# 10. Force Due to Liquid Pressure by Integration

by M. Bourne

The force F on an area A at a depth y in a liquid of density w is given by

F = wyA

The force will increase if the density increases, or if the depth increases or if the area increases.

So if we have an unevenly shaped plate submerged vertically in a liquid, the force on it will increase with depth. Also, if the shape of the plate changes as we go deeper, we have to allow for this.

So we have:

Now, the total force on the plate is given by

F=wint_a^bxy\ dy

where

x is the length (in m) of the element of area (expressed in terms of y)

y is the depth (in m) of the element of area

w is the density of the liquid (in N m-3)

(for water, this is w = 9800 N m-3)

a is the depth at the top of the area in question (in m)

b is the depth at the bottom of the area in question (in m)

### Example 1

Find the force on one side of a cubical container 6.0 cm on an edge if the container is filled with mercury. The weight density of mercury is 133 kN/m3.

### Example 2

A right triangular plate of base 2.0 m and height 1.0 m is submerged vertically in water, with the top vertex 3.0 m below the surface.

Find the force on one side of the plate.

top

### Online Algebra Solver

This algebra solver can solve a wide range of math problems.

### Calculus Lessons on DVD

Easy to understand calculus lessons on DVD. See samples before you commit.