A solid sphere of radius $R$ and density $\rho$ 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)

Step 1: Define the parameters of the system.

Let $V$ be the volume of each sphere of radius $R$.

$$V = \frac{4}{3}\pi R^3$$

The masses of the light sphere ($m_1$) and the heavy sphere ($m_2$) are determined by their densities.

$$m_1 = \rho V$$ $$m_2 = 3\rho V$$

Step 2: Determine the buoyant forces assuming both spheres are fully submerged.

The density of the liquid is $2\rho$.

$$B_1 = B_2 = 2\rho V g$$

Step 3: Write the equilibrium equations for each sphere.

For the light sphere, the upward buoyant force balances the downward gravitational force and the spring tension.

$$B_1 = m_1 g + T$$ $$2\rho V g = \rho V g + T$$ $$T = \rho V g$$

For the heavy sphere, the upward buoyant force and spring tension balance the downward gravitational force.

$$B_2 + T = m_2 g$$ $$2\rho V g + T = 3\rho V g$$ $$T = \rho V g$$

Step 4: Calculate the net elongation of the spring.

Apply Hooke's law to find the elongation $x$.

$$x = \frac{T}{k}$$ $$x = \frac{\rho V g}{k}$$

Substitute the volume $V$ into the expression.

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

Step 5: Verify the submergence of the light sphere.

Let $f$ be the fraction of the light sphere submerged in the liquid.

$$B_{\text{tot}} = f(2\rho)Vg + (2\rho)Vg$$ $$B_{\text{tot}} = 2\rho V g (f + 1)$$

The total weight of the system is the sum of the weights of both spheres.

$$W_{\text{tot}} = \rho V g + 3\rho V g$$ $$W_{\text{tot}} = 4\rho V g$$

Equate the total buoyant force to the total weight for overall equilibrium.

$$2\rho V g (f + 1) = 4\rho V g$$ $$f + 1 = 2$$ $$f = 1$$

Since $f = 1$, the light sphere is completely submerged.

Answer: The correct statements are that the net elongation of the spring is $\dfrac{4\pi R^3 \rho g}{3k}$ and the light sphere is completely submerged.