Torque and levers

TL;DR:

• Torque is the turning effect of a force: τ = F × d, where d is the perpendicular distance from the pivot.

• An object stays balanced when clockwise torques equal counterclockwise torques.

• A lever trades force for distance. Its class depends on where the pivot, load and effort sit.

Two children want to balance on a seesaw, but one is heavier. Where should each sit? Anyone who has used a long wrench on a stuck bolt already knows the answer's flavour: a longer handle means less effort. Torque is the precise version of that feeling.

The idea in plain words

A force can make something move, but it can also make it turn. How well it turns depends on two things: the size of the force and how far from the pivot it acts. Torque combines them: τ = F × d, where d is measured at a right angle to the force, from the pivot to the line along which the force acts. The unit is the newton-meter, N·m.

If the force is not perpendicular to the lever, only the perpendicular part counts: τ = F d sin θ, with θ the angle between the force and the lever arm.

Why distance matters

Imagine pushing a door near its hinge. It barely budges. Push at the handle, far from the hinge, and it swings easily. The force is the same; only the distance changed. A 150 N push at 0.25 m from a bolt gives 150 × 0.25 = 37.5 N·m, and the same push on a handle 0.5 m long gives 75 N·m.

Worked example: balancing a seesaw

A child of mass 30 kg sits 2.0 m to the left of the pivot. A second child of mass 40 kg sits on the right. How far from the pivot must the second child sit for balance? Use g = 9.8 m/s².

1. Weights: 30 × 9.8 = 294 N left, 40 × 9.8 = 392 N right.

2. Left torque: 294 × 2.0 = 588 N·m (counterclockwise).

3. Balance condition: clockwise torque equals counterclockwise torque, so 392 × d = 588.

4. Solve: d = 588 / 392 = 1.5 m.

Notice that g appears on both sides and cancels, so you can skip it: 30 × 2.0 = 40 × d, giving d = 1.5 m directly. Heavier child, shorter distance.

The pivot also holds up the total weight: 294 + 392 = 686 N (or 70 × 9.8 = 686 N). That upward force from the pivot acts at the pivot itself, with zero distance, so it contributes no torque. This is why choosing the pivot as your reference point makes problems easier.

The three classes of lever

• First class: the pivot lies between the effort and the load, like a seesaw, scissors or a crowbar.

• Second class: the load lies between the pivot and the effort, like a wheelbarrow or a nutcracker. The effort arm is always longer, so it multiplies force.

• Third class: the effort lies between the pivot and the load, like tweezers or a fishing rod. It multiplies distance and speed instead, with a larger effort than load.

The idea is the same for all: torque balance. Effort × effort arm = load × load arm gives the mechanical advantage effort arm / load arm. A crowbar with an effort arm of 1.2 m and a load arm of 0.2 m has an advantage of 1.2 / 0.2 = 6.

This is also why a seesaw with equal masses balances only when both sit at equal distances, and why a very heavy child can balance a much lighter one only by sitting close in. A 60 kg adult at 1.0 m balances a 30 kg child at 2.0 m, since both give 60 × 1.0 = 30 × 2.0 = 60 (in kg·m, with g dropped from both sides).

A good habit is to draw the plank as a line, mark the pivot with a triangle, and label each force arrow with its distance from the triangle. Then writing the balance equation is just reading the picture.

How to check your answer

• Compute both torques. 392 × 1.5 = 588 and 294 × 2.0 = 588. Equal.

• Units. N times m is N·m. The distances must be to the pivot, not between the children.

• Direction. Pick one rotation direction as positive, say counterclockwise, and be consistent. At balance, the signed torques sum to zero.

• Sense. The heavier child should be closer to the pivot, and 1.5 m is less than 2.0 m.

Common mistakes

• Using the straight-line distance rather than the perpendicular distance.

• Measuring from the wrong point. For balance, any pivot works, but you must use the same one for every torque.

• Forgetting the weight of the plank itself. If it is uniform and pivoted at its middle, its weight acts at the pivot and drops out.

• Confusing torque (N·m) with work or energy (J). They share units but are different quantities.

Safety: Keep hands and fingers clear of pivot points on real levers, seesaws and tools. A lever multiplies force, which includes the force that can pinch.

Where you will use it next

Torque is the rotational version of force. It leads to angular acceleration, the stability of structures and the arm of a robot. Back in circular motion, the centripetal force article covers the inward force that torque does not produce on its own.

Frequently asked questions

Does a lever create energy?

No. A lever trades force for distance. A small push over a long distance matches a big push over a short distance, so the work in equals the work out, apart from friction.

Why does a longer wrench help?

With the same force, a longer handle gives a bigger torque. To get the same torque you can use a smaller force.