Question

A container with a pin hole at the bottom is filled with water and kerosene (specific gravity 0.8). The height of the water layer is 10 cm and the kerosene layer is 20 cm. The velocity of efflux of water is:

• A. 2.3 m/s
• B. 4.5 m/s
• C. 1.5 m/s
• D. 3.2 m/s

Hint:

When dealing with multiple fluids, convert all layers into the equivalent height of the liquid that is actually flowing out (the bottom layer) using the formula: \(h_{eq} = h_{other} \times \frac{\rho_{other}}{\rho_{bottom}}\).

Answer 1

The Correct Option is A

To solve this problem, we need to determine the velocity of efflux of water from a container that has two layers of liquids: water and kerosene. We'll use Torricelli's theorem, which relates the velocity of efflux (v) to the height (h) of the liquid column above the hole.

The effective height (h) of the liquid column can be calculated by considering the pressure contribution of both water and kerosene.

1. Calculate the pressure due to the water column:

\(P_{\text{water}} = \rho_{\text{water}} \cdot g \cdot h_{\text{water}}\)

where:

- \(\rho_{\text{water}} = 1000 \, \text{kg/m}^3\) (density of water)

- \(g = 9.8 \, \text{m/s}^2\) (acceleration due to gravity)

- \(h_{\text{water}} = 0.1 \, \text{m}\) (height of the water column)

1. Calculate the pressure due to the kerosene column:

\(P_{\text{kerosene}} = \rho_{\text{kerosene}} \cdot g \cdot h_{\text{kerosene}}\)

where:

- \(\rho_{\text{kerosene}} = 0.8 \cdot 1000 \, \text{kg/m}^3 = 800 \, \text{kg/m}^3\) (density of kerosene considering specific gravity)

- \(h_{\text{kerosene}} = 0.2 \, \text{m}\) (height of the kerosene column)

1. Find the total pressure at the bottom due to both liquids. The effective height (h) is given by the sum of the pressures divided by the density of water:

\(h_{\text{eff}} = \frac{P_{\text{water}} + P_{\text{kerosene}}}{\rho_{\text{water}} \cdot g}\)

This simplifies to:

\(h_{\text{eff}} = h_{\text{water}} + \frac{\rho_{\text{kerosene}}}{\rho_{\text{water}}} \cdot h_{\text{kerosene}}\)

1. Substitute the values:

\(h_{\text{eff}} = 0.1 + \frac{800}{1000} \cdot 0.2 = 0.1 + 0.16 = 0.26 \, \text{m}\)

1. Calculate the velocity of efflux using Torricelli's theorem:

\(v = \sqrt{2 \cdot g \cdot h_{\text{eff}}}\)

Substitute the values:

\(v = \sqrt{2 \cdot 9.8 \cdot 0.26} \approx 2.3 \, \text{m/s}\)

Thus, the velocity of efflux of water is 2.3 m/s, which matches option: \(2.3 \, \text{m/s}\).