Question

A circular swimming pool has a diameter of 20 meters, the sides are 3 meters high, and the depth of the water is 2.5 meters. The acceleration due to gravity is 9.8 m/s2 and the density of water is 1000 kg/m3.
How much work is required to:
(a) pump all of the water over the side?
(b) pump all of the water out of an outlet 2 m over the side?

Answer 1

the volume of water is

10^2 * π * 2.5 = 785.4 m^3
That much water weighs
785.4 * 1000 * 9.8 = 7696920 N

The center of mass is 1.75m below the top of the pool, so the work required to lift it all is

7696920N * 1.75m = 13,470,000 J

But, if you want to use calculus, consider a thin sheet of water of thickness dx. Its volume is

100π dx m^3
So, its weight of water is

100π*1000*9.8 = 3078761 N

So, add up the work required to lift each sheet up to the top:

∫[0,2.5] 3078761(3-x) dx = 3,470,000