Force multiplied, and nothing gained

Push a piston into a closed vessel of liquid and the pressure everywhere in the liquid goes up by the same amount. Not by an amount that fades with distance, not by an amount that depends on which way round a corner the liquid has to go — by the same amount, everywhere. That statement is Pascal’s principle, and it is one of the few results in physics whose engineering consequence arrives in the same sentence.

A person leaning on the small piston with the weight of a bag of shopping lifts a third of a tonne. Nothing in the apparatus resembles a lever, a gear or a pulley; there is a tube of oil.

Why the increase does not fade #

The previous rung established that pressure in a still fluid varies with depth, as $\rho g h$, and with nothing else. That result is doing all the work here, and it is worth seeing why.

Take any two points in a connected body of still fluid. The pressure difference between them is fixed by their depths — it is $\rho g \Delta h$, and it depends on the fluid’s density and on gravity, neither of which the piston has touched. So if the pressure at one point goes up by $\Delta p$ and the fluid is still afterwards, the pressure at every other point must have gone up by the same $\Delta p$, because otherwise their difference would have changed, and their difference is not free to change.

It is worth being clear what the principle does not say. The pressure in a standing column of fluid is already different at every depth before any piston is touched, because the fluid has weight. Pushing on the top adds the same amount to every one of those pressures; it does not make them equal. A press is designed against the sum of the two effects, and only one of them is Pascal’s — which is why a tall hydraulic system has to account for the head of fluid in it separately from anything the pump does.

This is the point at which the principle is usually stated slightly wrongly, and the error is worth naming. Pressure is not equal throughout a fluid. It varies with depth, everywhere, always. What is transmitted undiminished is a change in pressure. In a press half a metre tall the depth term is about five kilopascals, which against a working pressure of ten megapascals is a twentieth of a per cent and can be forgotten; in a hydraulic system with a thirty-metre standpipe it is three hundred kilopascals and cannot.

The multiplication, and where it comes from #

Given a pressure $p$ common to both cylinders, the forces are whatever the areas make them:

$$F_1 = p A_1, \qquad F_2 = p A_2, \qquad \frac{F_2}{F_1} = \frac{A_2}{A_1}.$$

Because the areas go as the square of the radius, the ratio goes as the square of the radius ratio. Four to one in radius is sixteen to one in force, which is what the figure draws. Ten to one in radius is a hundred to one in force, which is a car jack — and the ratio is a pure number, in the sense that the falloff of a field is decided by an arrangement rather than by a strength.

The multiplication is therefore geometric in the purest sense: it is decided entirely by the ratio of two areas and not at all by the fluid, the pressure, or how the tube between the cylinders is routed. That last point is the one that distinguishes this machine from every mechanical one. A lever’s ratio is fixed by where the fulcrum is and both ends must be part of the same rigid bar; a hydraulic ratio survives the working fluid being taken round a corner, up a wall and through a flexible hose, which is why the brakes on a car are hydraulic and not levered.

What is paid, and why it must be #

Now push the small piston in by a distance $d_1$. It sweeps out a volume $A_1 d_1$. That liquid has to go somewhere, and if the liquid does not compress, the only place it can go is into the other cylinder, raising the large piston by $d_2$ with $A_2 d_2 = A_1 d_1$. So

$$\frac{d_1}{d_2} = \frac{A_2}{A_1} = \frac{F_2}{F_1},$$

and multiplying through,

$$F_1 d_1 = F_2 d_2.$$

The work in equals the work out. The figure computes both sides separately and asserts their equality before drawing anything, because a picture of a machine that appeared to produce more joules than it consumed would be a picture of the wrong universe, and it is the kind of arithmetic slip that a well-drawn diagram would hide perfectly.

What has actually been bought is not energy but shape. The same number of joules is delivered either as a small force through a long distance or as a large force through a short one, and which of those is convenient depends entirely on the job. A car jack needs the second; a bicycle’s hydraulic brake, where the lever’s travel is generous and the pad’s is a millimetre, needs it too.

This is the same accounting that the energy landscape makes for a conservative force, and the same one a lever makes for a rigid body. Nothing on this page is a new conservation law. It is the old one, arriving through a fluid.

The assumption doing the work #

Everything above turns on the liquid conserving volume, and that is an approximation rather than a fact. Water’s bulk modulus is about 2.2 gigapascals: raise the pressure by 22 megapascals — a plausible working pressure for an industrial press — and it shrinks by one per cent.

One per cent sounds negligible and it is not, for a reason that has nothing to do with the force. The forces are unaffected: pressure is still transmitted undiminished and $F_2 = pA_2$ still holds exactly. What suffers is the travel. The volume swept by the small piston now partly goes into squeezing the liquid rather than into moving the large one, so the large piston moves less than the geometry predicts, and the shortfall grows with pressure. In a system with a long, slightly-compliant hose it is worse still, because the hose expands, and a driver feels that as a spongy brake pedal.

The energy is not lost — it is stored, elastically, in the compressed fluid, and comes back out when the pressure is released. That makes a hydraulic system a spring as well as a machine, and a stiff one: the same compressibility that spoils the travel is what lets a hydraulic accumulator hold useful energy.

The assumption doing the work is incompressibility, and it is worth naming because it is the one that fails first. A real fluid compresses a little under load, so some of the small piston’s travel goes into squeezing the oil rather than into moving the large piston — the missing travel is stored in the fluid as a spring stores it, and returned when the pressure falls. That is the same exchange a mechanical spring makes, obeying the same differential equation when it is allowed to oscillate, and it is why a hydraulic system with air trapped in it feels spongy: air is a far better spring than oil, and the travel disappears into it.

Two cylinders, or a hundred #

Nothing in the argument limits the machine to two pistons, and the extension is where hydraulics stops being a curiosity and becomes an industrial method.

Connect four cylinders of different areas to one pressurised line and every one of them produces a force proportional to its own area, simultaneously, from the same supply. There is no distribution problem to solve and no linkage to design: the pressure is a single number that the whole circuit shares, and each actuator converts it into whatever force its area dictates. An excavator with five independent joints, each needing a different force and a different travel, is one pump and five areas.

The same reasoning is what makes a car’s braking system work the way it does. One master cylinder feeds four slave cylinders; the fronts are larger in area than the rears, so the front brakes receive proportionally more force from the same pedal, without any valve, cable or balance bar arranging it. The brake bias is set by choosing two numbers at manufacture, and it is then as stable as the geometry.

There is a safety consequence in the same fact, and it is the reason a modern car’s brakes are split into two independent circuits. Because the pressure is shared, a single leak anywhere lets the pressure everywhere fall to atmospheric, and every actuator on that circuit stops. The property that distributes force perfectly also distributes failure perfectly, and the two cannot be separated because they are the same property.

The pressure that has to be held in #

There is a design consequence that follows from the areas and catches people out. Making the large piston larger multiplies the force, and it multiplies nothing else — the pressure is unchanged. So the cylinder walls do not care how much force the machine produces. They care about the pressure.

This is why a hydraulic system’s rating is quoted in pascals rather than newtons, and why the failure mode of a press is not the piston being pushed through the end but the hose bursting, far from anything that looks like the load. Every part of the circuit sees the same pressure, which is precisely the principle at the top of this page, read as a warning instead of as an opportunity.

The stresses in the wall of a cylindrical vessel are worth one line, because they are not what intuition suggests: the hoop stress is twice the longitudinal one, so a pressurised tube splits along its length rather than snapping across. That is a statics result rather than a fluids one: it follows from the geometry of the section carrying the load, not from anything the fluid does.

What a real machine does that this one does not #

The idealised press has two pistons and no valves, and can therefore lift the large piston exactly once before the small one runs out of travel. Every real hydraulic machine solves that with a pump: a small piston that strokes repeatedly, with a pair of one-way valves arranging that each stroke draws from a reservoir and delivers to the cylinder.

That changes nothing about the physics and everything about the usefulness. The work per stroke is still conserved, and the total work is the sum over strokes; what the valves buy is the ability to accumulate many small displacements into one large one. A bottle jack’s handle sweeping through fifty strokes is fifty applications of the arithmetic above, and the load rises by fifty times the per-stroke amount.

The other real-world addition is a relief valve, which exists because the pressure a system can generate is not bounded by anything in the physics. A long enough lever on the pump handle produces whatever pressure the operator’s patience allows, and something will fail at a pressure decided by the weakest component. The relief valve makes that failure a choice rather than a discovery.

Why the pressure keeps going up #

The observation that the cylinder walls care about pressure and not about force has a design consequence that has driven one industry’s choices for eighty years.

Since $F = pA$, a required force can be got from a large cylinder at low pressure or a small one at high. The forces are identical; the masses are not. A cylinder at twice the pressure needs half the bore for the same force, so its walls enclose a quarter of the volume, the pipes feeding it are narrower, and the fluid filling the whole system is less. Everything shrinks except the wall thickness, which grows in proportion to the pressure and to the radius — and the radius has just halved.

For anything that has to be carried, that arithmetic is decisive. Aircraft hydraulics settled on about two hundred bar in the 1940s and stayed there for half a century; the newest large aircraft run at three hundred and fifty, and the reason is the weight of everything downstream of the pump. The saving on a large airliner is measured in hundreds of kilograms, which is passengers.

What stops the pressure rising further is not the physics on this page, which is indifferent, but the seals. Every joint has to hold the pressure with something soft enough to conform and hard enough not to be extruded through the gap it is sealing, and the gap grows as the metal around it strains. That is the same constraint Bramah’s press waited on, two centuries later and three orders of magnitude further up.

The animal that has no extensor muscles #

The most unexpected user of this page’s arithmetic is a spider.

A spider’s leg has flexor muscles that pull it in and, at two of its joints, no extensor muscles at all. What straightens the leg is pressure: muscles in the body squeeze the haemolymph, the pressure is transmitted undiminished into the legs, and each joint extends because the fluid pushes on the larger of the two areas presented to it. The whole animal is a hydraulic system with one pump and eight sets of actuators, sharing a pressure exactly as an excavator does.

The pressures involved are substantial for an animal — several times the resting value during a jump, a fair fraction of an atmosphere — and they are what a jumping spider leaps with. It has no extensor muscle to leap with, and does not need one: the muscle that supplies the energy is in the body, and the legs convert a pressure into a force through their own cross-section, which is the first equation on this page.

The trade is the second equation. A leg is a narrow cylinder, so its area is small, so the force it can produce is modest and the extension is fast and long — which is the arrangement a jumping animal wants, and the opposite of the one a press wants.

And the failure mode is the one the braking section names. A spider’s legs are held out only while the pressure is maintained, so a spider that dies loses its pressure and its flexors pull unopposed. That is why a dead spider is always found curled up, and it is the shared-pressure property of this page seen from its least cheerful side.

Where the model stops #

The fluid is still. Every statement here is hydrostatic. The moment fluid actually flows through the connecting pipe there is a pressure drop along it that depends on the flow rate and the pipe’s radius — steeply, as the fourth power of the radius — and that flow is the same momentum-diffusion problem viscosity always is — and the two cylinders no longer see quite the same pressure. For a press moving slowly this is negligible; for a hydraulic system driving a fast actuator it is a large part of the design.

The pistons are frictionless. They are not, and the seals that keep the fluid in are exactly what makes them not. Real presses reach eighty to ninety-five per cent efficiency, and the missing fraction goes into heat in the seals. That loss is unavoidable in the same sense that the conservation of work is: a seal that did not rub would not seal.

The fluid does not cavitate. On the suction stroke of a pump the pressure can fall low enough for the liquid to boil at room temperature, and the collapsing vapour cavities are destructive. That is a phase change driven by falling pressure rather than by rising temperature, and it sets a floor on how fast a pump may draw.

The fluid is a liquid. Filling the same apparatus with air would leave every force relation on this page intact and destroy the machine, because air’s bulk modulus is its own pressure — five orders of magnitude below oil’s. The small piston would travel its full stroke compressing the gas and the large one would barely move. That is not a defect of pneumatics but a description of what they are for: a pneumatic actuator is deliberately springy, which is why a pneumatic tool can hammer and a hydraulic one cannot.

Gravity is ignored. As noted above, that is a choice about the ratio $\rho g h$ to the working pressure, and it is a good one at ten megapascals and a bad one at ten kilopascals.

The history, and the machine that came late #

Pascal stated the principle in the 1650s, and the hydraulic press was not built until Joseph Bramah patented one in 1795 — a gap of nearly a century and a half for a device whose principle is one line long.

The delay was not conceptual. It was the seal. A press works only if the fluid does not leak past the piston, and at the pressures that make the machine worth building, leakage past a plain fitted piston is total. Bramah’s press became practical when his employee Henry Maudslay devised a leather cup washer that the pressure itself forces against the cylinder wall — so the harder the machine works, the better it seals. The principle had been available since the seventeenth century and the machine waited on a piece of shaped leather.

That pattern recurs often enough to be worth naming. A physical principle states what is permitted; whether a machine exists is usually a question about materials, and the two can be separated by a very long time. The same gap sits between knowing that light of the right frequency ejects electrons and having a photocell, and between knowing what a lens does and being able to grind one.

It is worth adding what the press then did to everything else. Bramah’s machine made it possible to apply forces of hundreds of tonnes in a workshop, and forging, extrusion and plate-bending followed from it; the leather cup washer therefore sits some way upstream of the industrial nineteenth century. A one-line principle, a piece of shaped hide, and a change in what could be made.

The ladder from here #

Later rungs on this anchor: the manometer and the differential pressure gauge, where the same law is run backwards to turn a pressure into a length. The free surface of a rotating fluid, whose paraboloid is exact and is used to cast telescope mirrors. Stratified fluids, where density varies with height on purpose. The siphon, which is more subtle than it looks and whose usual explanation is incomplete. And the atmosphere as a hydrostatic column, which is where the exponential came from.

The neighbouring ladder starts where this one left the block submerged: the pressures on a body’s faces do not cancel, and what survives is a force upward.