Question

Water rises upto a height of $4 \text{ cm}$ in a capillary tube. The lower end of the capillary tube is at a depth of $8 \text{ cm}$ below the water level. The mouth pressure required to blow an air bubble at the lower end of the capillary will be 'X' cm of water, where X is equal to

• A. $10$
• B. $8$
• C. $6$
• D. $12$

Hint:

To blow a bubble at a depth $d$ from a tube that exhibits a capillary rise $h$, the net pressure head is always simply the direct sum of the two heights: $X = d + h$. Adding $8 + 4$ immediately gives $12$.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
A capillary tube is submerged to a depth below the free water surface. Due to capillary action, water naturally tends to stand at a certain height inside. To push this liquid down and force an air bubble out of the submerged bottom tip, an external mouth pressure must overcome both the hydrostatic water head and the capillary pressure.

Step 2: Key Formula or Approach:
The total pressure required ($P_{\text{total}}$) expressed in terms of equivalent columns of water is given by:
$$P_{\text{total}} = P_{\text{hydrostatic}} + P_{\text{capillary}}$$ The capillary pressure head is equal to the natural capillary rise height ($h_1 = 4 \text{ cm}$). The hydrostatic pressure head is equal to the depth of immersion ($h_2 = 8 \text{ cm}$).

Step 3: Detailed Explanation:
Let's analyze the pressure components using cm of water as our working unit:
1. The pressure required to counter the liquid column from the surface to the lower submerged tip is:
$$P_{\text{hydrostatic}} = 8 \text{ cm of water}$$ 2. The excess pressure required to overcome the curvature forces to initiate bubble formation is equivalent to the pressure supporting the capillary rise:
$$P_{\text{capillary}} = 4 \text{ cm of water}$$ Summing these individual pressures together yields the minimum total pressure $X$:
$$X = h_2 + h_1 = 8 \text{ cm} + 4 \text{ cm} = 12 \text{ cm of water}$$ Therefore, the value of $X$ is exactly $12$.

Step 4: Final Answer:
The required mouth pressure value $X$ is $12$, which corresponds to option (D).