Question
A tank has the shape of a cylinder with height 7m and base radius 3m. It is filled with water to a height of 5m. Find the work required to empty the tank by pumping all of the water to the top of the tank. (The density of water is 1000kg/m^3, and use g=9.8m/s^2.)
Answers
the volume of water is πr^2 h = 45π m^3
the weight of water is volume * density = 45000π kg * 9.8 = 1,386,856 N
The distance to raise the center of mass is 7-(5/2) = 4.5 m
work = force * distance = 6,240,852 J
or, if you insist on using calculus, you can go the long way, considering the work to raise a thin slice of water, which has thickness dy and is y m from the bottom of the tank:
∫[0,5] π*3^2*9.8*1000 * (7-y) dy
the weight of water is volume * density = 45000π kg * 9.8 = 1,386,856 N
The distance to raise the center of mass is 7-(5/2) = 4.5 m
work = force * distance = 6,240,852 J
or, if you insist on using calculus, you can go the long way, considering the work to raise a thin slice of water, which has thickness dy and is y m from the bottom of the tank:
∫[0,5] π*3^2*9.8*1000 * (7-y) dy
LOL do it the first way :)