calculating work done

Question

A tank is full of water. Find the work W required to pump the water out of the spout. Use the fact that water weighs 62.5 lb/ft³.

W =______  ft-lb

a	= 8
b	= 11
c	= 15

Answer 1

Sketch it out:

Using similar triangles, you can set up a relationship between y and the z-direction width of the cross-section of water shown in red. Since 15/z = 8/(8 - y) then 15(8 - y) = 8z or 15(8 - y)/8 = z or z = 15 - 1.875y

In the x-direction, the width of the cross section is 11.

So the area of the surface of the red cross section is 11(15 - 1.875y) which is 165 - 20.625y. The depth of the cross section is "dy", so the volume of the cross section is (165 - 20.625y)dy. The weight of the cross section is its volume times its density so its weight (which is equal to the force required to lift the cross section) = 62.5(165 - 20.625y)dy so:

F = 10312.5 - 1289.0625y

And Work (W) = Fd = ₀∫⁸ (10312.5 - 1289.0625y)y dy = ₀∫⁸ (10312.5y - 1289.0625y²) dy

You can integrate this manually or you can use a calculator. My calculator shows the answer is 110,000 ft lbs