The skater who pulls in her arms

The most famous example of conservation of angular momentum is the skater who spins on the spot and speeds up by pulling in her arms. It’s a non-rigid system: it changes its own shape, and hence its own moment of inertia, while rotating.

Example — Conservation in a non-rigid system

A skater spins with her arms out at $\omega_A = 2\;\mathrm{rad/s}$ and moment of inertia $I_A = 8\;\mathrm{kg\,m^2}$. Pulling her arms in to her body, the moment of inertia drops to $I_B = 3\;\mathrm{kg\,m^2}$. What is the final angular velocity?

No external torque acts (friction of the ice about the vertical axis is negligible), so $\vec{L}$ is conserved: $I_A\omega_A = I_B\omega_B \;\Rightarrow\; \omega_B = \frac{I_A}{I_B}\omega_A = \frac{8}{3}\cdot 2 \approx 5{,}3\;\mathrm{rad/s}$ The skater spins more than twice as fast!

There’s a counter-intuitive detail, though. While the angular momentum stays constant, the rotational kinetic energy increases. Writing it as $E_\text{cin,rot} = \tfrac{1}{2}I\omega^2 = \dfrac{L^2}{2I}$, you can see that as $I$ decreases (at constant $L$) the energy grows. Where does this extra energy come from? From the muscular work of the arms: to bring them in towards the axis, the skater must pull them against the apparent centrifugal force, and that work ends up in the rotational kinetic energy.

The pirouette effect

This is the celebrated “pirouette effect”: shortening the radius of rotation reduces $I$ and, to conserve $L = I\omega$, increases $\omega$. Conservation of angular momentum doesn’t give energy for free — the body supplies it through its own work — but it redistributes the rotation spectacularly. The same principle explains the diver who tucks into a ball.

Connections

Topics: Dinamica rotazionale Concepts: Conservazione del momento angolare · Momento d’inerzia · Energia cinetica rotazionale

Related exercises: Problema — Asta incernierata che cade dall’orizzontale · Problema — La pattinatrice che si stringe · Problema — Perché il pattinatore ruota più veloce