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Surface Tension

Two solid sph...

Question

Two solid spheres are of equal volumes, but of different densities $d_{A}$ and $d_{B}$ and are connected by a string. They are fully immersed in a fluid of density $d_{F}$ . They get arranged into an equilibrium state as shown in the figure with a tension in the string. The arrangement is possible only if

A

d_{A} < d_{F}

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B

d_{B} > d_{F}

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C

d_{A} > d_{F}

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D

d_{A} + d_{B} = 2 d_{F}

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Solution

The correct options are
A $d_{A} < d_{F}$
B $d_{B} > d_{F}$
D $d_{A} + d_{B} = 2 d_{F}$
Let $V$ be the volume of each sphere and $T$ be the tension in each string.

For the string to be taut,
$d_{F} V g > d_{A} V g \Rightarrow d_{F} > d_{A}$
and $d_{B} V g > d_{F} V g \Rightarrow d_{B} > d_{F}$
For equilibrium,
$d_{F} V g + d_{F} V g + T = T + d_{A} V g + d_{B} V g \Rightarrow d_{A} + d_{B} = 2 d_{F}$

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