Solid Spheres Connected by Spring Submerged in Liquid Equilibrium

To determine the correct statement(s), let us analyze the forces acting on the two spheres in liquid equilibrium.

1. System Parameters

• Volume of each solid sphere of radius $R$: $V = \frac{4}{3}\pi R^3$
• Mass of the lighter sphere (density $\rho$): $m_1 = V\rho = \frac{4}{3}\pi R^3 \rho$
• Mass of the heavier sphere (density $3\rho$): $m_2 = V(3\rho) = 4\pi R^3 \rho$
• Density of the liquid: $\rho_L = 2\rho$

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2. Overall Equilibrium of the System

The total weight of the system consisting of both spheres is: $W_{\text{total}} = m_1 g + m_2 g = V\rho g + 3V\rho g = 4V\rho g$

If both spheres are completely submerged, the maximum total buoyant force exerted by the liquid is: $F_{B,\text{max}} = V \rho_L g + V \rho_L g = V(2\rho)g + V(2\rho)g = 4V\rho g$

Since $W_{\text{total}} = F_{B,\text{max}}$, the system achieves equilibrium when both spheres are completely submerged in the liquid.

If the lighter sphere were only partially submerged (say, a fraction $f < 1$), the total buoyant force would be: $F_B = f V(2\rho)g + V(2\rho)g = (2f + 2)V\rho g$

Setting $F_B = W_{\text{total}}$ gives: $(2f + 2)V\rho g = 4V\rho g \implies 2f + 2 = 4 \implies f = 1$

Thus, the lighter sphere must be completely submerged in equilibrium. Therefore, Option (D) is correct and Option (C) is incorrect.

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3. Spring Elongation

Since the density of the lighter sphere ($\rho$) is less than that of the liquid ($2\rho$), it tends to float to the top. Conversely, the heavier sphere ($3\rho$) tends to sink to the bottom. As a result, the spring between them is in a state of extension (elongation $x$).

Considering the equilibrium of the heavier sphere (bottom sphere):

1. Downward gravitational force: $W_2 = m_2 g = 3V\rho g$
2. Upward buoyant force: $F_{B2} = V\rho_L g = 2V\rho g$
3. Upward spring force: $F_s = kx$

For equilibrium of the heavier sphere: $F_{B2} + F_s = W_2$ $2V\rho g + kx = 3V\rho g$ $kx = V\rho g$

Substituting $V = \frac{4}{3}\pi R^3$: $kx = \frac{4}{3}\pi R^3 \rho g \implies x = \frac{4\pi R^3 \rho g}{3k}$

This matches Option (A).

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Conclusion

• Option (A) is correct.
• Option (D) is correct.