Angular Velocity Examples in Physics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Angular Velocity.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Physics.

Concept Recap

The rate at which an object rotates about an axis, measured in radians per second, with a direction along the axis.

How fast something is spinning — a car wheel spinning fast has high angular velocity.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Angular Velocity starts by naming what changes, over what time interval, and whether direction matters.

Common stuck point: Students often know a formula related to angular velocity but skip the recognition step: Am I describing motion over time with position, distance, direction, speed, velocity, or acceleration clearly separated? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Am I describing motion over time with position, distance, direction, speed, velocity, or acceleration clearly separated?

Worked Examples

Example 1

medium

A car wheel of radius

0.30\,\text{m}

rolls without slipping at

v = 18\,\text{m/s}

. Find its angular velocity.

Answer

\omega = 60 \text{ rad/s}

First step

Rolling without slipping:

v = \omega r

, so

\omega = v/r

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Example 2

medium

Two gears mesh: gear A (radius

5\,\text{cm}

) spins at

\omega_A = 8\,\text{rad/s}

and drives gear B (radius

2\,\text{cm}

). Find

\omega_B

Example 3

hard

A wheel turns

200\,\text{rad}

while slowing from

\omega_0 = 30\,\text{rad/s}

\omega = 10\,\text{rad/s}

. Find

\alpha

Example 4

challenge

A car drives at

v = 25\,\text{m/s}

. Its tires have radius

0.32\,\text{m}

. How many full rotations does each tire make over a

5\,\text{km}

trip?

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A wheel turns

\pi

radians in

2

s. Find its angular velocity.

Example 2

easy

Convert

360°

to radians.

Example 3

easy

A disk spins at

\omega=4

rad/s. A point at radius

0.5

m has what linear speed?

Example 4

easy

A turntable completes one revolution in

2

s. Find its angular velocity.

Example 5

easy

On a rigid spinning disk, do all points share the same angular velocity?

Example 6

easy

What direction does the angular velocity vector point for a spinning wheel (right-hand rule)?

Example 7

easy

A fan spins at

2

rev/s. Find its angular velocity in rad/s.

Example 8

easy

Convert

90°

to radians.

Example 9

medium

A wheel of radius

0.4

m rolls so its center moves at

8

m/s. Find the wheel's angular velocity.

Example 10

medium

A disk at

\omega=10

rad/s. Find the linear speed of a point at radius

0.2

m and one at

0.6

Example 11

medium

A motor accelerates from rest to

\omega=20

rad/s in

5

s. Find the angular acceleration.

Example 12

medium

A wheel spins at

\omega=6

rad/s. How many radians in

10

s (constant

\omega

Example 13

medium

A point at radius

2

m has centripetal acceleration

8

m/s

^2

. Find the angular velocity.

Example 14

medium

A clock's second hand makes one revolution per minute. Find its angular velocity in rad/s.

Example 15

challenge

A wheel of radius

0.5

m starts at rest and reaches

\omega=12

rad/s in

4

s. Find the linear speed of the rim at

t=4

s and the rim's tangential acceleration.

Example 16

challenge

Two gears mesh: gear A (radius

4

cm) drives gear B (radius

2

cm). If A spins at

\omega_A=6

rad/s, find

\omega_B

Example 17

challenge

A point on a

0.25

m radius wheel moves at linear speed

5

m/s. Find how many full revolutions it makes in

10

Example 18

medium

A disk spins at

\omega=5

rad/s. A point at radius

0.4

m has what linear speed?

Example 19

medium

A wheel completes

3

revolutions in

6

s. Find its angular velocity in rad/s.

Example 20

medium

A motor speeds from rest to

\omega=18

rad/s in

6

s. Find the angular acceleration.

Example 21

easy

A wheel turns through

6\pi

rad in

3\,\text{s}

. Find its angular velocity.

Example 22

easy

A bike wheel of radius

0.35\,\text{m}

spins at

\omega = 12\,\text{rad/s}

. Find the rim's linear speed.

Example 23

easy

A CD makes

5

revolutions in

2\,\text{s}

. Find

\omega

in rad/s.

Example 24

easy

Convert

\omega = 60\,\text{rpm}

to rad/s.

Example 25

easy

A wheel at

\omega = 8\,\text{rad/s}

rotates for

3\,\text{s}

at constant rate. Through what angle has it turned?

Example 26

easy

A turntable spins at

\omega = 5\,\text{rad/s}

. Find the linear speed of a point at

r = 0.15\,\text{m}

Example 27

medium

A motor speeds up from

\omega_0 = 10\,\text{rad/s}

\omega = 30\,\text{rad/s}

4\,\text{s}

. Find

\alpha

Example 28

medium

A point at

r = 0.5\,\text{m}

moves with centripetal acceleration

a_c = 18\,\text{m/s}^2

. Find

\omega

Example 29

medium

A fan blade has period

T = 0.25\,\text{s}

. Find

\omega

Example 30

medium

A clock's minute hand goes around once per hour. Find

\omega

in rad/s.

Example 31

medium

A pulley of radius

0.10\,\text{m}

raises a load via a cable moving up at

v = 1.5\,\text{m/s}

. Find

\omega

of the pulley.

Example 32

medium

A wheel starts at rest and reaches

\omega = 24\,\text{rad/s}

after

6\,\text{s}

at constant

\alpha

. Through how many radians did it turn?

Example 33

medium

A ceiling fan spins at

\omega = 6\pi\,\text{rad/s}

. How many revolutions does it make in

10\,\text{s}

Example 34

medium

A wheel slows from

\omega_0 = 15\,\text{rad/s}

\omega = 3\,\text{rad/s}

4\,\text{s}

. Find

\alpha

and state its sign relative to

\omega

Example 35

hard

A point on a wheel of radius

r = 0.4\,\text{m}

has tangential acceleration

a_t = 2\,\text{m/s}^2

and centripetal acceleration

a_c = 8\,\text{m/s}^2

. Find

\omega

and

\alpha

at that instant.

Example 36

hard

A wheel at

\omega_0 = 20\,\text{rad/s}

has constant

\alpha = -2\,\text{rad/s}^2

. After how long does it stop, and through what angle?

Example 37

hard

A cyclist accelerates from rest to

v = 12\,\text{m/s}

8\,\text{s}

. Wheel radius is

0.30\,\text{m}

. Find the wheel's angular acceleration.

Example 38

hard

A rotor's angular velocity follows

\omega(t) = 4 + 3t^2\,\text{rad/s}

. Find

\alpha

t = 2\,\text{s}

Example 39

hard

A point on a

0.5\,\text{m}

radius disk has linear speed

v = 3\,\text{m/s}

. How many revolutions does the disk complete in

20\,\text{s}

at this rate?

Example 40

hard

A propeller spins at

\omega_0 = 50\,\text{rad/s}

and is turned off; friction provides

\alpha = -2.5\,\text{rad/s}^2

. How many revolutions until it stops?

Example 41

challenge

A satellite in low Earth orbit has period

T = 90\,\text{min}

. Find its angular velocity in rad/s.

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Related Concepts

Circular Motion Velocity Angular Momentum

Background Knowledge

These ideas may be useful before you work through the harder examples.

circular motionvelocity