1. Angular Momentum

Angular Momentum

Angular momentum describes the rotational motion of an object. It depends on both how difficult the object is to rotate and how quickly it is rotating.

For a rigid object rotating about a fixed axis:

L = Iω

where:

• L = angular momentum
• I = moment of inertia
• ω = angular velocity

The SI unit of angular momentum is:

kg·m²/s

Angular momentum is the rotational counterpart of linear momentum.

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Understanding Angular Momentum

An object has angular momentum when it is rotating.

The amount of angular momentum depends on:

• the object's mass
• how that mass is distributed around the axis
• how quickly the object rotates
• the direction of rotation

For a rigid rotating object:

L = Iω

This means that angular momentum increases if either:

• I increases
• ω increases

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Moment of Inertia and Angular Momentum

Moment of inertia describes the object's resistance to changes in rotational motion.

For example, a thin ring rotating around its centre has:

I = MR²

Therefore:

L = Iω

becomes:

L = MR²ω

A larger mass, larger radius, or larger angular velocity produces greater angular momentum.

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Worked Example: Rotating Disk

A disk has:

I = 2.5 kg·m²

ω = 8 rad/s

Calculate its angular momentum.

L = Iω

L = (2.5)(8)

L = 20 kg·m²/s

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Effect of Angular Velocity

For an object whose moment of inertia remains constant:

L ∝ ω

Suppose:

I = 4 kg·m²

At:

ω = 5 rad/s

L = (4)(5)

L = 20 kg·m²/s

If angular velocity doubles:

ω = 10 rad/s

L = (4)(10)

L = 40 kg·m²/s

Doubling angular velocity doubles angular momentum.

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Effect of Moment of Inertia

For a fixed angular velocity:

L ∝ I

Suppose two objects both rotate at:

ω = 6 rad/s

Object A:

I_A = 2 kg·m²

Object B:

I_B = 5 kg·m²

Then:

L_A = (2)(6)

L_A = 12 kg·m²/s

L_B = (5)(6)

L_B = 30 kg·m²/s

The object with the greater moment of inertia has greater angular momentum.

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Mass Distribution and Angular Momentum

Moment of inertia depends strongly on the distribution of mass.

Moving mass farther from the rotation axis increases I.

Since:

L = Iω

moving mass outward increases angular momentum if angular velocity is held constant.

However, in an isolated system where angular momentum is conserved, moving mass outward causes ω to decrease.

This distinction is important.

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Angular Momentum of a Point Mass

Angular momentum is not limited to spinning rigid objects.

A point mass moving around an axis can also have angular momentum.

For a point mass moving perpendicular to the radius:

L = mvr

where:

• m = mass
• v = tangential speed
• r = distance from the axis

Since:

v = rω

then:

L = mr(rω)

L = mr²ω

Since:

I = mr²

we obtain:

L = Iω

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Worked Example: Orbiting Point Mass

A 2.0 kg mass moves at:

5.0 m/s

around a circular path of radius:

3.0 m

Its angular momentum is:

L = mvr

L = (2.0)(5.0)(3.0)

L = 30 kg·m²/s

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Angular Momentum Depends on the Axis

Just as moment of inertia depends on the axis of rotation, angular momentum must be considered relative to a chosen axis or point.

Changing the reference axis can change:

• r
• I
• L

Therefore, always identify the axis when analyzing angular momentum.

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Linear Momentum and Angular Momentum

Linear momentum is:

p = mv

Angular momentum for a rigid rotating object is:

L = Iω

These equations show an important parallel.

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Comparing Linear and Angular Momentum

Linear momentum describes an object's tendency to continue moving in a straight-line sense.

Angular momentum describes rotational motion about an axis.

For linear momentum:

more mass → more p

greater velocity → more p

For angular momentum:

greater moment of inertia → more L

greater angular velocity → more L

Both quantities also have direction.

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Direction of Angular Momentum

Angular momentum is a vector quantity.

For a rotating object, its direction lies along the rotation axis.

The right-hand rule can be used to determine the direction.

Curl the fingers of your right hand in the direction of rotation.

Your thumb points in the direction of angular momentum.

For example, when viewed from above:

• counterclockwise rotation → angular momentum points upward
• clockwise rotation → angular momentum points downward

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Positive and Negative Angular Momentum

In one-dimensional rotational problems, we often use signs rather than full vector notation.

A common convention is:

counterclockwise = positive

clockwise = negative

Then:

ω > 0 → L > 0

ω < 0 → L < 0

provided I is positive, which it always is.

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Worked Direction Example

A wheel has:

I = 3 kg·m²

and:

ω = −4 rad/s

Then:

L = Iω

L = (3)(−4)

L = −12 kg·m²/s

The negative sign indicates angular momentum in the chosen negative rotational direction.

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Torque Changes Angular Momentum

Force changes linear momentum.

Similarly:

torque changes angular momentum.

The relationship is:

Στ = ΔL/Δt

For an instantaneous rate of change:

Στ = dL/dt

This is a more general form of Newton's Second Law for rotation.

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Connecting Torque to Στ = Iα

If I remains constant:

L = Iω

Therefore:

ΔL = IΔω

Divide by Δt:

ΔL/Δt = I(Δω/Δt)

Since:

Δω/Δt = α

then:

Στ = Iα

So Newton's Second Law for rotation can be understood as the rate at which torque changes angular momentum.

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Angular Impulse

The product of torque and time is called angular impulse.

For constant torque:

angular impulse = τΔt

and:

τΔt = ΔL

This is the rotational equivalent of linear impulse:

FΔt = Δp

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Worked Example: Angular Impulse

A constant torque of:

6 N·m

acts for:

4 s

The change in angular momentum is:

ΔL = τΔt

ΔL = (6)(4)

ΔL = 24 kg·m²/s

If the object initially had:

L_i = 10 kg·m²/s

then:

L_f = L_i + ΔL

L_f = 10 + 24

L_f = 34 kg·m²/s

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Finding Angular Velocity from Angular Momentum

The equation:

L = Iω

can be rearranged:

ω = L/I

Suppose:

L = 45 kg·m²/s

I = 5.0 kg·m²

Then:

ω = 45/5.0

ω = 9.0 rad/s

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Finding Moment of Inertia

The same equation can be rearranged:

I = L/ω

Suppose:

L = 30 kg·m²/s

ω = 12 rad/s

Then:

I = 30/12

I = 2.5 kg·m²

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Angular Momentum of a Solid Disk

For a solid disk:

I = 1/2 MR²

Therefore:

L = 1/2 MR²ω

Suppose:

M = 8 kg

R = 0.50 m

ω = 10 rad/s

Moment of inertia:

I = 1/2(8)(0.50²)

I = 4(0.25)

I = 1.0 kg·m²

Angular momentum:

L = (1.0)(10)

L = 10 kg·m²/s

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Angular Momentum of a Thin Ring

For a thin ring:

I = MR²

Suppose the ring has the same:

M = 8 kg

R = 0.50 m

ω = 10 rad/s

Then:

I = (8)(0.50²)

I = 2.0 kg·m²

Angular momentum:

L = (2.0)(10)

L = 20 kg·m²/s

The ring has twice the angular momentum of the disk at the same angular velocity because it has twice the moment of inertia.

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Angular Momentum of a Rotating Rod

For a uniform rod rotating about its centre:

I = 1/12 ML²

Suppose:

M = 6 kg

L = 2.0 m

ω = 9 rad/s

Moment of inertia:

I = 1/12(6)(2²)

I = 2.0 kg·m²

Angular momentum:

L = Iω

L = (2.0)(9)

L = 18 kg·m²/s

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Changing the Axis

The same rod rotating about one end has:

I = 1/3 ML²

For:

M = 6 kg

L = 2.0 m

I = 1/3(6)(4)

I = 8 kg·m²

At the same angular velocity:

L = (8)(9)

L = 72 kg·m²/s

The axis of rotation strongly affects moment of inertia and therefore angular momentum.

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Conservation of Angular Momentum

One of the most important principles in rotational mechanics is the conservation of angular momentum.

If the net external torque on a system is zero:

L_initial = L_final

or:

L_i = L_f

For a rotating rigid system:

I_i ω_i = I_f ω_f

This principle explains many dramatic examples of rotational motion.

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Figure Skater Example

A figure skater spins with:

I_i = 6 kg·m²

ω_i = 2 rad/s

The skater pulls their arms inward, reducing moment of inertia to:

I_f = 2 kg·m²

Assuming external torque is negligible:

I_i ω_i = I_f ω_f

(6)(2) = (2)ω_f

12 = 2ω_f

ω_f = 6 rad/s

The moment of inertia decreased by a factor of three.

The angular velocity increased by a factor of three.

Angular momentum remained:

12 kg·m²/s

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Why the Skater Speeds Up

The skater does not need an external torque to increase angular velocity.

Instead:

I decreases

so:

ω increases

in order to keep:

L = Iω

constant.

This is an example of angular momentum conservation rather than angular acceleration caused by an external torque.

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Extending the Arms

The reverse also occurs.

If a skater extends their arms:

• I increases
• ω decreases
• L remains constant if external torque is negligible

Therefore:

mass inward → faster rotation

mass outward → slower rotation

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Diver Example

A diver leaves the platform with angular momentum.

While airborne, external torque may be relatively small.

The diver tucks:

• moment of inertia decreases
• angular velocity increases

Before entering the water, the diver extends:

• moment of inertia increases
• angular velocity decreases

This allows the diver to control rotation without needing a large external torque while airborne.

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Gymnastics

Gymnasts use the same principle during:

• somersaults
• flips
• twists
• dismounts

Changing body position changes moment of inertia.

This changes angular velocity while approximately conserving angular momentum during airborne motion.

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Rotating Chair Experiment

Imagine a person sitting on a freely rotating chair while holding weights.

Initially:

I_i = 5 kg·m²

ω_i = 3 rad/s

Initial angular momentum:

L_i = (5)(3)

L_i = 15 kg·m²/s

The person pulls the weights inward so that:

I_f = 3 kg·m²

Then:

15 = (3)ω_f

ω_f = 5 rad/s

Angular momentum remains constant.

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Angular Momentum and Planets

Angular momentum is important in astronomy.

Planets orbit stars and possess angular momentum associated with their orbital motion.

When external torque is negligible, orbital angular momentum is conserved.

This helps explain why planets move faster when closer to the Sun and more slowly when farther away.

This behaviour is closely related to Kepler's Second Law.

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Collapsing Stars

Angular momentum conservation can produce extreme changes in angular velocity.

Suppose a rotating star contracts.

Its mass moves much closer to its rotation axis.

Therefore:

I decreases dramatically

If external torque is small:

L remains approximately constant

so:

ω increases dramatically

This contributes to the rapid rotation of compact objects such as neutron stars.

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Angular Momentum of Wheels

A spinning wheel has angular momentum along its axle.

The larger its:

• moment of inertia
• angular velocity

the larger its angular momentum.

A rapidly spinning wheel therefore resists changes to the direction of its rotation axis.

This behaviour is important in gyroscopes.

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Gyroscopes

A gyroscope contains a rapidly rotating object.

Its angular momentum helps it resist changes in orientation.

Gyroscopic effects are used in:

• aircraft
• spacecraft
• ships
• navigation systems
• smartphones
• drones
• stabilization systems

Changing the direction of angular momentum requires a torque.

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Angular Momentum in Sports

Angular momentum appears throughout sports.

Examples include:

• figure skating
• gymnastics
• diving
• cycling
• discus
• hammer throw
• baseball
• golf
• ice hockey

Athletes control angular momentum by applying torques and changing body configuration.

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Throwing and Swinging

During a swing or throw, muscles apply torque to body segments.

The torque changes their angular momentum:

τΔt = ΔL

Applying a larger torque or applying torque for a longer time can produce a larger change in angular momentum.

This is the rotational counterpart of impulse in linear motion.

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Angular Momentum in a Collision

Angular momentum can also be conserved during rotational collisions if external torque is negligible.

Suppose a rotating platform has:

I₁ = 4 kg·m²

ω₁ = 6 rad/s

A stationary object lands on it, increasing the total moment of inertia to:

I₂ = 10 kg·m²

Initial angular momentum:

L_i = (4)(6)

L_i = 24 kg·m²/s

Conservation gives:

24 = (10)ω_f

ω_f = 2.4 rad/s

The system rotates more slowly because its moment of inertia increased.

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Rotational Collisions and Energy

Angular momentum may be conserved during a collision even when rotational kinetic energy is not.

Before:

E_rot = 1/2 I_iω_i²

After:

E_rot = 1/2 I_fω_f²

Some kinetic energy may be transferred into:

• heat
• sound
• deformation

Therefore:

conservation of angular momentum does not automatically mean conservation of kinetic energy.

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Worked Energy Comparison

Using the previous example:

Before:

E_i = 1/2(4)(6²)

E_i = 72 J

After:

E_f = 1/2(10)(2.4²)

E_f = 28.8 J

Angular momentum is conserved, but rotational kinetic energy decreases.

The difference is:

72 − 28.8 = 43.2 J

transferred to other forms.

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External Torque and Conservation

Angular momentum is conserved only when:

Στ_external = 0

Internal forces can redistribute angular momentum within the system, but they cannot change the total angular momentum of an isolated system.

This is similar to linear momentum:

no net external force → linear momentum conserved

no net external torque → angular momentum conserved

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Internal and External Torques

Suppose a skater pulls their arms inward.

Muscles exert internal forces and torques within the skater's body.

These can change the distribution of mass and individual motions.

However, if external torque is negligible:

total angular momentum remains constant.

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A Complete Angular Momentum Problem

A rotating platform has:

I_i = 8.0 kg·m²

ω_i = 3.0 rad/s

A person changes position, reducing the system's moment of inertia to:

I_f = 5.0 kg·m²

Assume external torque is negligible.

Initial Angular Momentum

L_i = I_iω_i

L_i = (8.0)(3.0)

L_i = 24 kg·m²/s

Final Angular Momentum

By conservation:

L_f = 24 kg·m²/s

Final Angular Velocity

I_iω_i = I_fω_f

(8.0)(3.0) = (5.0)ω_f

ω_f = 24/5

ω_f = 4.8 rad/s

The system spins faster because its moment of inertia decreased.

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A Problem Involving External Torque

A wheel has:

I = 3.0 kg·m²

and initially rotates at:

ω_i = 4.0 rad/s

A constant net torque of:

6.0 N·m

acts for:

5.0 s

Initial Angular Momentum

L_i = Iω_i

L_i = (3.0)(4.0)

L_i = 12 kg·m²/s

Change in Angular Momentum

ΔL = τΔt

ΔL = (6.0)(5.0)

ΔL = 30 kg·m²/s

Final Angular Momentum

L_f = L_i + ΔL

L_f = 12 + 30

L_f = 42 kg·m²/s

Final Angular Velocity

ω_f = L_f/I

ω_f = 42/3.0

ω_f = 14 rad/s

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Checking with Newton's Second Law

We can verify the previous answer.

α = τ/I

α = 6.0/3.0

α = 2.0 rad/s²

Then:

ω_f = ω_i + αt

ω_f = 4.0 + (2.0)(5.0)

ω_f = 14 rad/s

Both methods agree.

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Linear and Angular Conservation Laws

There is a powerful symmetry between translational and rotational mechanics.

This parallel makes angular momentum easier to understand if linear momentum is already familiar.

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Choosing the Right Angular Momentum Equation

For a rigid object rotating around a fixed axis:

L = Iω

For a point mass moving perpendicular to the radius:

L = mvr

For angular momentum conservation:

L_i = L_f

or commonly:

I_iω_i = I_fω_f

For a change caused by constant torque:

ΔL = τΔt

Choosing the correct relationship depends on the situation.

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Problem-Solving Strategy

First identify the system and rotation axis.

Then determine whether you are dealing with:

• a rotating rigid body
• a point mass
• a changing mass distribution
• an external torque
• a rotational collision

If I and ω are known, use:

L = Iω

If a point mass moves perpendicular to the radius, use:

L = mvr

If external torque is negligible, use:

L_i = L_f

If torque acts over a known time, use:

ΔL = τΔt

Finally, check:

• units
• signs
• rotation direction
• whether angular momentum should be conserved

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Common Mistakes

Confusing angular momentum with linear momentum

Linear momentum is p = mv. Angular momentum describes motion about an axis.

Using mass instead of moment of inertia

For a rigid rotating body:

L = Iω

not mω.

Ignoring mass distribution

Changing mass distribution changes I and can therefore change ω when angular momentum is conserved.

Assuming angular momentum is always conserved

It is conserved only when the net external torque is zero.

Thinking pulling the arms inward creates angular momentum

It changes I and ω, but total L remains constant when external torque is negligible.

Forgetting direction

Angular momentum is a vector quantity.

Assuming zero torque means zero angular momentum

Zero net torque means angular momentum does not change. It may already be nonzero.

Assuming conservation of angular momentum means conservation of kinetic energy

The two are separate conservation principles.

Ignoring the axis of rotation

Angular momentum depends on the chosen reference axis.

Confusing torque with angular momentum

Torque describes the rate at which angular momentum changes.

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Key Terms

Angular momentum (L) — quantity describing rotational motion, commonly calculated for a rigid body using L = Iω.

Moment of inertia (I) — measure of an object's resistance to changes in rotational motion and its mass distribution relative to an axis.

Angular velocity (ω) — rate at which angular position changes.

Linear momentum (p) — quantity of translational motion calculated using p = mv.

Torque (τ) — turning effect of a force and the rate of change of angular momentum.

Conservation of angular momentum — principle stating that total angular momentum remains constant when net external torque is zero.

Angular impulse — product of torque and time, equal to the change in angular momentum.

External torque — torque produced by forces originating outside the chosen system.

Internal torque — torque produced by interactions between parts of the chosen system.

Rotation axis — axis relative to which rotational quantities are measured.

Right-hand rule — method for determining the direction of angular momentum.

Rotational collision — interaction between objects in which rotational motion and angular momentum are important.

Gyroscope — rotating device whose angular momentum gives it resistance to changes in orientation.

Isolated system — system with no relevant external interaction capable of changing its total momentum.

Rotational kinetic energy — energy of rotation, calculated using E_rot = 1/2 Iω².

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Key Takeaways

• Angular momentum describes rotational motion.
• For a rigid body rotating around a fixed axis, L = Iω.
• Angular momentum is measured in kg·m²/s.
• Greater moment of inertia produces greater angular momentum at the same angular velocity.
• Greater angular velocity produces greater angular momentum for the same moment of inertia.
• Mass distribution affects angular momentum through moment of inertia.
• Angular momentum has direction and is therefore a vector.
• The right-hand rule can determine its direction.
• Linear momentum and angular momentum are closely related concepts.
• Linear momentum uses p = mv, while rotational angular momentum uses L = Iω.
• Torque changes angular momentum.
• Στ = ΔL/Δt describes the relationship between torque and angular momentum.
• Angular impulse is τΔt = ΔL for constant torque.
• If net external torque is zero, total angular momentum is conserved.
• Conservation of angular momentum is commonly expressed as I_iω_i = I_fω_f.
• Moving mass inward decreases I and can increase ω.
• Moving mass outward increases I and can decrease ω.
• Figure skaters, divers, gymnasts, rotating platforms, planets and stars demonstrate angular momentum principles.
• Angular momentum can be conserved even when kinetic energy is not.
• Conservation of angular momentum is one of the fundamental conservation laws of physics.

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Check Your Understanding

1. Define angular momentum and give its SI unit.
2. A wheel has I = 3.0 kg·m² and rotates at 8.0 rad/s. Calculate its angular momentum.
3. The angular velocity of the wheel in Question 2 doubles while I remains constant. Calculate its new angular momentum.
4. Two objects rotate at the same angular velocity. Object A has I = 2 kg·m² and Object B has I = 7 kg·m². Compare their angular momenta.
5. A 3.0 kg point mass travels at 6.0 m/s around a circular path of radius 2.0 m. Calculate its angular momentum about the centre.
6. A solid disk has mass 10 kg, radius 0.40 m, and angular velocity 12 rad/s. Calculate its moment of inertia and angular momentum.
7. A thin ring has the same mass, radius, and angular velocity as the disk in Question 6. Calculate its angular momentum and explain why it differs.
8. A rotating object has angular momentum 36 kg·m²/s and moment of inertia 4.0 kg·m². Calculate its angular velocity.
9. Explain how torque is related to the rate of change of angular momentum.
10. A constant torque of 8.0 N·m acts for 3.0 s. Calculate the change in angular momentum.
11. A skater initially has I = 5.0 kg·m² and ω = 3.0 rad/s. The skater pulls their arms inward so that I = 2.0 kg·m². Assuming angular momentum is conserved, calculate the final angular velocity.
12. Explain why pulling the arms inward makes a spinning skater rotate faster without increasing the system's total angular momentum.
13. A rotating platform initially has I = 6.0 kg·m² and ω = 5.0 rad/s. A stationary object lands on it, increasing the total moment of inertia to 10 kg·m². Calculate the final angular velocity.
14. Explain why angular momentum can be conserved during a rotational collision even though rotational kinetic energy decreases.
15. A wheel has I = 4.0 kg·m² and initially rotates at 6.0 rad/s. A constant net torque of 8.0 N·m acts in the direction of rotation for 5.0 s. Calculate the initial angular momentum, angular impulse, final angular momentum, and final angular velocity.