Angular Momentum Examples in Physics

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

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 rotational equivalent of linear momentum, measuring the quantity of rotational motion in a spinning or orbiting object.

A spinning skater pulling their arms in spins faster — they're conserving angular momentum.

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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 Momentum works by defining the interacting system and comparing motion before and after the interaction.

Common stuck point: Students often know a formula related to angular momentum but skip the recognition step: Is the interaction short, collision-like, or rotational, and have I checked whether external forces or torques can be ignored? That leads to a correct-looking substitution attached to the wrong physical model.

Sense of Study hint: Ask: Is the interaction short, collision-like, or rotational, and have I checked whether external forces or torques can be ignored?

Worked Examples

Example 1

medium

A skater has

I_1 = 8\,\text{kg·m}^2

\omega_1 = 1.5\,\text{rad/s}

. They tuck in to

I_2 = 3\,\text{kg·m}^2

. Find

\omega_2

Answer

\omega_2 = 4 \text{ rad/s}

First step

Conserve

L

I_1 \omega_1 = I_2 \omega_2

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Example 2

hard

Show that if KE is conserved when a skater pulls in their arms (

I

halves), then

L

cannot be conserved. (Reach a contradiction.)

Practice Problems

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

Example 1

easy

A disk has moment of inertia

I = 4\,\text{kg·m}^2

and spins at

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

. Find its angular momentum.

Example 2

easy

A 2 kg mass moves at

5\,\text{m/s}

in a circle of radius

3\,\text{m}

. Find its angular momentum about the center.

2 kg mass in circular motion: r = 3 m, v = 5 m/s

Example 3

easy

Angular momentum

L = 18\,\text{kg·m}^2/\text{s}

and

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

. Find the moment of inertia.

Example 4

easy

A spinning skater pulls in their arms, reducing

I

. What happens to their spin rate

\omega

(no external torque)?

Example 5

easy

Is angular momentum a vector or a scalar?

Example 6

easy

A skater has

I = 2\,\text{kg·m}^2

spinning at

4\,\text{rad/s}

. Find their angular momentum.

Example 7

easy

If no external torque acts on a system, what happens to its total angular momentum?

Example 8

easy

A 0.5 kg ball whirls at

2\,\text{m/s}

on a string of radius

4\,\text{m}

. Find its angular momentum.

0.5 kg ball whirling on a 4 m string at 2 m/s

Example 9

medium

A skater spins at

2\,\text{rad/s}

with

I = 6\,\text{kg·m}^2

, then pulls in arms to

I = 2\,\text{kg·m}^2

. Find the new spin rate.

Example 10

medium

A merry-go-round (

I = 100\,\text{kg·m}^2

) spins at

1\,\text{rad/s}

. A child adds

I = 25\,\text{kg·m}^2

by stepping on at the rim. Find the new angular speed.

Example 11

medium

A ball on a string moves at

4\,\text{m/s}

at radius

2\,\text{m}

. The string is pulled in to radius

1\,\text{m}

. Find the new speed (conserve

L = mvr

Example 12

medium

A 3 kg point mass at

r = 2\,\text{m}

rotates at

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

. Find its angular momentum (

I = mr^2

Example 13

medium

Two disks: one spinning (

I = 4\,\text{kg·m}^2

6\,\text{rad/s}

) drops onto a stationary one (

I = 2\,\text{kg·m}^2

); they couple. Find the common angular speed.

Example 14

medium

A planet at perihelion moves at

60\,\text{km/s}

r = 1

unit; at aphelion

r = 4

units. Find its aphelion speed (conserve

L = mvr

Example 15

medium

A wheel slows from

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

to rest in

5\,\text{s}

with

I = 2\,\text{kg·m}^2

. Find the average torque applied.

Example 16

medium

A disk with

I = 5\,\text{kg·m}^2

spins at

4\,\text{rad/s}

, then a brake changes its inertia arrangement so

I = 10\,\text{kg·m}^2

(mass moved outward, no external torque). Find the new angular speed.

Example 17

medium

A 4 kg point mass moves at

3\,\text{m/s}

along a line passing

2\,\text{m}

from a reference point (perpendicular distance). Find its angular momentum about that point.

L = mvd: velocity × perpendicular moment arm

Example 18

challenge

A 0.5 kg bug lands on the rim of a spinning disk (

I = 1.5\,\text{kg·m}^2

\omega = 4\,\text{rad/s}

) at radius

r = 2\,\text{m}

. Find the new angular speed after the bug sticks.

Example 19

challenge

A rod (

I = 3\,\text{kg·m}^2

) spins at

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

. A torque of

6\,\text{N·m}

opposes it. How long until it stops?

τ · t = ΔL: opposing torque removes all angular momentum

Example 20

challenge

A child (mass

30\,\text{kg}

) runs at

4\,\text{m/s}

tangent to the rim of a stationary merry-go-round (

I = 200\,\text{kg·m}^2

, radius

2\,\text{m}

) and jumps on. Find the resulting angular speed.

Example 21

easy

A wheel has

I = 5\,\text{kg·m}^2

and spins at

\omega = 4\,\text{rad/s}

. Find

L

Example 22

easy

1\,\text{kg}

point mass orbits at

r = 2\,\text{m}

with

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

. Find

L

Example 23

easy

True/false: external torque is needed to change a system's angular momentum.

Example 24

easy

A hoop of mass

2\,\text{kg}

and radius

0.5\,\text{m}

rolls so its center moves at

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

. Find

L_{spin}

. (

I_{hoop} = MR^2

Example 25

easy

A solid disk has

I = \tfrac12 M R^2

. For

M = 4\,\text{kg}

R = 0.5\,\text{m}

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

, find

L

Example 26

easy

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

and

L = 20\,\text{kg·m}^2/\text{s}

, find

I

Example 27

easy

A planet's angular momentum about the Sun is approximately conserved. True or false?

Example 28

medium

A solid disk (

M = 6\,\text{kg}

R = 0.4\,\text{m}

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

) is suddenly joined by a coaxial

4\,\text{kg}

disk of the same radius initially at rest. Find the new

\omega

Example 29

medium

Earth has

I \approx 8.0 \times 10^{37}\,\text{kg·m}^2

and rotates at

\omega \approx 7.3 \times 10^{-5}\,\text{rad/s}

. Estimate Earth's spin angular momentum.

Example 30

medium

A neutron star forms when a stellar core of radius

7 \times 10^5\,\text{km}

collapses to

10\,\text{km}

, keeping

M

constant (

I \propto R^2

for a uniform sphere). If it spun at one rev/month before, find its new period.

Example 31

medium

A merry-go-round of

I = 200\,\text{kg·m}^2

is at rest. A

50\,\text{kg}

child runs tangentially at

3\,\text{m/s}

and jumps on at radius

2\,\text{m}

. Find the new angular speed.

Example 32

medium

A child of mass

30\,\text{kg}

stands on a

I = 150\,\text{kg·m}^2

turntable spinning at

\omega = 2\,\text{rad/s}

r = 3\,\text{m}

. They walk to

r = 1\,\text{m}

. Find the new

\omega

Example 33

medium

A net torque of

\tau = 4\,\text{N·m}

acts on a disk for

3\,\text{s}

. Find the change in

L

Example 34

medium

A flywheel of

I = 25\,\text{kg·m}^2

initially at rest reaches

\omega = 40\,\text{rad/s}

10\,\text{s}

. Find the average torque applied.

Example 35

medium

A particle moves at constant velocity along a straight line that does NOT pass through point

O

. Is its angular momentum about

O

constant?

Example 36

medium

0.4\,\text{kg}

puck on a frictionless air hockey table moves at

5\,\text{m/s}

along a line

2\,\text{m}

from the table's center. Find its angular momentum about the center.

L = mvd: straight-line motion still has angular momentum about a point

Example 37

medium

200\,\text{kg}

flywheel disk of radius

1\,\text{m}

stores

L = 5000\,\text{kg·m}^2/\text{s}

. Find its spin rate. (

I = \tfrac12 M R^2

Example 38

hard

A bullet (

m = 0.05\,\text{kg}

v = 400\,\text{m/s}

) embeds in the rim of a stationary disk (

M = 5\,\text{kg}

R = 0.5\,\text{m}

I = \tfrac12 M R^2

). Find the disk's angular speed after.

Example 39

hard

A rod of length

L = 1\,\text{m}

, mass

M = 2\,\text{kg}

, pivots at its center. A

0.1\,\text{kg}

clay ball at

10\,\text{m/s}

hits the rod's end and sticks. Find the angular speed after. (

I_{rod,center} = \tfrac{1}{12}M L^2

Example 40

hard

A spinning bicycle wheel (

L_{spin}

along its axle, horizontal) is held by a person on a stationary turntable. The person flips the wheel

180^\circ

. What is the person+turntable's final

L

if their

I = 8\,\text{kg·m}^2

and the wheel's

L_{spin} = 4\,\text{kg·m}^2/\text{s}

Example 41

hard

Two skaters on ice, each of mass

60\,\text{kg}

, hold a

4\,\text{m}

pole and rotate about its center at

\omega = 1.2\,\text{rad/s}

. They pull together so they are now

1\,\text{m}

apart. Find the new

\omega

(treat them as point masses, pole massless).

Example 42

hard

A comet's perihelion (closest approach) is

0.5\,\text{AU}

at speed

60\,\text{km/s}

. At aphelion it is

50\,\text{AU}

from the Sun. Find its aphelion speed.

Example 43

hard

3\,\text{m}

uniform rod (

M = 4\,\text{kg}

) lies on frictionless ice. A

0.2\,\text{kg}

puck moving at

8\,\text{m/s}

strikes it perpendicularly

1\,\text{m}

from the center and sticks. Find the angular speed of the rod+puck about their new center of mass. (

I_{rod,CM} = \tfrac{1}{12}M L^2

Example 44

challenge

A figure skater spins at

1.0\,\text{rev/s}

with arms out (

I = 5\,\text{kg·m}^2

). She tucks to

I = 1\,\text{kg·m}^2

. Find her new rotational KE and the work she did.

Example 45

challenge

A solid sphere (

I = \tfrac25 M R^2

M = 2\,\text{kg}

R = 0.1\,\text{m}

) rolls without slipping at

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

. Find its total angular momentum about a point on the ground directly below its current center.

Example 46

challenge

A platform (

I_p = 100\,\text{kg·m}^2

) rotates at

\omega_0 = 2\,\text{rad/s}

. A

50\,\text{kg}

person stands at rim

r = 2\,\text{m}

. They jump off radially outward. Find the platform's new angular speed.

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Related Concepts

Torque Momentum

Background Knowledge

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

torquemomentum