Torque and rotational motion (Physics HL)

The big idea: Watch a merry-go-round spin up, or a wheel roll faster and faster — that is rotational motion: straight-line motion gone round in a circle. Every quantity has a spinning twin:

- distance → angle turned θ (the angular displacement, in radians) - velocity → angular velocity ω (the angle turned per second, rad s⁻¹) - acceleration → angular acceleration α (how fast ω changes, rad s⁻²)

What is a radian?: One radian is the angle whose arc length equals the radius. A full turn = 2π rad ≈ 6.28 rad = 360°. Always work in radians for these formulas.

On an angular-velocity–time (ω–t) graph you read two things, exactly like a v–t graph: the slope is the angular acceleration α, and the area under the line is the angle turned θ. For constant α the motion obeys the rotational suvat equations.

What makes something spin?: A torque (τ) is the turning effect of a force — the rotational version of force. It depends on the force F, the distance r from the pivot, and the angle θ between them. Only the part of the force perpendicular to the arm turns the body.

When is a body balanced?: A rigid body is in rotational equilibrium when the total torque about any point is zero:

clockwise torques = anticlockwise torques.

(For full equilibrium the forces must also balance, ΣF = 0.)

How this is tested — rotational motion and torque are HL only (A.4):

The classic trap: Working in degrees, or forgetting to convert revolutions to radians (× 2π) before using the formulas — always work in radians.

Three easy marks: (1) Convert any revolutions to radians (× 2π) first. (2) For torque use the perpendicular distance (the sin θ). (3) Balanced ⇒ set clockwise = anticlockwise.