Question

A solid sphere of radius $1\text{m}$ and density $1000{\text{kg/m}}^3$ is attached to one end of a massless spring of force constant $5000\text{N/m}$. The other end of the spring is connected to another solid sphere of radius $1\text{m}$ and density $3000{\text{kg/m}}^3$. The complete arrangement is placed in a liquid of density $2000{\text{kg/m}}^3$ and is allowed to reach equilibrium. The correct statement(s) is(are):

• A. The net elongation of the spring is $\frac{8\pi }{3}\text{m}$
• B. The net elongation of the spring is $\frac{4\pi }{3}\text{m}$
• C. The light sphere is partially submerged
• D. The light sphere is completely submerged

Answer 1

Answer: (A), (D)

The light sphere is completely submerged

Let us suppose, the net elongation of the spring is $x$.
According to question,
For heavier sphere, on applying $\sum F=ma$
$kx+F_b-Mg=0$ [$F_b\to$ buoyancy force]
$\Rightarrow kx+2\rho gV-3\rho gV=0$ [$V\to$ volume of sphere]
$\Rightarrow kx=\rho gV$ ... (1)
$\Rightarrow x=\frac{\rho gV}{k}$
$\Rightarrow x=\frac{\rho g}{k}\times \frac{4}{3}\pi R^3$
$\Rightarrow x=\frac{1000\times 10}{5000}\times \frac{4}{3}\times \pi \times 1^3$
$\Rightarrow x=\frac{8\pi }{3}\text{m}$
For lighter sphere, on applying $\sum F=ma$
$F_b^{\prime }-kx-mg=0$ [$F_b^{\prime }\to$ buoyancy force]
$\Rightarrow F_b^{\prime }=kx+mg$
$=\rho gV+\rho gV$ [from(1)]
$\Rightarrow F_b^{\prime }=2\rho gV$ ... (2)
Also, force of buoyancy on lighter sphere is maximum when it is fully submerged and is equal to $2\rho gV$.
From (2), as force of buoyancy is maximum,
therefore, it is completely submerged.