Torque
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 4 practice questions.
Torque measures how effectively a force twists something about an axis: τ = r × F, with magnitude τ = rF sin θ. Equivalently, multiply the force by its lever arm r⊥, the perpendicular distance from the axis to the force’s line of action.
1. Lever arm
2. Direction and net torque
The right-hand rule gives τ’s direction: curl your fingers from r toward F; your thumb points along τ. For problems in the xy-plane, τ points along ±z: +k̂ is counterclockwise.
Add torques with their signs to find the net torque about a chosen axis.
To get the most torque from a force: push perpendicular to the handle, as far from the axis as you can. A force along the handle, toward or away from the axis, does nothing to turn it.
3. Example, then try it
A force F = (3 î + 4 ĵ) N acts at r = (2 î + 1 ĵ) m from an axis at the origin. Find the torque.
τz = xFy − yFx = (2)(4) − (1)(3) = 5 N·m, along +k̂ (counterclockwise).
A 50 N force acts at the end of a 0.30 m wrench at 30° to the handle, as in the figure below. Find the lever arm and the torque. What torque would the same force give perpendicular to the handle, or along it?
Lever arm: r⊥ = r sin 30° = (0.30)(0.50) = 0.15 m.
Torque: τ = r⊥F = (0.15)(50) = 7.5 N·m.
Perpendicular: τ = (0.30)(50) = 15 N·m, the maximum.
Along the handle: sin 0° = 0, so no torque.
Answer: 0.15 m and 7.5 N·m; 15 N·m perpendicular; 0 along the handle.
4. Common mistakes
The same force is applied perpendicular to a door, first at the edge and then halfway to the hinge. How does the torque change?
τ = rF with the force perpendicular. Half the distance gives half the torque, which is why door handles are far from the hinges.
Ready to practice?
4 questions on Torque, each with a hint and an explanation for every wrong answer.