In every equilibrium of any forces whatever, in whatever way they are applied, the sum of the positive energies will be equal to the sum of the negative energies.

— Johann Bernoulli, letter to Pierre Varignon (26 January 1717)

Since Archimedes, each simple machine had had its own explanation. The lever obeyed the law of the arms; the inclined plane, a decomposition of forces invented by Stevin; the pulley, an argument about the ropes; the screw, yet another. In 1717, in a letter to his French friend Pierre Varignon, Johann Bernoulli showed that all these laws are cases of a single principle.

The idea: move it just a little

Imagine a machine at rest in equilibrium. Now imagine that you displace it, only in thought, by a tiny distance, respecting its constraints: the lever turns about its fulcrum, the rope does not stretch, the block does not leave the ramp. This imaginary displacement is called virtual, because it does not really happen.

At each point where a force $F_i$ acts, the point moves a tiny amount $\delta s_i$ in the direction of the force. The product $F_i\,\delta s_i$ is the virtual work of that force. Johann called it energy, and it is the first time the word appears with a meaning close to today’s. The principle says:

$$\boxed{\ \text{in equilibrium,}\quad\sum_i F_i\,\delta s_i=0\ \text{ for every virtual displacement.}\ }$$

The work of the forces that help the motion equals that of the forces that resist it. No machine at rest can create work out of nothing.

Archimedes’s lever

A lever has weights $P_1$ and $P_2$ at distances $a$ and $b$ from the fulcrum. Turn the lever through a tiny angle $\delta\theta$. The side of $P_1$ goes down $a\,\delta\theta$ and the side of $P_2$ goes up $b\,\delta\theta$. The weight that goes down does positive work; the one that goes up, negative work:

$$P_1\,a\,\delta\theta-P_2\,b\,\delta\theta=0\quad\Longrightarrow\quad P_1\,a=P_2\,b.$$

It is Archimedes’s law, without any figure of similar triangles: the $\delta\theta$ simply cancels.

The pulley

In a system of pulleys, a weight $W$ is supported by $n$ segments of rope, and a person pulls the free end with a force $F$. If the weight rises $\delta h$, each of the $n$ segments shortens by $\delta h$, and all that rope comes out at the free end, which moves $n\,\delta h$. So

$$F\cdot n\,\delta h-W\,\delta h=0\quad\Longrightarrow\quad F=\frac Wn.$$

With four segments of rope, you lift 100 kg with the force of 25 kg. But you pay in distance: for the weight to rise 1 metre, you must pull 4 metres of rope.

The inclined plane

A block of weight $W$ is on a ramp of inclination $\alpha$, held by a force $F$ parallel to the ramp. Push the block up the ramp a distance $\delta d$. It rises vertically by $\delta d\sin\alpha$:

$$F\,\delta d-W\,\delta d\sin\alpha=0\quad\Longrightarrow\quad F=W\sin\alpha.$$

On a 30° ramp, half the weight is enough to hold the block. Stevin had reached the same result in 1586 with an ingenious argument about a necklace of beads draped over two inclined planes. Johann got there in one line.

Exercise: a screw car jack has a pitch of 5 mm (it rises 5 mm per turn) and a 25 cm handle. What force on the handle lifts 1,000 kg?

In one full turn, the hand travels $2\pi\cdot0.25\approx1.571$ m and the car rises $0.005$ m. By the principle,

$$F\cdot1.571=1{,}000\cdot0.005\quad\Longrightarrow\quad F\approx3.2\ \text{kgf}.$$

A force of a little over 3 kg lifts a tonne. In practice, friction in the thread requires much more, but the order of magnitude explains why the jack works.

Exercise: a well windlass has an axle of radius 10 cm and a crank of radius 40 cm. What force holds a 20 kg bucket?

Turn the crank through an angle $\delta\theta$. The hand moves $0.40\,\delta\theta$ and the bucket rises $0.10\,\delta\theta$. So $F\cdot0.40=20\cdot0.10$, and $F=5$ kgf. It is a lever in disguise: the ratio of the radii, 4, is the mechanical advantage.

Sources: Pierre Varignon, Nouvelle mécanique ou statique (Paris, 1725), volume II, which contains Johann’s letter · René Dugas, A History of Mechanics (Dover, 1988), ch. 2 · biography of Johann Bernoulli, MacTutor.

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