Question

A trough is 10 meters long, 1 meters wide, and 2 meters deep. The vertical cross-section of the trough parallel to an end is shaped like an isoceles triangle (with height 2 meters, and base, on top, of length 1 meters). The trough is full of water (density 1000kg/m3 ). Find the amount of work in joules required to empty the trough by pumping the water over the top. (Note: Use g

Answer 1

The amount of work required to empty the trough by pumping the water over the top is approximately 98,000 J

Step-by-step explanation:

The length of the trough = 10 meters

The width of the through = 1 meter

The depth of the trough = 2 meters

The vertical cross section of the through = An isosceles triangle

The density of water in the through = 1000 kg/m³

Let 'x' represent the width of the water at a depth

x/y = 1/2

∴x = y/2

The volume of a layer of water, dV, is given as follows;

dV = 10 × y/2 × dy = 5·y·dy

The mass of the layer of water, m = ρ × dV

∴ m = 1000 kg/m³ × 5·y·dy m³ = 5,000·y·dy kg

The work done, W = m·g·h

Where;

h = The the depth of the trough from which water is pumped

g = The acceleration due to gravity ≈ 9.8 m/s²

`\therefore \, W \approx \int\limits^2_0 {5,000 * y * 9.8 \, dy} = \left[24,500\cdot y^2 \right]^2_0 = 98,000`

The work done by the pump to pump all the water in the trough, over the top W ≈ 98,000 J