Pascal's Law

Pascal's law states that the pressure applied at any point of an enclosed fluid at rest is transmitted equally and undiminished to every point of the fluid, and also to the walls of the container holding it, provided the effect of gravity on the fluid itself can be neglected.

This can be demonstrated experimentally using a vessel with four arms, A, B, C and D, each fitted with a frictionless, water-tight piston, and the whole vessel filled with an incompressible fluid. If the cross-sectional areas of A, B, C and D are a, 2a, 3a and a/2 respectively, applying a force F on piston A creates a pressure $p = F/a$ in the fluid, and the other three pistons B, C and D are observed to move outward as a result. To hold each of these three pistons in its original position requires applying forces of 2F, 3F and F/2 to B, C and D respectively — which means the pressure on each of them, $p_B = 2F/2a = F/a$, $p_C = 3F/3a = F/a$, and $p_D = (F/2)/(a/2) = F/a$, works out to exactly the same value p as on piston A. Since $p_B = p_C = p_D = p$, this confirms directly that the pressure applied at A is transmitted equally and undiminished to every other part of the fluid and to the vessel walls, exactly as Pascal's law states.

Applications of Pascal's Law:

i) Hydraulic lift. A hydraulic lift is a machine, based directly on Pascal's law, used to lift a heavy object using only a comparatively small applied force. A tank of fluid is fitted with two pistons, a smaller one $S_1$ of cross-sectional area $A_1$ and a much larger one $S_2$ of cross-sectional area $A_2$ ($A_2 \gg A_1$). Applying a downward force $F_1$ on the smaller piston $S_1$ generates a pressure $p = F_1/A_1$ in the fluid, which by Pascal's law is transmitted undiminished to the larger piston $S_2$, producing an upward force there of $F_2 = pA_2$, i.e.

Since $A_2 \gg A_1$, the output force $F_2$ is very much larger than the input force $F_1$ — a heavy load can therefore be placed on $S_2$ and lifted up or lowered down simply by applying a much smaller force on $S_1$. This is the working principle of a hydraulic lift. …