5.1 Torque

What is Torque?

Torque (τ) measures a force's ability to rotate an object about a pivot or axis. It depends on the force's magnitude, the distance from the axis to the point of application (lever arm), and the angle between them: τ = r F sinθ. Torque is a vector, measured in N·m, and causes angular acceleration.

Lever Arm and Direction

The lever arm is the perpendicular distance from the axis of rotation to the line of action of the force — maximizing torque requires the force to be applied perpendicular to r (θ = 90°). Torque direction follows the right-hand rule: curl fingers from r to F, thumb points along τ (out of or into the page).

Torque as a Vector: τ = r × F

Formally, torque is the cross product of the position vector and force vector: τ = r × F. Its magnitude is rFsinθ, and its direction is perpendicular to the plane containing r and F. Multiple torques about the same axis add as vectors (signed scalars in 2D), which is essential for analyzing rotational equilibrium and dynamics.

Key terms

Torque formula

τ = rFsinθ, where r is the position vector magnitude, F is force, and θ is the angle between them.

Lever arm

The perpendicular distance from the axis of rotation to the line of action of the force.

Torque as a vector

τ = r × F; direction found via the right-hand rule, perpendicular to the plane of r and F.

Units of torque

Newton-meters (N·m).

Condition for maximum torque

Force applied perpendicular (θ = 90°) to the position vector r, maximizing sinθ.

Try an AP-style question

r = (2î) m and F = (3ĵ) N. What is the torque τ = r × F?

1. 6k̂ N·m
2. −6k̂ N·m
3. 6î N·m
4. 0