Properties of Matter 2 Question 14
14. A solid sphere of radius $R$ and density $\rho$ is attached to one end of a massless 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)
(2013 Adv.)
(a) the net elongation of the spring is $\frac{4 \pi R^{3} \rho g}{3 k}$
(b) the net elongation of the spring is $\frac{8 \pi R^{3} \rho g}{3 k}$
(c) the light sphere is partially submerged
(d) the light sphere is completely submerged
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Answer:
Correct Answer: 14. (a, d)
Solution:
On small sphere
$$ \frac{4}{3} \pi R^{3}(\rho) g+k x=\frac{4}{3} \pi R^{3}(2 \rho) g \cdots(i) $$
On second sphere (large)
$$ \frac{4}{3} \pi R^{3}(3 \rho) g=\frac{4}{3} \pi R^{3}(2 \rho) g+k x \cdots(ii) $$
By Eqs. (i) and (ii), we get
$$ x=\frac{4 \pi R^{3} \rho g}{3 k} $$
