An open cubic tank, side length L is filled with water to height H. I know that the pressure p in the water as a function of height h is
p(h) = p_0 + (H - h)g*(rho)
where p_0 is air pressure at the surface and (rho) is the density of the water. But I can't work out the force the water exerts on a single side of the tank?

Question

An open cubic tank, side length L is filled with water to height H. I know that the pressure p in the water as a function of height h is 
p(h) = p_0 + (H - h)g*(rho)
where p_0 is air pressure at the surface and (rho) is the density of the water. But I can't work out the force the water exerts on a single side of the tank?

anonymous · Answer

Step 1. Think of the side of the tank divided into infinitesimal horizontal strips of height dh. Each strip has an area L*dh
2. Look at any one of those strips, and let us say it is at a height h. The pressure all along that strip is the same since it is a horizontal strip. That pressure is the function p(h) that you wrote.
3. Multiply that pressure with the area of the strip to get the force on that strip. That force is p(h)*L*dh
4. Integrate that from h = 0 to h = H. That is the total force