Torque

Multiply the force by the lever arm: M = F·r when the force acts perpendicular to the lever. Example: 80 N perpendicular at the end of a 25 cm wrench gives M = 80 × 0.25 = 20 N·m. If the force acts at an oblique angle θ, only its perpendicular component counts: M = F·r·sin(θ). At 60° you get only 20 × sin(60°) ≈ 17.3 N·m instead of 20 N·m. Watch the units: r in metres, F in newtons, the result in newton metres. The lever arm is the distance from the axis to the line of action of the force, not just any distance on the tool.

The law of the lever, force times force arm equals load times load arm (F₁·r₁ = F₂·r₂), is nothing but the torque balance: a lever does not rotate when the counterclockwise and clockwise moments are equal. Seesaw example: a 30 kg child 2 m from the pivot creates the moment 30 × 9.81 × 2 ≈ 589 N·m. A 40 kg child balances it at r = 589/(40 × 9.81) = 1.5 m. Because g appears on both sides it cancels, and you can directly compute m₁·r₁ = m₂·r₂. The beam balance, crowbar, wheelbarrow and all pliers rest on this principle.

For a required torque M the relation F = M/r holds: the necessary force is inversely proportional to the lever arm. If a stuck bolt needs 40 N·m, a 20 cm wrench requires 200 N, a 40 cm wrench only 100 N. Extending the wrench with a pipe lowers the force further. The price is a longer path: the handle must travel a larger arc, and the work done W = F·s stays the same; there is no free energy (the golden rule of mechanics). This is exactly why wheel-nut wrenches have long arms and door handles sit far from the hinge.

No, the matching unit is a coincidence of dimensions; the quantities are fundamentally different. In work W = F·s force and path point in the same direction, and the scalar product yields an amount of energy in joules. In torque M = r × F lever arm and force are perpendicular, and the cross product yields a directed turning action. A torque of 20 N·m does not mean 20 J of stored energy; energy only flows when something actually rotates: W = M·φ with the rotation angle φ in radians. That is why torques are deliberately written N·m and energies J, although the units are formally identical.

Bolted joints need a defined preload: bolts tightened too loosely work free under vibration, bolts tightened too hard overstretch the thread or snap. Since the preload is directly linked to the tightening torque, manufacturers specify torques, for instance 110 N·m for car wheel bolts or only 5 N·m for delicate carbon bicycle parts. The torque wrench measures the applied moment via a calibrated spring and audibly breaks over or clicks once the set value is reached. The physics behind it is M = F·r: the mechanism limits the product of hand force and lever length, no matter where you grip.