How Is Work Calculated When Pumping Muddy Water from a Barrel?

[eestep]

Homework Statement

A cylindrical barrel, standing upright on its circular end, contains muddy water. The top of barrel, which has diameter 1 meter, is open. Height of barrel is 1.8 meter and it is filled to a depth of 1.5 meter. Density of water at a depth of h meters below surface is given by \delta(h)=1+kh kg/m³, where k is a positive constant. Find total work done to pump muddy water to top rim of barrel. Answer is .366(k+1.077)g\pi joules

Homework Equations

force of gravity on slice=density*g*volume

The Attempt at a Solution

volume of slice\approx\pi(1/2)²\Deltah m³
1+kh(g)(\pi/4)\Deltah nt
work done on slice\approxforce*distance=1+kh(g)(\pi/4)(1.8-h)\Deltah joules
total work=\int₀^1.5\pi/4g(1+kh)(1.8-h)dh joules=\pi/4g(1.8h-h²/2+1.8kh²/2-kh³/3)

 

[Mark44]

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