An ice skater spins with arms extended, then pulls her arms tightly to her body. A student argues: 'Her moment of inertia decreases, so her angular velocity stays the same and her angular momentum decreases.' What is wrong with this claim?
ANothing is wrong — a more compact body does rotate with less angular momentum.
BWhen no external torque acts, L = Iω is conserved. Decreasing I requires ω to increase proportionally — she spins faster, not at the same rate. Angular momentum does not decrease.
CThe student is right that ω stays constant, but wrong about angular momentum — it stays the same, not decreases.
DThe student is right that angular momentum decreases, but wrong about ω — it also decreases.
In the absence of external torque (friction with the ice is negligible), angular momentum L = Iω is conserved. If the skater reduces her moment of inertia I by pulling her arms in, ω must increase so that the product Iω remains constant. This is the classic demonstration of conservation of angular momentum: the same total rotational quantity is now distributed over a body with lower rotational inertia, so it spins faster. The student confused 'more compact' with 'less momentum,' but L is conserved — its redistribution between I and ω is the whole point.
Question 2 Multiple Choice
The rotational equation τ_net = dL/dt is described as the exact rotational analog of Newton's second law. Which pairing best captures this analogy?
AForce ↔ torque, mass ↔ angular velocity, linear acceleration ↔ moment of inertia.
BForce ↔ torque, mass ↔ moment of inertia, linear momentum ↔ angular momentum.
CWork ↔ torque, kinetic energy ↔ angular momentum, power ↔ angular velocity.
DForce ↔ angular velocity, mass ↔ moment of inertia, linear acceleration ↔ angular acceleration.
Newton's second law: F_net = dp/dt, where p = mv (mass × velocity). The rotational analog: τ_net = dL/dt, where L = Iω (moment of inertia × angular velocity). The correspondence is: force ↔ torque (causes change), mass ↔ moment of inertia (resistance to change), linear momentum ↔ angular momentum (quantity of motion). For constant I, this becomes τ = Iα, mirroring F = ma with I playing the role of m and α playing the role of a.
Question 3 True / False
If no external torque acts on a spinning rigid body, its angular momentum remains constant in both magnitude and direction.
TTrue
FFalse
Answer: True
τ_net = dL/dt implies that when τ_net = 0, L is constant — not just its magnitude but the entire vector, including direction. This is why a gyroscope resists tipping: applying a torque changes the direction of L (causing precession) rather than reducing its magnitude, but without any torque, both the spin rate and the spin axis remain fixed. Conservation of angular momentum is a direct consequence of zero net external torque.
Question 4 True / False
For any rotating rigid body, the angular momentum vector L generally points in the same direction as the angular velocity vector ω.
TTrue
FFalse
Answer: False
L ∥ ω only when the rotation is about a principal axis (an eigenvector of the inertia tensor). For rotation about an arbitrary axis, L = Iω where I is the full 3×3 inertia tensor matrix, and the matrix multiplication generally produces a vector not parallel to ω. This misalignment is why an asymmetric object thrown in the air wobbles: ω and L point in different directions, and ω precesses around L. Stable, non-wobbling rotation occurs only about the principal axes.
Question 5 Short Answer
Why does a spinning ice skater spin faster when she pulls her arms in? Name the physical principle and explain the mechanism.
Think about your answer, then reveal below.
Model answer: Conservation of angular momentum. When no external torque acts on the skater, L = Iω is constant. Pulling her arms inward brings mass closer to the rotation axis, reducing her moment of inertia I. Since L must remain fixed, ω = L/I must increase proportionally — the same total angular momentum is now carried by a body with lower rotational inertia, so it rotates faster. The mechanism is the direct trade-off between I and ω enforced by the conservation law.
This is the clearest everyday demonstration of angular momentum conservation. The same principle explains why a collapsing protostellar cloud spins up into a rapidly rotating star, and why a diver pulls into a tuck to spin faster before unfolding for entry.