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Properties of Matter

Section: Physics

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Question : 32 of 46

Marks: +1, -0

A solid sphere of radius R and density ρ is attached to one end of a mass-less spring of force constant k. The other end of the spring is connected to another solid sphere of radius R and density 3

\rho

. The complete arrangement is placed in a liquid of density 2

\rho

and is allowed to reach equilibrium. The correct statement(s) is (are)

[JEE Adv 2013 P1]

the net elongation of the spring is $\frac{4\pi R^{3}\rho g}{3k}$
the net elongation of the spring is $\frac{8\pi R^{3}\rho g}{3k}$
the light sphere is partially submerged.
the light sphere is completely submerged.

Solution:

On small sphere

\frac{4}{3}\pi R^{3} (\rho) g \; + k x = \frac{4}{3}\pi R^{3} (2\rho) g \;\;

........(i)

on second sphere (large)

\frac{4}{3}\pi R^{3} (3\rho) g \; = \frac{4}{3}\pi R^{3} (2\rho) g + k x \;\;

......(ii)

by equation (i) and (ii)

x = \frac{4\pi R^{3}\rho g}{3k}

For equilibrium of the complete system, net force of buoyancy must be equal tothe total weight of the sphere which holds true in the given problem. So both the spheres are completely submerged

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