University · Mechanics Round 1/6 · Score 0

Spin Lab

Race rolling shapes, change an axle offset, investigate a massive pulley and conserve angular momentum. Gravity is 9.81 m/s².

Rolling-shape race

Changing the rotation axis changes moment of inertia, even when mass stays constant.

Acceleration vs inertia factor

Challenge

The physics

Inertia: a solid disc has I = ½mr²; a hoop has I = mr²; a solid sphere has I = ⅖mr²; a hollow sphere has I = ⅔mr²; a rod about its centre has I = mL²/12. The parallel-axis theorem gives I = Icentre + md².

Torque: τ = Iα. Rolling without slipping: a = g sin θ/(1 + k), with k = I/(mr²). The frictionless block slides with k = 0. The speed after descending a height h follows v² = 2gh/(1 + k).

Angular momentum: I₁ω₁ = I₂ω₂ when external torque is zero. Rotational kinetic energy is ½Iω². A skater can add kinetic energy by doing work while drawing her arms inward. For the pulley, the light string is inextensible, does not slip, and axle friction is neglected.