow cylinder, both with the same mass and radius, roll down an incline without slipping. Which one reaches the bottom first and why?

Accepted Answer

The solid cylinder, because it has a smaller moment of inertia.

Question 14

A rotating object has an angular velocity $$\omega = 6\ \mathrm{rad/s}$$ and a tangential acceleration $$a_t = 3\ \mathrm{m/s^2$} at a $$radius $$r = 0.5\ \mathrm{m}. $$What is the magnitude of the total acceleration?

Accepted Answer

$$a_{	ext{total}} = \sqrt{3^2$ + (6 	imes 0.5)^2$} = \sqrt{9 + 9} = 4.24\ \mathrm{m/s^2$}$$

Question 15

A point mass $$m is $$attached to a massless rod of length $$L$$ and rotated about one end. What is the moment of inertia?

Accepted Answer

$$I = mL^2$

Question 16

A compact disk (CD) has mass $$0.02\ \mathrm{kg}$$ and radius $$0.06\ \mathrm{m}. $$What is its moment of inertia about its center?

Accepted Answer

$$I = \dfrac{1}{2} 	imes 0.02 	imes 0.06^2$ = 3.6 	imes 10$$^{-5}$$\ \mathrm{kg\cdot m^2$}$$

Question 17

A force of $$25\ \mathrm{N} is $$applied at $$0.2\ \mathrm{m}$$ from the axis of rotation at an angle of $$90^\circ. $$What is the torque?

Accepted Answer

$$	au = 25 	imes 0.2 = 5\ \mathrm{N\cdot m}$$

Question 18

A solid sphere of mass $$2\ \mathrm{kg}$$ and radius $$0.1\ \mathrm{m} is $$rotating at $$10\ \mathrm{rad/s}. $$What is its rotational kinetic energy?

Accepted Answer

$$K = \dfrac{1}{2} I \$$omega^2$$ = \dfrac{1}{2} 	imes \dfrac{2}{5} 	imes 2 	imes 0.1^2$ 	imes 10^2$ = 0.4\ \mathrm{J}$$

Question 19

A uniform thin rod of mass $$3\ \mathrm{kg}$$ and length $$1.2\ \mathrm{m}$$ rotates about its center. What is its moment of inertia?

Accepted Answer

$$I = \dfrac{1}{12} 	imes 3 	imes 1.2^2$ = 0.36\ \mathrm{kg\cdot m^2$}$$

Circular and Rotational Motion: Study Notes

Study Guide - Practice Questions

Study Guide - Flashcards