Levers

The ancient Greek Archimedes made a bold claim: "Give me a place to stand, and I will move the Earth." It sounds like bragging, but what he held was a sound piece of mathematics — the lever.

The lever principle: force times arm

A lever is a rigid bar that turns about a fixed point, the fulcrum. The perpendicular distance from the fulcrum to a force is its arm. The balance condition is a single line:

$F_1 \times L_1 = F_2 \times L_2$

Force and arm are in inverse proportion: the longer the arm, the smaller the force needed. To save effort, lengthen your own arm. Pushing a door is a lever in action: the handle always sits as far from the hinges as possible, because that is where the arm is longest — nobody mounts a handle next to its hinge.

Try it: balancing the seesaw

A 30 kg child sits 2 m from the fulcrum. How far away should a 60 kg adult sit? From $30 \times 2 = 60$ , the adult sits at $60 \div 60 = 1$ m. The heavier person sits closer, the lighter one farther — every balancing act on the playground solves this little equation. And if both sit equally far out? Only equal weights will do.

Force-saving and force-wasting levers

By where the fulcrum, the effort and the load sit, levers come in three classes:

Fulcrum in the middle: seesaws, scissors. Who saves force depends on whose arm is longer;
Load in the middle: bottle openers, wheelbarrows, nutcrackers. The effort arm always beats the load arm — force-saving;
Effort in the middle: tweezers, chopsticks, fishing rods. The effort arm is shorter than the load arm — force-wasting, yet fingers part a little and the tips open wide — distance-saving.

Try it: how much force to lift the rock

The rock presses down on a crowbar with 400 N at a point 20 cm from the fulcrum; your hands grip 100 cm from the fulcrum. The force needed is $F = 400 \times 20 \div 100 = 80$ N — one fifth of the load. Nudge the fulcrum closer to the rock, the load arm shrinks, and the bargain gets even better. A bottle opener pops its cap with a satisfying snap by running the same bargain: trade distance for force.

Wasting force is a good deal too

Why would anyone use a force-wasting lever? For distance and precision. Tweezers designed to save force would need your fingers to sweep a long way to part the tips a little — useless. Your forearm is a force-wasting lever too: the muscle contracts a few centimetres and the hand sweeps a wide arc. A fishing rod pushes the account to the extreme — your hand flicks ten centimetres and the line flies a whole metre, at the price of the fish's whole weight landing on your arm. Force-saving levers are slow and steady; force-wasting ones are quick and nimble. Each has its own job. Scissors are two levers hinged together: sharpen the blades, lengthen the handles, and both moves save force.

Levers, pulleys and ramps all belong to the "simple machines": they create no energy, they just redistribute force and distance. To meet a family member, visit Force & Newton's Laws; to see where the rock's downward push comes from, the answer waits in Gravity & Free Fall.

Check yourself

Quick quiz

1. A 30 kg child sits 2 m from the fulcrum. How far from it must a 60 kg adult sit to balance?
2. Which of these is a force-wasting lever?
3. A crowbar has an effort arm of 100 cm and a load arm of 20 cm. How much force lifts a 400 N rock?