spherical tank with radius 4 m is half full of a liquid that has a density of 900 kg/m3. The tank has a 2 m spout at the top. Find the work W required to pump the liquid out of the spout.

Answer 1

I'll set up the integral for you.

ρ = 900 kg/m³

gravity, g = 9.8 m/s²

radius, r = 4 m

Need to sum up the volumes of circular disks from y = 0 to 4

Volume of one disk, V = πr²h = πx²Δy

x²+(y-4)² = 4² ==> x² = 8y-y²

V = π(8y-y²)Δy

Distance to move each disk starting from the bottom of the sphere, D = 2 + r + r - y = 10-y

Work = ρgπ∫^r₀DV = ρgπ∫₀⁴(10-y)(8y-y²)dy

Which you should be able to evaluate