Problem — Spinning top and precession

Problem

A tilted spinning top does not fall over at once, but precesses about the vertical. Explain why angular momentum makes the top rotate horizontally instead of falling.

Solution

The top has a large angular momentum $\vec{L}$ directed along its axis. The weight, applied at the centre of mass, produces a torque $\vec{\tau}$ about the point of support. This torque is perpendicular to $\vec{L}$ (horizontal, directed sideways), not downward.

The law $\dfrac{d\vec{L}}{dt} = \vec{\tau}$ says that $\vec{L}$ changes in the direction of $\vec{\tau}$: since $\vec{\tau}\perp\vec{L}$, the angular momentum does not grow or shrink in magnitude, but rotates in the horizontal plane. The top’s axis thus traces out a cone about the vertical instead of falling: this is precession. The same mechanism governs gyroscopes, satellite orientation, and the very slow precession of Earth’s equinoxes. $\ev{\vec\tau_\text{peso}\perp\vec L \;\Rightarrow\; \tfrac{d\vec L}{dt}=\vec\tau \text{ makes } \vec L \text{ rotate (precession), not fall}}$

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