A cubical block of co-efficient of linear expansion $\alpha_{s}$ is submerged partially inside a liq

Question

A cubical block of co-efficient of linear expansion $\alpha_{s}$ is submerged partially inside a liquid of co-efficient of volume expansion $\gamma_{l}$. On increasing the temperature of the system $\Delta T$, the height of the cube inside the liquid remains unchanged. Find the relation between $\alpha_{s}$ and $\gamma_{l}$.\

Professor PC Thomas Classes · Answer

When the temperature is increased by $\Delta T$, the volume of the cube increases but the density of liquid decreases. The depth upto which the cube is submerged in the liquid remains the same. Obviously the upthrust remains the same.\ Initial upthrust $=$ Final upthrust\ $V ho g=V^{\prime} ho^{\prime} g$\ where $V, V^{\prime}=$ volumes of cubical body immersed\ $ho, ho^{\prime}=$ densities of liquid at two temperatures\ $	herefore(A h) ho=\left[A\left(1+2 \alpha_{s} \Delta Tight) hight]\left(\frac{ho}{1+\gamma_{l} \Delta T}ight)$\ or $\left(1+\gamma_{l} \Delta Tight)=1+2 \alpha_{s} \Delta T$\ or $\gamma_{l} \Delta T=2 \alpha_{s} \Delta T$ or $\gamma_{l}=2 \alpha_{s}$.\