Hydrostatic paradox deals with the pressure of a liquid at all points of the same horizontal plane. The word ‘hydrostatic‘ means fluid velocity is zero, the pressure variation is due only to the weight of the fluid. There is no pressure change in the horizontal direction. And the word ‘Paradox‘ is used when something involves two or more facts or qualities that seem to contradict each other. Before going into Hydrostatic Paradox, we need to know about Hydrostatic Pressure and Hydrostatic law.
Hydrostatic pressure is the pressure that is exerted by a fluid at equilibrium at a given point within the fluid, due to the force of gravity. Hydrostatic pressure increases in proportion to the depth measured from the surface because of the increasing weight of fluid exerting downward force from above.
So, hydrostatic pressure at point A is given by, P = ρ×g×h.
Where, P = Hydrostatic pressure at point A
ρ = Density of the fluid
g = Acceleration due to gravity
h = Height of liquid column above point A
Hydrostatic law states that the rate of change of pressure in the vertically downward direction is directly proportional to the weight density of the fluid.
So, dP/dZ = – ρ×g, or we can write dP = – ρ×g×dZ .
Here, dP/dZ = Rate of change of pressure in vertical downward direction,
ρ×g = weight density of the fluid,
From here we can say Pressure(P) is directly proportional to the height of the liquid column(h) [as the density of the fluid(ρ) and acceleration due to gravity(g) is constant].
Now, the hydrostatic paradox tells that the fluid pressure at the bottom of any container only depends on the height of the fluid column above it and it doesn’t depend on the weight of the water above it.
Here we take three containers A, B, C. Bottom areas of every container (let’s say, S). We take liquid up to height h in each container and the density of the liquid is ρ
Now, pressure at the bottom of container A, P = ρ×g×h
pressure at the bottom of container B, P = ρ×g×h
pressure at the bottom of container C, P = ρ×g×h
So, the pressure at the bottom of each container is the same.
But the weight of the water carried by the three containers is not the same. Here the weight of the water is more in container B and the weight of the water is less in container C.
So, by weight comparison (W)b > (W)a > (W)c .
So, from here we can see pressure at the bottom of the containers is the same, but the water column’s weight above it is not the same.
So, pressure at the bottom of the containers only depends on the height of the water column, not on the weight of the water above it.