# 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/m^{3}.

### 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.

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