Torque is the turning effect of a force. For a force F applied at a distance r from the axle, with an angle φ between the radius and the force, the torque is

τ = r F sin φ = F r_⊥ (r_⊥ = r sin φ is the lever arm)

The lever arm is the shortest distance from the axle to the line along which the force acts (the dashed line in the picture). A force pointing straight at the axle or away from it (φ = 0° or 180°) has no lever arm and no torque. The largest torque comes from a force at right angles to the radius (φ = 90°), at the greatest distance. By the usual convention a torque that turns the wheel anticlockwise is positive.

Moment of inertia is the rotational equivalent of mass: how hard it is to change the rotation. It depends on the mass and also on where the mass is: I = ½MR² for a solid disc and I = MR² for a ring with all its mass at the rim. The radius is squared, so doubling it makes the wheel four times as hard to spin up.

Στ = I α, ω = ω₀ + α t, L = I ω, KE = ½ I ω², W = τ Δθ

Newton’s second law for rotation. The net torque equals the moment of inertia times the angular acceleration, Στ = Iα. With a steady torque the angular velocity grows in proportion to time (the straight line on the graph). When the forces stop, there is no friction on this wheel, so it keeps turning at a constant angular velocity. The work done by the torque, τΔθ, equals the gain in rotational kinetic energy: the two lines in the lower graph lie on top of each other.

Thruster or hand? If the forces stay fixed to the wheel and turn with it (like small rockets on the rim), the torque stays the same as it turns. If the force keeps a fixed direction in space (like a hand pushing in one direction), the angle φ and so the torque change as the wheel turns, and the wheel can even swing back and forth. Try both with the same force.

The radius is shown to scale on the ruler. The wheel is drawn from above, with no friction or axle drag. The forces are limited to 40 N.