(25%) Problem 2: The pump in a water tower lifts water with a density ofpw-1.00

Question

(25%) Problem 2: The pump in a water tower lifts water with a density ofpw-1.00 kg liter from a filtered lake at the bottom of the tower up hT 24 m to the highest point of the tower. The water begins at rest in the lake and comes to rest at the highest point of the tower. The pump is 100% efficient in lifting the water, and it lifts a volume of Tw 8.8 liters of water up the tower every second. The water tower sits on a cliff that is Te 35 m above sea level. Let gravitational potential energy be zero at ground level.

Explanation / Answer

Part (a)

The pump lifts water of volume Vw = 8.8 litres every second.

So, mass of water lifted every second by the pump is m = Vww = (8.8 litre) (1 kg/liter) = 8.8 kg

The work done by the tower in lifting mass m of water by a height h = 24 m is,

WT = mgh = (8.8 kg)(9.8 m/s2)(24 m)

or, WT = 2069.76 J

This mass is lifted every second, so power used by the pump is,

Pp = WT/(1 s) = 2069.76 J/s = 2069.76 W

or, Pp = 2069.76 Watts

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Part (b)

If the pump works for t = 2.1 hours = 3600 x 2.1 s = 7560 s, then energy used by the pump is,

E = Ppt = ( 2069.76 W)(7560 s)

or, E = 15.65 X 106 J

This energy is used to lift the water and the pump is 100% efficient, So total increase in potenrial energy of water is,

Uw = E

or, Uw = 15.65 X 106 J

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Part (c)

Mass of full tank of water is m = (106 liter)(1 kg/liter) = 106 kg

The water is at a height of H = (24 + 35)m = 59 m relative to sea level

So gravitational potential energy of water relative to sea level is,

Uw = mgH = (106 kg)(9.8 m/s2)(59 m)

or, Uw = 5.78 X 108 J

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