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The law of the lever

The short answer

A lever balances when the turns on its two sides are equal, each turn being a force times its distance from the pivot. A 400 N load 0.25 m from the pivot is balanced by 100 N at 1 m, because 100 × 1 = 400 × 0.25. The price is distance: pushing the long end down 0.20 m raises the load only 0.05 m, so the work, force times distance, is 20 J on both sides. A lever trades force for distance; it never makes energy.

1:46How things workBeginnerleverssimple machineswork and energyArchimedestorque

Chapters and transcript

0:00
A 100 N push lifts a 400 N load.
On a lever, a push of 100 N can lift a load of 400 N.
0:12
Each weight turns the bar: force × distance from the pivot.
Each weight turns the bar about its pivot. How hard it turns the bar is its force times its distance from the pivot.
0:19
Slide 100 N further out, and its turn grows.
Sliding the 100 N weight away from the pivot makes its turn larger. While the load's 400 N × 0.25 m = 100 N·m is bigger, the load side stays down.
0:30
At 1 m: 100 × 1 = 400 × 0.25. Balanced.
At 1 m from the pivot, the weight's turn is 100 N × 1 m = 100 N·m, equal to the load's 400 N × 0.25 m, and the bar balances.
0:36
Archimedes proved this rule.
Archimedes proved it in On the Equilibrium of Planes: two weights balance at distances inversely proportional to the weights.
0:44
But you pay in distance.
The smaller force does not come free. It is paid for in distance.
0:49
Push the long end down 0.20 m; the load rises 0.05 m.
Each point on the bar moves in proportion to its distance from the pivot, so when the 1 m point goes down 0.20 m, the load at 0.25 m rises only 0.05 m.
0:56
Work is force × distance.
The work a force does is the force times the distance it moves.
1:01
100 N × 0.20 m = 20 J. 400 N × 0.05 m = 20 J.
The push does 100 N × 0.20 m = 20 J of work, and the load gains 400 N × 0.05 m = 20 J. On screen each is drawn as the area of a rectangle.
1:07
The same work. The lever makes no energy.
Work in equals work out on an ideal lever, with no friction and a weightless bar. The lever changes the force, not the energy.
1:14
Move the pivot closer to the load.
Moving the pivot towards the load shortens the load's arm and lengthens the push's arm, so a smaller push balances the same load.
1:22
17 N lifts 400 N, if you push 24 times as far.
With the pivot 0.05 m from the load and 1.20 m from the push, 400 N × 0.05 ÷ 1.20 = 16.7 N balances it, but the push must travel 24 times as far as the load rises.
1:34
A lever trades force for distance. Never energy.
A lever trades a small force over a long distance for a large force over a short one. It never makes energy.

Sources

Every figure in the film comes from these. The animation is illustrative.