MHT CET · Physics · Mechanical Properties of Fluids

When a large bubble rises from bottom of a water lake to its surface, then its radius doubles. If the atmospheric pressure is equal to the pressure of height H of a certain water column, then the depth of lake will be

A 2H
B H
C 7H
D 4H

Verified Solution

Answer & Solution

Correct Answer

(C) 7H

Step-by-step Solution

Detailed explanation

When a large bubble rises from bottom of water lake to its surface, then its radius becomes double.
i.e.,

r_{2} = 2 r_{1}

Since, volume

V \propto r^{3}

∴ \frac{V_{2}}{V_{1}} = {(\frac{r_{2}}{r_{1}})}^{3} = (\frac{2 r_{1}}{r_{1}}) = 8

\Rightarrow V_{2} = 8 V_{1}

… (i)
From Boyle’s law,

p_{1} V_{1} = p_{2} V_{2}

Where,

p_{2}

= atmospheric pressure.

p_{1} V_{1} = p_{2} 8 V_{1}

[from Eq. (ii)]

p_{2} = \frac{p_{1}}{8}

where,

p_{2} =

atmospheric pressure
and

p_{1}

= pressure at depth d.

∴ p_{1} = p_{a t o m s p h e r i c} + p g d

p_{1} = p_{2} + p g d

… (iii)
From Eqs. (ii) and (iii), we have

p_{2} = \frac{p_{2} + p g d}{8}

7 p_{2} = p g d

7 . p g H = p g d

7 H = d \Rightarrow d = 7 H

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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