Pressure Depends on Depth and Density, Not on Shape
Static fluid questions rest on two facts: pressure at a given depth depends only on the depth and the density, and an external pressure applied to an enclosed fluid is transmitted unchanged. Together they cover vessels, U-tubes, syringes and hydraulic presses.
Motion and force questions become simpler when direction, starting condition, time and the acting forces are marked first.
Pressure at depth and the thrust on a base
The thrust on the base is force = pressure x base area, so two vessels with the same base area and the same water height exert the same thrust on their bases even if one holds four times as much water; the ratio is 1 : 1. Extra mass in a wider vessel is carried by the sloping walls, not the base.
U-tubes with two liquids
In a U-tube, points at the same level in the same continuous liquid are at the same pressure. Pick the interface between the two liquids (or a level in the denser liquid) and equate pressures from both arms: ρ_oil h_oil = ρ_water h_water, so the heights are in inverse ratio to the densities, and the lighter liquid stands higher.
Pascal's law, hydraulic press and syringes
Pascal's law states that pressure applied to an enclosed fluid is transmitted undiminished everywhere in it.
For two syringes joined by a water-filled tube, the pressure is the same in both, so the force on the larger piston is larger in proportion to its area and the displacement is smaller in the inverse proportion.
OlympIQ exam lens
- ✓P = P0 + ρ g h; depends on depth and density only, not on the shape of the vessel
- ✓Thrust on the base = P x base area
- ✓U-tube: equate pressures at the same level in the same liquid; ρ1 h1 = ρ2 h2
- ✓Hydraulic press: F1/A1 = F2/A2 and A1 d1 = A2 d2
- ✓Pascal's law: pressure in an enclosed fluid is transmitted equally in all directions
Do not compare the thrust on the base of two vessels by comparing the weights of liquid in them. The thrust depends only on pressure times base area, so a vessel with more liquid does not press harder on its base if the height and base area are the same.
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