Conservation of angular momentum: the rotating stool

The angular momentum of a body rotating about a fixed axis is L = I·ω, where I is the moment of inertia about that axis and ω is the angular velocity. The law of conservation of angular momentum says that if the total torque of external forces about the axis is zero, L does not change with time. A rotating stool — a round seat on an almost frictionless bearing — is exactly such a system: gravity and the normal force act along the rotation axis and produce no torque.

In the “Dumbbells” mode a person holding a dumbbell of mass m in each hand spins at angular velocity ω₀ with arms stretched out. The moment of inertia of the system is I = I₀ + 2·m·r², where I₀ is the moment of inertia of the person together with the stool and r is the distance of the dumbbells from the axis. Press “Pull arms in” or move the r slider while the stool is spinning: r decreases, I drops and the angular velocity grows so that the product I·ω stays the same — I₁·ω₁ = I₂·ω₂. In the 3D scene the purple arrow L keeps its length while the green arrow ω gets longer; the graphs show the same. This is how a figure skater speeds up by pulling in the arms during a spin, and how a diver tucks into a somersault.

Look at the energy: KE = I·ω²/2 = L²/(2I). With L constant and I decreasing, the rotational kinetic energy grows — the gain equals the work done by the person’s muscles pulling the dumbbells towards the axis against their “centrifugal” tendency. Stretching the arms out, the person gives that energy back. The measurements panel shows I, ω, the rotation frequency, L, KE and the work A = ΔKE relative to the initial state.

In the “Wheel” mode the person sits still holding a bicycle wheel above the head, spinning about a vertical axis — the angular momentum of the whole system L = L_w points up. Flip the wheel: its angular momentum turns downwards, and to keep the total vertical component unchanged the stool with the person starts rotating in the direction the wheel was spinning at first, with angular momentum 2·L_w. This is a clear demonstration that angular momentum is a vector. Turn on “Friction in the axle” to see an external torque gradually “eat up” L: the conservation law holds only for a closed system. The “Check yourself” box asks you to predict the outcome of an experiment first and then shows it.