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If pressure at half the depth of a lake is equal to 2/3 pressure at the bottom of the lake, the depth of the lake is? [Take g = 10 m/s2 atmospheric pressure=105N/m2]

Question

If pressure at half the depth of a lake is equal to 2/3 pressure at the bottom of the lake, the depth of the lake is? [Take g = 10 m/s2 atmospheric pressure=105N/m2]

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Solution

The pressure at any point in a fluid is given by the equation P = Po + ρgh, where:

  • P is the pressure at the point
  • Po is the atmospheric pressure
  • ρ is the density of the fluid
  • g is the acceleration due to gravity
  • h is the height of the fluid column above the point

Given that the pressure at half the depth (h/2) is 2/3 the pressure at the bottom (h), we can set up the following equation:

Po + ρg(h/2) = 2/3(Po + ρgh)

Solving for h, we get:

Po + ρgh/2 = 2Po/3 + 2ρgh/3

Subtract Po from both sides:

ρgh/2 = 2Po/3 - Po

Multiply by 2:

ρgh = 4Po/3 - 2Po

Simplify:

ρgh = 2Po/3

Substitute the given values (ρ = 1000 kg/m³ for water, g = 10 m/s², Po = 105 N/m²):

1000 kg/m³ * 10 m/s² * h = 2 * 105 N/m² / 3

Solve for h:

h = (2 * 105 N/m² / 3) / (1000 kg/m³ * 10 m/s²)

h = 7 m

So, the depth of the lake is 7 meters.

This problem has been solved

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