Angular Momentum and Energy
2. Conservation of Angular Momentum
Learning outcomes
- I can state the principle of conservation of angular momentum.
- I can explain why angular momentum is conserved.
- I can analyze systems with changing moment of inertia.
- I can predict outcomes using conservation principles.
- I can solve conservation problems involving rotation
- Law of conservation: in the absence of external torques.
- Examples: Ice skaters spinning, collapsing stars, gyroscopes.
Conservation of Angular Momentum
The principle of conservation of angular momentum states that the total angular momentum of a system remains constant when there is no net external torque acting on the system.
In simple form:
L_initial = L_final
For a rotating rigid system:
I_iω_i = I_fω_f
where:
- I_i = initial moment of inertia
- ω_i = initial angular velocity
- I_f = final moment of inertia
- ω_f = final angular velocity
This principle explains why figure skaters spin faster when they pull in their arms, why divers rotate faster when they tuck, and why collapsing stars can rotate extremely rapidly.
Angular Momentum
For a rigid object rotating around a fixed axis:
L = Iω
Angular momentum depends on:
- the object's moment of inertia
- its angular velocity
- its direction of rotation
Moment of inertia depends on how mass is distributed relative to the axis.
Therefore, angular momentum can remain constant even while I and ω individually change.
The Conservation Principle
If the net external torque on a system is zero:
Στ_external = 0
then:
ΔL = 0
Therefore:
L_i = L_f
This does not mean that nothing in the system can change.
The system may change:
- shape
- moment of inertia
- angular velocity
- distribution of mass
What remains constant is the total angular momentum.
Why Angular Momentum Is Conserved
Torque is related to the rate of change of angular momentum:
Στ_external = ΔL/Δt
If:
Στ_external = 0
then:
ΔL/Δt = 0
Therefore:
ΔL = 0
and angular momentum remains constant.
This is the rotational equivalent of conservation of linear momentum.
For linear motion:
ΣF_external = 0 → p remains constant
For rotational motion:
Στ_external = 0 → L remains constant
Internal and External Torques
The distinction between internal and external torques is important.
Internal forces and torques act between parts of the chosen system.
External forces and torques come from outside the system.
Internal interactions can redistribute angular momentum between parts of a system, but they do not change the system's total angular momentum.
For example, a figure skater uses muscles to pull their arms inward. These internal forces change the skater's mass distribution, but if external torque is negligible, total angular momentum remains constant.
Changing Moment of Inertia
For a rotating system:
L = Iω
If angular momentum is conserved:
I_iω_i = I_fω_f
This produces an important relationship:
I increases → ω decreases
I decreases → ω increases
This inverse relationship appears throughout rotational mechanics.
Figure Skater
A spinning figure skater provides a classic example.
With arms extended:
- mass is distributed farther from the axis
- moment of inertia is relatively large
- angular velocity is relatively small
When the skater pulls their arms inward:
- mass moves closer to the axis
- moment of inertia decreases
- angular velocity increases
If external torque is negligible:
I_iω_i = I_fω_f
Worked Example: Figure Skater
A skater initially has:
I_i = 6.0 kg·m²
ω_i = 2.0 rad/s
The skater pulls their arms inward so that:
I_f = 2.0 kg·m²
Find the new angular velocity.
I_iω_i = I_fω_f
(6.0)(2.0) = (2.0)ω_f
12 = 2ω_f
ω_f = 6.0 rad/s
The moment of inertia decreased by a factor of three, so the angular velocity increased by a factor of three.
Checking the Angular Momentum
Before:
L_i = I_iω_i
L_i = (6)(2)
L_i = 12 kg·m²/s
After:
L_f = I_fω_f
L_f = (2)(6)
L_f = 12 kg·m²/s
The angular velocities are different, but the angular momentum is unchanged.
Extending the Arms
The process works in reverse.
Suppose:
I_i = 2 kg·m²
ω_i = 8 rad/s
The skater extends their arms so:
I_f = 5 kg·m²
Conservation gives:
(2)(8) = (5)ω_f
16 = 5ω_f
ω_f = 3.2 rad/s
Increasing the moment of inertia decreases the angular velocity.
Predicting Changes Without Calculating
Many conservation problems can be solved using proportional reasoning.
Since:
I_iω_i = I_fω_f
if I doubles:
ω becomes half as large
if I triples:
ω becomes one-third as large
if I becomes half as large:
ω doubles
if I becomes one-quarter as large:
ω becomes four times larger
These relationships assume angular momentum is conserved.
Divers
Divers use conservation of angular momentum to control their rotation.
After leaving the diving board, the diver may experience relatively little external torque.
When the diver tucks:
- mass moves closer to the rotation axis
- I decreases
- ω increases
When the diver straightens before entering the water:
- I increases
- ω decreases
The diver can therefore change rotational speed without requiring a large external torque while airborne.
Gymnastics
Gymnasts use the same principle during:
- somersaults
- flips
- twists
- dismounts
A tucked position generally produces a smaller moment of inertia than an extended position.
Therefore, for approximately constant angular momentum:
tuck → faster rotation
extend → slower rotation
This allows the gymnast to control the timing of a landing.
Rotating Chair Demonstration
A person sits on a low-friction rotating chair holding masses.
Initially:
I_i = 8 kg·m²
ω_i = 1.5 rad/s
The person pulls the masses inward, reducing the moment of inertia to:
I_f = 3 kg·m²
Initial angular momentum:
L = (8)(1.5)
L = 12 kg·m²/s
Therefore:
12 = (3)ω_f
ω_f = 4 rad/s
The person rotates much faster.
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Moving Point Masses
The conservation principle can also be applied to systems containing point masses.
For a point mass:
I = mr²
Suppose two identical masses move closer to the rotation axis.
Because:
I ∝ r²
even a relatively small change in radius can produce a large change in moment of inertia.
Worked Example: Changing Radius
Two masses rotate around an axis.
Initially each mass is:
0.80 m
from the axis.
They are moved inward to:
0.40 m
The mass does not change.
For each mass:
I ∝ r²
Therefore:
I_f/I_i = (0.40/0.80)²
I_f/I_i = (1/2)²
I_f = 1/4 I_i
Conservation requires:
I_iω_i = I_fω_f
Therefore:
ω_f = 4ω_i
Halving the radius causes the angular velocity to become four times larger.
Why Radius Has Such a Strong Effect
Moment of inertia depends on the square of the distance from the axis.
For a point mass:
I = mr²
Therefore, moving mass inward can dramatically reduce I.
This is why pulling the arms inward can have such a noticeable effect on rotational speed.
Rotational Collisions
Conservation of angular momentum is especially useful when objects collide or join together while rotating.
If external torque during the collision is negligible:
L_before = L_after
This works even when the collision is inelastic.
Object Landing on a Rotating Platform
A platform has:
I_p = 4.0 kg·m²
and rotates at:
ω_i = 6.0 rad/s
A stationary object lands on the platform.
The total final moment of inertia becomes:
I_f = 10 kg·m²
Before:
L_i = I_pω_i
L_i = (4)(6)
L_i = 24 kg·m²/s
After:
L_f = I_fω_f
Conservation gives:
24 = 10ω_f
ω_f = 2.4 rad/s
The system slows because its moment of inertia increased.
Why the Platform Slows Down
The stationary object increases the system's moment of inertia.
Since no significant external torque acts:
angular momentum must remain constant
Therefore, increasing I requires ω to decrease.
This is the same principle as a skater extending their arms.
Angular Momentum Can Be Shared
Consider two rotating disks that are brought into contact and eventually rotate together.
Initially:
Disk A has angular momentum.
Disk B may be stationary or rotating differently.
Friction between the disks produces internal torques that redistribute angular momentum.
If external torque is negligible:
L_total before = L_total after
Worked Example: Two Disks
Disk A:
I₁ = 2 kg·m²
ω₁ = 10 rad/s
Disk B:
I₂ = 3 kg·m²
ω₂ = 0
They are coupled together and rotate at the same final angular velocity.
Initial angular momentum:
L_i = I₁ω₁ + I₂ω₂
L_i = (2)(10) + (3)(0)
L_i = 20 kg·m²/s
Final moment of inertia:
I_f = 2 + 3
I_f = 5 kg·m²
Therefore:
20 = 5ω_f
ω_f = 4 rad/s
Opposite Rotational Directions
Direction must be considered when conserving angular momentum.
Choose one direction as positive.
For example:
counterclockwise = positive
clockwise = negative
Suppose:
I₁ = 3 kg·m²
ω₁ = +6 rad/s
and:
I₂ = 2 kg·m²
ω₂ = −4 rad/s
Initial angular momentum:
L_i = I₁ω₁ + I₂ω₂
L_i = (3)(6) + (2)(−4)
L_i = 18 − 8
L_i = +10 kg·m²/s
If they join:
I_f = 3 + 2
I_f = 5 kg·m²
Then:
10 = 5ω_f
ω_f = +2 rad/s
The final rotation is counterclockwise.
When Opposing Angular Momenta Cancel
Suppose two identical disks have equal angular speeds in opposite directions.
Disk A:
L_A = +20 kg·m²/s
Disk B:
L_B = −20 kg·m²/s
Then:
L_total = 0
If they are coupled together:
ω_f = 0
They stop rotating.
The individual disks initially had angular momentum, but the total angular momentum of the system was zero.
Conservation Does Not Mean Each Object Keeps Its Momentum
Angular momentum conservation applies to the total system.
Individual objects can:
- gain angular momentum
- lose angular momentum
- reverse rotational direction
provided the total angular momentum remains constant when external torque is zero.
Angular Momentum and Rotational Kinetic Energy
Angular momentum conservation does not necessarily mean rotational kinetic energy is conserved.
Rotational kinetic energy is:
E_rot = 1/2 Iω²
During collisions or coupling:
- angular momentum may remain constant
- kinetic energy may decrease
Energy can be transferred into:
- thermal energy
- sound
- deformation
Worked Energy Comparison
Consider the two-disk example:
Before:
I₁ = 2 kg·m²
ω₁ = 10 rad/s
Disk B is stationary.
Initial energy:
E_i = 1/2(2)(10²)
E_i = 100 J
After:
I_f = 5 kg·m²
ω_f = 4 rad/s
E_f = 1/2(5)(4²)
E_f = 2.5(16)
E_f = 40 J
Angular momentum was conserved, but:
100 − 40 = 60 J
of mechanical energy was transferred to other forms.
Angular Momentum and Energy When Shape Changes
A surprising situation occurs when a skater pulls their arms inward.
Angular momentum remains constant:
L = Iω
But rotational kinetic energy can increase.
How is this possible?
The skater's muscles do work while pulling the arms inward.
Chemical energy stored in the body is transferred into rotational kinetic energy.
Therefore:
conservation of angular momentum does not require constant rotational kinetic energy.
Example: Energy of a Spinning Skater
Initially:
I_i = 6 kg·m²
ω_i = 2 rad/s
Initial energy:
E_i = 1/2(6)(2²)
E_i = 12 J
After pulling inward:
I_f = 2 kg·m²
ω_f = 6 rad/s
Final energy:
E_f = 1/2(2)(6²)
E_f = 36 J
Angular momentum remains:
12 kg·m²/s
but rotational kinetic energy increases from:
12 J to 36 J
The additional energy came from work done by the skater.
External Torque Breaks the Simple Conservation Condition
If a significant external torque acts:
L_i ≠ L_f
Instead:
ΔL = τ_external Δt
for constant external torque.
Therefore, before applying conservation of angular momentum, always ask:
Is the net external torque approximately zero?
Example with External Torque
A wheel initially has:
L_i = 20 kg·m²/s
An external torque of:
5 N·m
acts for:
4 s
Change in angular momentum:
ΔL = τΔt
ΔL = (5)(4)
ΔL = 20 kg·m²/s
Therefore:
L_f = 20 + 20
L_f = 40 kg·m²/s
Angular momentum was not conserved because an external torque acted.
Planets and Orbital Motion
Angular momentum conservation is also important in orbital motion.
A planet moves faster when closer to the Sun and slower when farther away.
The gravitational force points approximately toward the Sun.
Because this force acts along the radius, its torque about the Sun is zero:
τ = rF sin 180° = 0
Therefore, the planet's angular momentum about the Sun is conserved.
Connection to Kepler's Second Law
Kepler's Second Law states that a line from a planet to the Sun sweeps out equal areas in equal times.
Angular momentum conservation provides the physical explanation.
Near the Sun:
- r is smaller
- orbital speed is greater
Farther away:
- r is larger
- orbital speed is smaller
The changing speed allows angular momentum to remain constant.
Collapsing Stars
One of the most dramatic applications occurs when stars collapse.
A large star can shrink to form a compact object such as a neutron star.
As the radius decreases:
- mass moves closer to the rotation axis
- moment of inertia decreases dramatically
- angular velocity increases
This is the same basic principle as a skater pulling in their arms, but on an astronomical scale.
Simple Model of a Contracting Star
Treat a star approximately as a sphere.
Moment of inertia has the form:
I = kMR²
where k depends on mass distribution.
If mass and k remain approximately constant:
I ∝ R²
Suppose the radius decreases to:
1/100
of its original value.
Then:
I_f/I_i = (1/100)²
I_f = I_i/10,000
If angular momentum is conserved:
ω_f ≈ 10,000ω_i
The actual behaviour of stars is more complicated, but the example shows how powerful the radius-squared dependence can be.
Gyroscopes
A rapidly spinning gyroscope has substantial angular momentum.
Changing the direction of that angular momentum requires torque.
This property makes gyroscopic systems useful for:
- navigation
- spacecraft orientation
- aircraft instruments
- drones
- stabilization systems
Angular Momentum in Sports
Athletes regularly use conservation of angular momentum.
Examples include:
- figure skating
- gymnastics
- diving
- freestyle skiing
- snowboarding
- trampoline
- aerial sports
Once an athlete is airborne, the external torque may be relatively small.
Changing body configuration allows the athlete to control angular velocity.
Why an Athlete Cannot Create Angular Momentum in Mid-Air
Once airborne, an athlete can change:
- moment of inertia
- angular velocity
- body orientation
But without significant external torque, the athlete cannot change the total angular momentum of the system.
The initial angular momentum must therefore usually be generated during contact with:
- the ground
- a diving board
- ice
- sporting equipment
External forces during takeoff provide the torque that establishes the initial angular momentum.
Reaction Wheels in Spacecraft
Spacecraft can change orientation using reaction wheels.
A motor spins a wheel inside the spacecraft.
If the wheel gains angular momentum in one direction, the rest of the spacecraft gains angular momentum in the opposite direction.
Total angular momentum of the spacecraft-wheel system remains constant if external torque is negligible.
This allows spacecraft to rotate without continuously firing external thrusters.
Worked Reaction-Wheel Example
Suppose a spacecraft is initially stationary.
A reaction wheel gains:
+30 kg·m²/s
of angular momentum.
The spacecraft body must gain:
−30 kg·m²/s
so that:
L_total = 0
If the spacecraft body has:
I = 150 kg·m²
then:
L = Iω
−30 = (150)ω
ω = −0.20 rad/s
The spacecraft rotates opposite to the reaction wheel.
Conservation in a System of Several Objects
For multiple rotating objects:
L_total = L₁ + L₂ + L₃ + ...
If external torque is zero:
ΣL_initial = ΣL_final
Signs must be included to represent rotational direction.
This is especially useful for:
- coupled disks
- gears
- spacecraft systems
- rotating platforms
- rotational collisions
A Complete Conservation Problem
A rotating platform has:
I_p = 5.0 kg·m²
ω_i = 8.0 rad/s
A stationary object with an effective moment of inertia of:
I_o = 3.0 kg·m²
is dropped onto the platform and sticks.
Initial Angular Momentum
L_i = I_pω_i
L_i = (5.0)(8.0)
L_i = 40 kg·m²/s
Final Moment of Inertia
I_f = 5.0 + 3.0
I_f = 8.0 kg·m²
Apply Conservation
L_i = L_f
40 = (8.0)ω_f
ω_f = 5.0 rad/s
The angular velocity decreases because the moment of inertia increases.
Checking the Energy
Initial rotational energy:
E_i = 1/2(5)(8²)
E_i = 2.5(64)
E_i = 160 J
Final rotational energy:
E_f = 1/2(8)(5²)
E_f = 4(25)
E_f = 100 J
Angular momentum is conserved, but kinetic energy decreases by:
60 J
The lost mechanical energy is transferred to other forms during the inelastic interaction.
Predicting Outcomes
Before calculating, conservation principles can often predict what will happen.
Mass moves inward
I decreases.
Therefore:
ω increases
Mass moves outward
I increases.
Therefore:
ω decreases
Stationary mass joins a rotating system
I increases.
Therefore:
ω usually decreases
Two objects rotating in the same direction join
Their angular momenta add.
Two objects rotating in opposite directions join
Their angular momenta partially or completely cancel.
External torque acts
Total angular momentum changes.
Problem-Solving Strategy
First, define the system.
Then choose a positive rotational direction.
Next, determine whether significant external torque acts.
If:
Στ_external ≈ 0
use conservation:
L_i = L_f
Write the angular momentum of every relevant part of the system.
For rigid rotating objects:
L = Iω
Then construct:
ΣI_iω_i = ΣI_fω_f
Solve for the unknown.
Finally, check:
- units
- direction/sign
- whether the change in ω makes sense
- whether I increased or decreased
Do not automatically assume kinetic energy is conserved.
Conservation of Linear and Angular Momentum
The two conservation laws follow very similar patterns.
| Linear Motion | Rotational Motion |
|---|---|
| p = mv | L = Iω |
| Net external force changes p | Net external torque changes L |
| ΣF_ext = 0 | Στ_ext = 0 |
| p_i = p_f | L_i = L_f |
| m₁v₁ + m₂v₂ = ... | I₁ω₁ + I₂ω₂ = ... |
Both are fundamental conservation principles.
Common Mistakes
Assuming angular momentum is always conserved
It is conserved only when net external torque is zero or negligible.
Conserving angular velocity
Angular velocity does not have to remain constant. Angular momentum does.
Ignoring changes in moment of inertia
If mass distribution changes, I changes.
Thinking pulling inward creates angular momentum
It changes I and ω while L remains constant.
Ignoring rotational direction
Clockwise and counterclockwise angular momenta must have opposite signs.
Conserving each object's angular momentum separately
It is usually the total angular momentum of the chosen system that is conserved.
Assuming kinetic energy must also be conserved
Angular momentum can be conserved while kinetic energy changes.
Forgetting stationary objects
An initially stationary object has zero initial angular momentum, but it can still change the final moment of inertia.
Using mass instead of moment of inertia
For rigid-body rotation, use:
L = Iω
Forgetting external torque
Friction, motors, brakes, supports, and contact forces can sometimes provide external torque.
Using the wrong axis
All angular momenta and moments of inertia must be calculated about the same reference axis.
Key Terms
Conservation of angular momentum — principle stating that total angular momentum remains constant when net external torque is zero.
Angular momentum (L) — rotational quantity commonly calculated using L = Iω.
Moment of inertia (I) — measure of mass distribution and resistance to changes in rotational motion.
Angular velocity (ω) — rate of change of angular position.
External torque — torque produced by an interaction outside the chosen system.
Internal torque — torque produced by interactions between parts within the chosen system.
Isolated rotational system — system experiencing no significant net external torque.
Rotational collision — interaction involving rotating objects in which angular momentum may be transferred between components.
Inelastic rotational collision — rotational collision in which objects may join and rotational kinetic energy is not conserved.
Reaction wheel — rotating wheel used to change spacecraft orientation through angular momentum exchange.
Mass distribution — arrangement of mass relative to a rotation axis.
Rotational kinetic energy — energy associated with rotation, calculated using E_rot = 1/2 Iω².
Angular impulse — product of external torque and the time over which it acts, equal to the change in angular momentum.
Key Takeaways
- Angular momentum is conserved when net external torque is zero.
- The conservation principle can be written as L_i = L_f.
- For many rotating systems, this becomes I_iω_i = I_fω_f.
- Conservation does not require I or ω to remain constant individually.
- Decreasing moment of inertia causes angular velocity to increase when L is conserved.
- Increasing moment of inertia causes angular velocity to decrease.
- Moving mass closer to an axis decreases moment of inertia.
- Moving mass farther from an axis increases moment of inertia.
- Radius has a strong effect because moment of inertia often depends on r².
- Internal forces can redistribute angular momentum without changing the total.
- External torque changes the total angular momentum.
- Rotational collisions can conserve angular momentum even when kinetic energy is not conserved.
- Rotational direction must be included when combining angular momenta.
- Figure skaters, divers, gymnasts, planets, stars and spacecraft all demonstrate angular momentum conservation.
- Athletes can change their angular velocity in mid-air by changing moment of inertia, but they cannot create total angular momentum without external torque.
- Reaction wheels allow spacecraft to exchange angular momentum internally and change orientation.
- Conservation of angular momentum is one of the fundamental conservation laws of physics.
Check Your Understanding
- State the principle of conservation of angular momentum and the condition required for it to apply.
- A skater has I_i = 6.0 kg·m² and ω_i = 3.0 rad/s. The skater pulls their arms inward until I_f = 2.0 kg·m². Calculate the final angular velocity.
- Explain why the skater in Question 2 rotates faster even though no external torque is required.
- A rotating system's moment of inertia doubles while angular momentum remains constant. What happens to its angular velocity?
- Two masses are moved from 0.60 m to 0.30 m from a rotation axis. Assuming the masses can be treated as point masses, by what factor does their moment of inertia change? What happens to angular velocity if angular momentum is conserved?
- A platform has I = 4.0 kg·m² and rotates at 10 rad/s. A stationary object lands on it, increasing the total moment of inertia to 10 kg·m². Calculate the final angular velocity.
- Two disks are coupled together. Disk A has I = 3.0 kg·m² and ω = 8.0 rad/s. Disk B has I = 5.0 kg·m² and is initially stationary. Calculate their common final angular velocity.
- Disk A has I = 4.0 kg·m² and rotates at +5.0 rad/s. Disk B has I = 2.0 kg·m² and rotates at −6.0 rad/s. They join together. Calculate their final angular velocity and state its direction.
- Explain why angular momentum can be conserved in an inelastic rotational collision even though rotational kinetic energy decreases.
- A diver decreases their moment of inertia from 12 kg·m² to 4.0 kg·m² while airborne. If the initial angular velocity is 1.5 rad/s, calculate the final angular velocity.
- Explain why a diver extends their body before entering the water.
- A rotating platform initially has I = 8.0 kg·m² and ω = 4.0 rad/s. A person changes position so that the total moment of inertia becomes 5.0 kg·m². Calculate the final angular velocity and verify that angular momentum is conserved.
- A spacecraft reaction wheel gains +24 kg·m²/s of angular momentum. What change in angular momentum must the rest of the spacecraft experience? If the spacecraft body has I = 120 kg·m² and initially has zero angular velocity, calculate its resulting angular velocity.
- Explain why a planet moves faster near the Sun and slower farther away using conservation of angular momentum.
- A rotating disk has I = 5.0 kg·m² and ω = 8.0 rad/s. A stationary ring with I = 3.0 kg·m² is dropped onto it and the two rotate together. Calculate the final angular velocity. Then calculate the initial and final rotational kinetic energies and determine how much mechanical energy was transferred to other forms.