A black coloured solid sphere of radius R and mass M is inside a cavity with vacuum inside. The walls of the cavity are maintained at temperature…

A black coloured solid sphere of radius $\displaystyle R$ and mass M is inside a cavity with vacuum inside. The walls of the cavity are maintained at temperature $\displaystyle \mathrm{T}_0$. The initial temperature of the sphere is $\displaystyle 3 \mathrm{~T}_0$. If the specific heat of the material of the sphere varies as $\displaystyle \alpha \mathrm{T}^3$ per unit mass with the temperature T of the sphere, where $\displaystyle \alpha$ is a constant, then the time taken for the sphere to cool down to temperature $\displaystyle 2 \mathrm{~T}_0$ will be ( $\displaystyle \sigma$ is Stefan Boltzmann constant)