Problem — Slowing top and nutation

Problem

Explain why a spinning top, as friction reduces $\omega$, begins to wobble (nutation) before falling.

Solution

When the top spins fast, its angular momentum $L = I\omega$ is large. The torque of the weight $\tau$ produces steady precession at rate $\Omega \approx \tau/L$: since $L$ is large, $\Omega$ is small and the motion is smooth and stable.

Friction progressively reduces $\omega$, and with it $L$. The ratio $\tau/L$ increases: precession becomes faster and less stable, and an up-and-down oscillation, nutation, superimposes on the axis. Below a threshold speed, the angular momentum is no longer sufficient to “support” the weight through precession, and the top falls. $\ev{\omega\downarrow \Rightarrow L\downarrow \Rightarrow \tau/L\uparrow \Rightarrow \text{nutation, then fall}}$

Links

Topics: Rotational dynamics Concepts: Angular momentum Objects: Spinning top