SPIN-UP
τ = Iα, θ = ½αt², KE = τθ — F = ma in a spinning alphabet, audited to 1e-12.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A wheel on a frictionless axle, given a steady twist. It is the exact rotational echo of a block pushed by a constant force — and it obeys the exact rotational echo of Newton’s second law. Learn one and you have learned both; the only work is swapping the letters.
τ = Iα
Force becomes torque, mass becomes moment of inertia, acceleration becomes angular acceleration. τ = Iα is F = ma in a new alphabet. A constant torque gives a constant angular acceleration, so angular velocity climbs linearly and the angle turned grows as t-squared — every straight-line kinematic formula reappears with a spin.
Solve
Divide torque by inertia for the angular acceleration; multiply by time for the spin rate; the angle is one-half alpha t-squared. The kinetic energy is one-half I omega-squared — and, satisfyingly, it comes out exactly equal to the torque times the angle turned. Work in equals energy stored, on the nose.
Spinning up
Watch the two clocks of rotation run together: the angular-velocity graph climbs a perfectly straight line whose slope is alpha, while the wheel’s marker sweeps through an angle that curves upward as time squared. The energy bar fills as omega-squared — and every joule in it is the torque’s accumulated work.
Audit
The kinematics close to a part in a trillion: omega equals alpha-t, theta equals one-half alpha-t-squared, and the rotational v-squared law, omega-squared equals two alpha theta, all hold at every sample. And the work-energy theorem is exact — torque times angle equals one-half I omega-squared — because τ = Iα really is F = ma wearing a hat.