A hemispherical tank, placed so that the top is a circular region of radius 6ft, is filled with water to a depth of 4 ft. Find the work done in pumping the water to the top of the tank.
Ans. 50,185 lb-ft
A hemispherical tank, placed so that the top is a circular region of radius 6ft, is filled with water to a depth of 4 ft. Find the work done in pumping the water to the top of the tank.
Ans. 50,185 lb-ft
Work = weight * (delta height)
weight = volume * density
density of water is 62.4 lbs/cu.ft.
Hemisphere.
Diameter = 2(6)
If the origin (0,0) is the center of the circle of the hemisphere, the equation of the circle is
x^2 +y^2 = 6^2 --------------(i)
Our infinitesimal volume, dV--your rectangular element-- is a horizontal disc whose radius is x, and whose thickness is dy.
So, dV = pi(x^2) *dy
Its weight, dW, is
dW = (pi(x^2)*dy)(62.4) = (62.4pi)(x^2)dy
To raise this dW to the top of the bowl, the work done is, say, d(work).
d(work) = dW * (-y) <-----negative because the the y's below (0,0) are negative.
So,
d(work) = [(62.4pi)(x^2)dy]*(-y)
From (i), x^2 = 36 -y^2, so,
d(work) = [(62.4pi)(36 -y^2)dy]*(-y)
d(work) = (62.4pi)[y^3 -36y]dy
Because the water is 4ft deep, then the integration with dy is from y = -6 to y = -2.
Hence,
Work = (62.4pi)INT.(-6 --> -2)[y^3 -36y]dy
Work = (62.4pi)[(1/4)(y^4) -18y^2]|(-6 --> -2)
Work = (62.4pi)[{(1/4)(-2)^4 -18(-2)^2} - {(1/4)(-6)^4 -18(-6)^2}]
Work = (62.4pi)[{4 -72} - {324 -648}]
Work = (62.4pi)[4 -72 -324 +648]
Work = (62.4pi)[256]
Work = 50,185 ft-lbs. ---------------answer.