Pascal's Law

Pascal's law states that any change in pressure applied to an enclosed incompressible fluid is transmitted undiminished to every part of the fluid and to the walls of the container.

Concept

Mathematically, if pressure $P_1$ is applied at point 1 and pressure $P_2$ exists at point 2 at the same height: $P_1 = P_2.$

Practical applications:

• Hydraulic lift / jack
• Hydraulic brakes
• Hydraulic press

The key idea: a small force on a small piston produces a large force on a larger piston, multiplied by the area ratio.

Derivation

Consider a hydraulic lift with two pistons of areas $A_1$ (small) and $A_2$ (large) connected by an incompressible fluid. Applying force $F_1$ on piston 1 creates pressure $P = F_1/A_1$, which is transmitted to piston 2:

$F_2 = P \cdot A_2 = F_1 \cdot \frac{A_2}{A_1}.$

This is the mechanical advantage of the lift. Conservation of volume of incompressible fluid: $A_1 d_1 = A_2 d_2 \;\Longrightarrow\; d_2 = d_1 \cdot \frac{A_1}{A_2}.$

So the work input equals work output: $F_1 d_1 = F_2 d_2$ (ideal, frictionless).

Worked Example

In a hydraulic lift, the small piston has area $5$ cm$^2$ and the large piston has area $500$ cm$^2$. To lift a car of mass $1000$ kg, the force required on the small piston is:

$F_1 = F_2 \cdot \frac{A_1}{A_2} = (1000 \times 9.8) \cdot \frac{5}{500} = 98\,\text{N}.$

Common Confusions

• The lift gives force advantage but not energy gain — distance moved on the small piston is correspondingly larger.
• Pascal's law applies only to enclosed fluids; in an open container, hydrostatic pressure variation with height also matters.
• The fluid must be (nearly) incompressible for the transmission to be truly undiminished.

Key Takeaways

• Pressure applied is transmitted uniformly in an enclosed fluid.
• Hydraulic lift: $F_2 = F_1 (A_2/A_1)$.
• Force is multiplied, not energy.