In a long cylinder of liquid of density $\mathrm{\rho }$, 2 balls of density ${\text{ρ}}_1 \mathrm{and} {\mathrm{\rho }}_2$ are hung by an ideal pulley. String is taut. Both balls have same volume V. Assume ${\mathrm{\rho }}_1>\mathrm{\rho }>{\mathrm{\rho }}_2.$ The condition under which the string remains taut is:

$$\begin{gathered} \begin{array}{l}{\mathrm{\rho }}_1\mathrm{Vg}-\mathrm{T}-\mathrm{\rho Vg}={\mathrm{\rho }}_1\mathrm{Va}\dots \text{ (i) }\\ \mathrm{T}+\mathrm{\rho Vg}-{\mathrm{\rho }}_2\mathrm{Vg}={\mathrm{\rho }}_2\mathrm{Va}\dots \text{ (ii) }\\ \therefore \mathrm{a}=\frac{\left({\mathrm{\rho }}_1-{\mathrm{\rho }}_2\right)\mathrm{g}}{\left({\mathrm{\rho }}_1+{\mathrm{\rho }}_2\right)}\\ \therefore \mathrm{T}={\mathrm{\rho }}_1\mathrm{Vg}-\mathrm{\rho Vg}-{\mathrm{\rho }}_1\mathrm{V}\left(\frac{{\mathrm{\rho }}_1-{\mathrm{\rho }}_2}{{\mathrm{\rho }}_1+{\mathrm{\rho }}_2}\right)\mathrm{g}\end{array} \\ \begin{array}{l}=\left[{\mathrm{\rho }}_1-\mathrm{\rho }-\left(\frac{{\mathrm{\rho }}_1^2-{\mathrm{\rho }}_1{\mathrm{\rho }}_2}{{\mathrm{\rho }}_1+{\mathrm{\rho }}_2}\right)\right]\mathrm{Vg}\\ =\left[\frac{{\mathrm{\rho }}_1^2+{\mathrm{\rho }}_1{\mathrm{\rho }}_2-{\mathrm{\rho }}_1\mathrm{\rho }-{\mathrm{\rho \rho }}_2-{\mathrm{\rho }}_1^2+{\mathrm{\rho }}_1{\mathrm{\rho }}_2}{{\mathrm{\rho }}_1+{\mathrm{\rho }}_2}\right]\mathrm{Vg}\\ =\frac{2{\mathrm{\rho }}_1{\mathrm{\rho }}_2}{{\mathrm{\rho }}_1{\mathrm{\rho }}_2}-\frac{\mathrm{\rho }\left({\mathrm{\rho }}_1+{\mathrm{\rho }}_2\right)}{{\mathrm{\rho }}_1+{\mathrm{\rho }}_2}\end{array} \end{gathered}$$
For string to remain taut, T > 0
$\begin{array}{l}\therefore \frac{2{\mathrm{\rho }}_1{\mathrm{\rho }}_2}{{\mathrm{\rho }}_1+\mathrm{\rho }}-\mathrm{\rho }>0\\ \Rightarrow \frac{2{\mathrm{\rho }}_1{\mathrm{\rho }}_2}{{\mathrm{\rho }}_1+{\mathrm{\rho }}_2}>\mathrm{\rho }\end{array}$