Question #42519

A solid block of volume $V_0$ and density $d$ is completely submerged in a liquid and suspended by a vertical string. The liquid has an initial density $\rho_0$ (where $\rho_0 < d$), mass $m$, specific heat capacity $c$, and coefficient of volume expansion $\beta$. The block's thermal expansion is negligible. At $t=0$, a heater starts supplying heat to the liquid at a constant power $P$. The liquid loses heat to the surroundings at a rate $\lambda(T - T_0)$, where $T$ is the instantaneous temperature of the liquid, $T_0$ is the constant ambient temperature, and $\lambda$ is a positive constant. Assuming uniform temperature in the liquid and $\beta(T-T_0) \ll 1$, which of the following expressions represents the tension in the string as a function of time $t$?