A rectangular door with dimensions 7 m by 6.1 m is submerged at a depth of 8.4 meters in water. How much force is exerted by pressure against the door?
The pressure at depth 8.4 meters is
The area of the rectangle is
so the force exerted by a 82320 N/m^2 pressure over this area will be
The pressure at depth 8.4 meters is the weight per unit of cross-sectional area of a column of water of cross-sectional area A and altitude equal to the depth. The volume of such a column is
where y is the depth. The mass of the column is therefore
where `rho = 1000 kg / m^3 is the density of water, in units of mass per unit of volume. The weight of the water is therefore
where g = 9.8 m/s^2 is the acceleration of gravity.
The force exerted to support the column, when the column is at rest, is thus equal to the weight of the column. This force is exerted over the 'bottom' of the column, which has area A equal to the cross-sectional area of the column. Thus the pressure (force per unit area) must be
Pressure is force per unit area. We therefore multiply pressure, or force per unit area, by area to get force.
The force on area A due to pressure P is thus