In a Hare's apparatus, the height of water and a liquid X are 0.3m and 0.5m respectively. The relative density of x is?

1. Analyze the question

We are given the heights of two liquids (water and liquid X) in a Hare's apparatus and asked to find the relative density of liquid X.

2. Explain the concept

A Hare's apparatus works on the principle of balancing pressure. Since both liquids are subjected to the same atmospheric pressure at the top of the tubes, the pressure exerted by the columns of liquid must be equal at the common level:

Where:

• $\rho_w$ = density of water
• $\rho_x$ = density of liquid X
• $h_w$ = height of water column
• $h_x$ = height of liquid X column
• $g$ = acceleration due to gravity (which cancels out)

The relative density (or specific gravity) of a substance is defined as the ratio of its density to the density of water:

From our pressure equation, we can rearrange to find the ratio:

3. Show the solution

• Given: • Height of water ($h_w$) = 0.3 m • Height of liquid X ($h_x$) = 0.5 m

• Calculation: $$\text{Relative Density of X} = \frac{h_w}{h_x}$$ $$\text{Relative Density of X} = \frac{0.3}{0.5}$$ $$\text{Relative Density of X} = \frac{3}{5} = 0.6$$

4. Identify the answer

Comparing our result to the options provided, the correct value is 0.6. This matches option B.

5. Tip

In a Hare's apparatus, always remember the inverse relationship: the liquid with the greater height is the less dense liquid. Since the height of liquid X (0.5m) is greater than the height of water (0.3m), liquid X must be less dense than water (Relative Density < 1). This quick check helps you eliminate options C and D immediately!