Pascal's Law: Pascal's Ball

Pascal's law. Pressure applied to a liquid or gas is transmitted unchanged to every point of the fluid and in all directions. If the piston raises the pressure beneath it by 5 kPa, then at the bottom, at the side walls and at any hole the pressure has risen by exactly 5 kPa. The particles of a liquid are free to move: squeeze it from one side and they push one another in every direction until the pressure evens out. A solid cannot do this — it transmits a force only along its line of action.

Pascal's ball. It is a hollow ball with small holes, connected to a cylinder with a piston. The piston pushes the water in one direction only — along the cylinder — yet the water bursts out of the holes in every direction: up, down, sideways and even backwards, towards the piston. The jets are alike because the pressure at all the holes is almost the same. In the simulator the water is coloured by pressure, and the yellow arrows inside the ball show how it presses on the walls: arrows of equal length point every way.

Piston pressure and the weight of water. The piston pressure p = F/S depends not only on the force but also on the area: the same force on a piston half the size creates twice the pressure. On top of it comes the pressure of the water's own weight, ρgh, which grows with depth. So at a hole at height y the pressure is p = F/S + ρg(y_axis − y): a little more at the lower holes, a little less at the upper ones. With F = 15 N and S = 30 cm² the piston creates 5 kPa, while the weight of the water adds only 0.74 kPa at the bottom of the ball and takes the same away at the top. The graph shows both parts: the green rectangle F/S is the same for all holes, the blue wedge ρgh depends on height.

Jet speed. Water leaves a hole at the speed v = √(2p/ρ): the work of the pressure turns into the kinetic energy of the water. At 5 kPa this is 3.2 m/s, and a sideways jet from a height of 62 cm lands 1.1 m away. If the pressure comes only from the weight of the water, p = ρgh, the formula becomes Torricelli's law v = √(2gh) — water leaves the hole as fast as it would after falling from the height h. The drops in the simulator follow the exact projectile formula, and the outflow moves the piston: when the cylinder is empty, the jets stop.

Where Pascal's law works. It is the basis of the hydraulic press and jack, of car brakes, where pressing the pedal is transmitted by oil to all four wheels at once, and of the hydraulics of excavators and cranes. The same law explains why an inflated ball and a soap bubble are round: the gas presses equally on every part of the shell from inside.

Try it yourself. Reduce the force to zero — water then flows only from the holes below the axis of the ball, under its own weight, and not a drop leaves the upper ones. Double the force — the jets get faster and longer but stay the same in every direction. Reduce the piston area with the same force — the pressure grows although you push just as hard. Compare the speeds of the upper and lower jets in Measurements: the difference is exactly the contribution of ρgh.