Conservation of Angular Momentum
- State conservation of angular momentum for zero net external torque
- Explain the spinning-skater effect in terms of I and ω
- Solve I₁ω₁ = I₂ω₂ problems
No external torque means constant angular momentum
If the net external torque on a system is zero, its total angular momentum is conserved. This is the rotational version of momentum conservation, and it holds even when the object’s shape changes. Because L = Iω, if the rotational inertia I changes, the angular velocity ω must change in the opposite way to keep the product constant.
The spinning skater
When a spinning skater pulls her arms in, she brings mass closer to her axis, reducing I. With no external torque, L = Iω is fixed, so ω must increase — she spins faster. Extending her arms does the reverse. The same principle explains why a collapsing star spins up and why a diver tucks to rotate faster. Notice this speeds up the spin without any torque at all.
A skater spins with rotational inertia 6 kg·m² at 2 rad/s, then pulls her arms in, reducing her rotational inertia to 2 kg·m². Find her new angular speed.
- 1.No external torque, so I₁ω₁ = I₂ω₂.
- 2.Substitute: 6 × 2 = 2 × ω₂ → 12 = 2 × ω₂.
- 3.Solve: ω₂ = 12 ÷ 2 = 6 rad/s.
- 4.Her inertia dropped to one-third, so her spin sped up threefold.
Pulling arms in changes ω but the angular momentum L stays the same. Kinetic energy, though, actually increases — the skater’s muscles do work pulling the mass inward, so energy is not conserved even though L is.
A spinning skater pulls her arms in, cutting her rotational inertia to half its value. With no external torque, what happens to her angular speed?
A turntable with rotational inertia 4 kg·m² spins at 3 rad/s. A lump of clay is dropped on, raising the rotational inertia to 6 kg·m². What is the new angular speed? (no external torque)
Conservation of angular momentum needs zero external torque. Internal changes — pulling in arms, dropping on clay — do not violate it. Check that nothing outside the system is applying a torque before you set L constant.
Answer the 2 checkpoints as you read.
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