Equilibrium of Connected Spheres Immersed in Fluid

To determine the conditions for the system of two solid spheres $A$ and $B$ to remain in static equilibrium fully immersed in a fluid, we analyze the forces acting on each sphere individually and on the combined system.

Let:

• $V$ be the volume of each sphere (given that both have equal volumes).
• $d_A$ and $d_B$ be the densities of spheres $A$ and $B$, respectively.
• $d_F$ be the density of the fluid.
• $T$ be the tension in the string connecting the two spheres.
• $g$ be the acceleration due to gravity.

1. Equilibrium of Sphere A (Upper Sphere)

The forces acting on sphere $A$ are:

• Upward buoyant force: $F_{BA} = V d_F g$
• Downward gravitational force (weight): $W_A = V d_A g$
• Downward tension force exerted by the string: $T$

For sphere $A$ to be in equilibrium:
$F_{BA} = W_A + T$
$V d_F g = V d_A g + T$
$T = V g (d_F - d_A)$

For the string to be taut and under tension, we must have $T > 0$:
$V g (d_F - d_A) > 0 \implies d_A < d_F$

Hence, option (A) is correct.

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2. Equilibrium of Sphere B (Lower Sphere)

The forces acting on sphere $B$ are:

• Upward buoyant force: $F_{BB} = V d_F g$
• Upward tension force exerted by the string: $T$
• Downward gravitational force (weight): $W_B = V d_B g$

For sphere $B$ to be in equilibrium:
$F_{BB} + T = W_B$
$V d_F g + T = V d_B g$
$T = V g (d_B - d_F)$

For the tension to be strictly positive ($T > 0$):
$V g (d_B - d_F) > 0 \implies d_B > d_F$

Hence, option (B) is correct.

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3. Equilibrium of the Combined System (Spheres A + B)

For the entire system suspended in the fluid without accelerating vertically, the total buoyant force must equal the total weight of both spheres:
$\sum F_{\text{up}} = \sum F_{\text{down}}$
$F_{BA} + F_{BB} = W_A + W_B$
$V d_F g + V d_F g = V d_A g + V d_B g$
$2 V d_F g = V g (d_A + d_B)$
$d_A + d_B = 2 d_F$

Hence, option (D) is correct.

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Conclusion

The arrangement shown is possible with a non-zero tension in the string if and only if:

• $d_A < d_F$
• $d_B > d_F$
• $d_A + d_B = 2 d_F$

Correct Options: A, B, D