amount of work needed to pump gasoline out of a tank

Question

A gas station stores its gasoline in a tank under the ground. The tank is a cylinder lying horizontally on its side. (In other words, the tank is not standing vertically on one of its flat ends.) If the radius of the cylinder is 2 meters, its length is 6 meters, and its top is 1 meter under the ground, find the total amount of work needed to pump the gasoline out of the tank. (The density of gasoline is 673 kilograms per cubic meter; use g=9.8 m/s2.)

Work =

Answer 1

Density, ρ=673 kg/m³

g=9.8 m/s²

radius, r=2

Work = Force×Distance

Force = ρg(Volume)

Method 1:

Put a circle with the x-y origin at the bottom of the circle. r²=x²+(y-r)² --> 4 = x²+(y-2)²

Take a rectangular slice with volume

V_slice = 6(2x)Δy

x=√(4-(y-2)²)

Distance = 5-y

W = 12 ρg ∫⁴₀(5-y)xdy = 12 ρg ∫⁴₀(5-y)√(4-(y-2)²)dy = 72πρg = 72π(673)(9.8) = 474868.8 N

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Method 2:

Concentrate all the weight of the cylinder at the mass center of the cylinder, which is the center of the circle.

Force=Weight=ρg(Volume of cylinder) = ρg(6π2²) = 24πρg

Distance = r+1 = 3

Work = 3(24)πρg=72πρg