Angular momentum, explained with examples
Angular momentum is the rotational version of momentum. It explains why a spinning skater speeds up when pulling in their arms, why a bicycle is stable and why collapsing stars spin so fast. It is part of the rotation units in AP Physics 1 and AP Physics C: Mechanics.
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Linear and angular, side by side
Every idea in linear motion has a rotational partner. Mass becomes rotational inertia I, which depends on how mass is spread around the axis. Velocity becomes angular velocity ω, in radians per second. Force becomes torque τ. And momentum p = mv becomes angular momentum:
L = I ω
Angular momentum is measured in kg m²/s. Like linear momentum, it has a direction; for rotation about a fixed axis, we usually just call clockwise or counterclockwise positive or negative.
Angular momentum of a point particle
A single particle moving in a straight line can also have angular momentum about a point. For a particle of mass m moving with speed v, its angular momentum about a point is
L = m v r⊥
where r⊥ is the perpendicular distance from the point to the particle's line of motion. For a particle moving in a circle of radius r, r⊥ = r, so L = mvr, which equals Iω with I = mr² and v = ωr. AP Physics C writes this as the cross product L = r × p.
Torque changes angular momentum
Newton's second law for rotation can be written in terms of angular momentum:
τnet = ΔL / Δt
A net torque changes angular momentum, just as a net force changes linear momentum. The product τΔt is called the angular impulse, and it equals the change in angular momentum.
Conservation of angular momentum
If the net external torque on a system is zero, its total angular momentum stays constant:
I1ω1 = I2ω2
This is one of the most powerful conservation laws in physics. When a rotating system changes shape, its rotational inertia changes, and its angular speed changes to compensate. Pull mass in towards the axis and I decreases, so ω increases. Spread mass out and the rotation slows, exactly as conservation demands.
Worked examples
Example 1: calculating angular momentum
A solid disc of mass 2.0 kg and radius 0.30 m spins at 10 rad/s. Find its angular momentum. (I = ½MR² for a solid disc)
- I = ½ × 2.0 × 0.30² = 0.090 kg m².
- L = Iω = 0.090 × 10 = 0.90 kg m²/s.
Example 2: a spinning skater
A skater spins at 2.0 rev/s with arms out, with rotational inertia 3.0 kg m². Pulling in their arms reduces this to 1.2 kg m². Find the new spin rate.
- No external torque (ice is nearly frictionless), so I1ω1 = I2ω2.
- ω2 = 3.0 × 2.0/1.2 = 5.0 rev/s. Units of revolutions per second can be kept, since they appear on both sides.
Example 3: kinetic energy is not conserved
In Example 2, does the skater's rotational kinetic energy stay the same?
- K = ½Iω² = ½Lω. L is constant, but ω increases by a factor of 2.5, so K also increases by a factor of 2.5.
- The extra energy comes from the work the skater's muscles do pulling their arms inwards.
Example 4: a rotational collision
A turntable with I = 0.020 kg m² spins at 4.0 rad/s. A 0.10 kg lump of clay is dropped onto it, sticking 0.20 m from the axis. Find the new angular speed.
- The clay adds I = mr² = 0.10 × 0.04 = 0.004 kg m². New total I = 0.024 kg m².
- 0.020 × 4.0 = 0.024 × ω2, so ω2 ≈ 3.3 rad/s.
Example 5: angular impulse
A constant torque of 0.50 N m acts on a wheel for 4.0 s. By how much does its angular momentum change?
- ΔL = τΔt = 0.50 × 4.0 = 2.0 kg m²/s.
Example 6: a particle passing a point
A 0.050 kg ball moves at 8.0 m/s along a straight line that passes 0.40 m from a pivot. Find its angular momentum about the pivot.
- L = mvr⊥ = 0.050 × 8.0 × 0.40 = 0.16 kg m²/s. If the ball hits and sticks to a rod on the pivot, this angular momentum is shared by the ball and rod.
Angular momentum in nature and technology
- Divers and gymnasts tuck to spin faster and open out to slow down before entering the water or landing.
- Collapsing stars: when a large star collapses into a neutron star a few kilometres across, its rotational inertia drops enormously and it can spin hundreds of times per second.
- Planets sweep out equal areas in equal times (Kepler's second law), which is conservation of angular momentum for an orbit: they move faster when closer to the Sun.
- Gyroscopes and bicycles: a spinning wheel resists changes to the direction of its angular momentum, which helps keep a moving bicycle upright.
- Helicopters need a tail rotor, because without one, the body would spin the opposite way to the main rotor to conserve angular momentum.
Common mistakes
- Assuming kinetic energy is conserved whenever angular momentum is.
- Forgetting to include every object's rotational inertia after a collision.
- Mixing rev/s and rad/s in the same equation.
- Using r instead of the perpendicular distance r⊥ for a particle not moving in a circle.
- Applying conservation when an external torque, such as friction at the axle, is significant.
Practice questions
Try each one before opening the answer.
Q1. A wheel with I = 0.5 kg m² spins at 6 rad/s. Find L.
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Q2. A spinning object's rotational inertia halves with no external torque. What happens to ω?
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Q3. A 2 kg mass moves in a circle of radius 0.5 m at 3 m/s. Find its angular momentum about the centre.
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Q4. Why do divers tuck their bodies to somersault faster?
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Frequently asked questions
Is angular momentum on AP Physics 1?
Yes. In the current course framework, angular momentum and its conservation are part of the rotational energy and momentum unit of AP Physics 1. AP Physics C: Mechanics covers it with calculus and vectors.
Is angular momentum in A level Physics?
It appears mainly in the engineering physics option of some specifications, alongside rotational inertia and torque. Check your board's specification.
Key takeaways
- L = Iω for a rotating body; L = mvr⊥ for a particle.
- Net torque equals the rate of change of angular momentum.
- With no external torque, I1ω1 = I2ω2.
- Pulling mass inwards makes a system spin faster.
- Angular momentum can be conserved while kinetic energy changes.