Hydrostatic Pressure

Ever notice that your ears hurt when you swim down to the bottom of a pool, but not while standing in the shallow end? That pain is hydrostatic pressure — the weight of all the water sitting above you, pressing on your eardrums.

The deeper you go, the more water sits above you, the bigger the pressure. The relationship is dead simple: pressure increases linearly with depth, period. Same idea explains why dams are thicker at the bottom (more pressure to hold back), why deep-sea creatures have specialized anatomy (they live under hundreds of atmospheres of pressure), and why your blood pressure in the veins of your feet is higher than in your head when you’re standing up.

Hydrostatic Pressure

In a stationary fluid, pressure increases linearly with depth.

Three insights pop out of this equation:

1. Pressure depends only on depth, not on the shape or width of the container. A narrow tube and a giant lake at the same depth have the same pressure at the bottom. This sounds wrong (the lake has vastly more water above you!) — but the extra water also spreads over a much larger area at the bottom. Pressure is force per area, and the two cancel exactly.

2. Pressure is the same at all points at the same depth (within a single connected fluid). Move horizontally — pressure doesn’t change. Only vertical position matters.

3. Layered fluids: add each layer’s contribution. If you have oil floating on water, the pressure at the bottom isn’t $\rho_{water} g (h_{total})$. It’s $\rho_{oil} g h_{oil} + \rho_{water} g h_{water}$. Each layer above contributes independently.

Worked Example

A diver descends to 10 m in the ocean ($\rho_{seawater} = 1025$ kg/m³). What’s the absolute pressure at that depth?

• $P_{abs} = P_{atm} + \rho g h = 101{,}325 + (1025)(9.8)(10) = 101{,}325 + 100{,}450 \approx 201{,}775$ Pa.
• That’s about 2 atm.

So at just 10 m down, the diver experiences twice the surface pressure. At 20 m, ~3 atm. At 30 m, ~4 atm. Each 10 m of water adds another atmosphere — that’s the rule of thumb every scuba diver learns.

Pascal’s Law

Pascal’s law says that pressure applied to an enclosed fluid is transmitted equally and undiminished to every point in the fluid and to the walls of the container.

Plain English: push on one part of a confined fluid, and the entire fluid feels the same pressure increase.

Hydraulic Systems

Pascal’s law is the principle behind hydraulic lifts, hydraulic brakes, hydraulic presses, and even the hydraulic systems that move heavy excavator arms. A small force applied to a small piston produces a pressure increase that’s transmitted to a large piston, where the same pressure × bigger area = much bigger output force.

Worked Example: Hydraulic Lift

A mechanic pushes with 100 N on a piston of area 0.01 m². The car sits on a piston of area 1 m². What force lifts the car?

• $P_1 = F_1/A_1 = 100/0.01 = 10{,}000$ Pa.
• $F_2 = P \times A_2 = 10{,}000 \times 1 = 10{,}000$ N.

The mechanic’s 100 N becomes 10,000 N at the lifting piston — enough to easily lift a small car. The area ratio (100×) is the force multiplier.

The catch (Pascal’s law isn’t free): if the small piston gets pushed down 1 meter, the large piston only rises 0.01 m (1 cm). Same volume of fluid moves, but spread over a much larger piston. Work done is identical on both sides: $100 \times 1 = 10{,}000 \times 0.01 = 100$ J. Conservation of energy strikes again — there’s no free lunch in physics.