Torque & the Lever Law
The Lever Law
A force's ability to turn something around a pivot depends on both how hard you push and how far from the pivot you push. The same force applied farther from the pivot creates more turning power, which is why a long wrench loosens a stubborn bolt more easily than a short one.
The formula
τ = r · F
- τ — torque (N·m): the turning power of a force around a pivot
- r — lever arm (m): the distance from the pivot to where the force is applied
- F — force (N): the push or pull applied perpendicular to the lever arm
Worked example
You push perpendicular to a wrench at a point 0.5 meters from the bolt, with a force of 20 N. How much torque do you apply?
- r = 0.5 m
- F = 20 N
- τ = r · F
- τ = 0.5 m × 20 N
τ = 10 N·m
Test yourself
The same 20 N force is now applied at a lever arm of 1 meter instead of 0.5 meters. What is the new torque?
- 10 N·m
- Correct answer: 20 N·m
- 40 N·m
Right! Torque scales directly with the lever arm: 1 m × 20 N = 20 N·m.
A 40 N weight sits 1 meter from the pivot on one side of a see-saw. How much force is needed 2 meters from the pivot on the other side to balance it?
- Correct answer: 20 N
- 40 N
- 80 N
Exactly — balance means r₁F₁ = r₂F₂: 1 m × 40 N = 2 m × F₂, so F₂ = 20 N.
Where you see this
A long wrench loosens a stuck bolt more easily than a short one, a doorknob sits far from the hinge so a small push swings a heavy door, and a see-saw balances when a light person sits farther from the pivot than a heavy one. All three are the same rule: force applied farther from a pivot has more turning effect.
Common mistakes
It's easy to think a bigger force always wins at a pivot, but a smaller force applied farther out can beat a larger force applied close in — that's the entire point of a lever. Students also over-extend the idea: only the component of force perpendicular to the lever arm contributes to torque, so a force pushing straight along the arm toward the pivot turns nothing at all.
How it connects
Torque is Newton's second law's rotational twin — force causes linear acceleration, torque causes rotational acceleration, with the same underlying logic. Centripetal force, next, returns to straight-line Newtonian force but for the special case of something moving in a circle instead of turning around a fixed pivot.