Derivation of ∆p = ρg∆h
- Hydrostatic pressure is the pressure at any given point within a fluid, that is exerted by the weight of the fluid above that point
- If the fluid is at rest, then all the points within the fluid are in equilibrium
- Therefore, the pressure acts in all directions at each point
Pressure on a point in a fluid
The hydrostatic pressure on area A is due to the weight, W, of the volume of fluid above it
- The weight, W, of the fluid above area A is given by:
- Using the density equation, mass can be given as:
- Substituting this expression for mass into the weight equation gives:
- Therefore, the pressure exerted on area A can be given as:
- Within the volume of the cube V, there is a change in pressure between the top and bottom surfaces
Change in pressure through a volume of fluid
The pressure at the bottom of the cube with volume V is greater than the area A at the top of the cube, because there is an increasing amount of fluid above, which increases the force of weight W acting upon it
- The change in pressure can be found by considering the change in height of the volume of fluid above the lower surface
- This gives the equation for hydrostatic pressure:
- Where:
- Δp = change in pressure in pascals (Pa)
- ρ (Greek letter rho) = density of fluid in kilograms per metre cubed (kg m-3)
- Δh = change in height in metres (m)
- g = gravitational field strength in newtons per kg (N kg-1)