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Question

The diameter of a bubble at the surface of a lake is 4 mm and at the bottom of the lake is 1 mm. If the atmospheric pressure is 1 atm and the temperature of the lake and the atmosphere are equal, what is the depth of the lake?
(The density of the lake and mercury are 1 g/ml and 13.6 g/ml respectively. Ignore the surface tension)

A
453 m
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B
651 m
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C
784m
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D
102 m
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Solution

The correct option is B 651 m
We know that pressure =ρgh
Also the volume of spherical bubble =43πr3
As temperature of lake is constant, using Boyle's law:
P1V1=P2V2
(760 mm×13.6×g)43π(42 mm)3=(760 mm×13.6×g+h×1×g)43π(12 mm)3
760×13.6×64=(760×13.6+h)
h=64×760×13.6760×13.6
h=63×760×13.6 mm
h=63×760×13.61000×1000 km=0.6511 km=651.1 m

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