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

Step 1: Understanding the Concept
Torricelli's Law for efflux velocity is derived from Bernoulli's principle. When multiple immiscible liquids are present, the pressure at the hole is determined by the total "head" or the sum of the pressures exerted by each liquid layer.

Step 2: Key Formula or Approach
1. Pressure at bottom (\(P\)) = \(\rho_k g h_k + \rho_w g h_w\).
2. Velocity of efflux (\(v\)) = \(\sqrt{2gh_{eff}}\), where \(h_{eff}\) is the equivalent height of a single water column.

Step 3: Detailed Explanation
1. Calculate Equivalent Water Height (\(h_{eff}\)): - Height of water (\(h_w\)) = 10 cm = 0.1 m. - Height of kerosene (\(h_k\)) = 20 cm = 0.2 m. - Specific gravity of kerosene (\(s_k\)) = 0.8. - The kerosene layer is equivalent to a water layer of height \(h_k \times s_k = 0.2 \times 0.8 = 0.16\) m. - Total effective height \(h_{eff} = 0.1 + 0.16 = 0.26\) m.
2. Calculate Velocity: \[ v = \sqrt{2 \times g \times h_{eff}} \] \[ v = \sqrt{2 \times 9.8 \times 0.26} \approx \sqrt{5.096} \approx 2.257 \, \text{m/s} \]
3. Rounding: The value is approximately 2.3 m/s.

Step 4: Final Answer
The velocity of efflux is 2.3 m/s.