A fountain consists of a large pool of water and a column of waterprojected upwa

Question

A fountain consists of a large pool of water and a column of waterprojected upward by means of a pump. The column has across-sectional area of 123 cm2at its base and reaches a height of 3.15m. (a) Compute the speed of water as it leaves thepool. 1 m/s Compute the flow rate of water as it leaves the pool. 2 liters/s (b) Find the increase in mechanical energy per unit mass?E/?m, as water starts from rest inthe pool and then is expelled by the pump. 3 J/kg (c) Find the power supplied by the pump by computing the increasein mechanical energy per unit time. 4 W

Explanation / Answer

Step 1: Finding flow rate ------------------- v = 7.86 m/s A = 123 cm2 = 0.0123 m² Q = v A = (0.0123 m²)(7.86 m/s) = 0.0967 m³/s =96.678 l/s Remember 1 m³ = 1000 dm³ = 1 l Step 2: Finding energy rate --------------------------------- E = ½mvf² -½mv02 and m = m thenE/m = ½vf² -½mv02= ½(7.86)² -½(0)² = 30.89 m²/s² = 30.89 J/kg Step 3: Finding power: ---------------------------------- First of all we need time. If Q = V/t then t = V/Q = (0.0123 m²)(3.15 m)/(0.0967m³/s) = 0.4 s Then E=½mv² = ½(V)(v²)=½(1000kg/m³)(0.03875 m³)(7.86 m/s)² = 1196.83 J And P = E/t = 1196.83 J / 0.4 s = 2992.06 W

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