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 $T_0$. The initial temperature of the sphere is $3T_0$. If the specific heat of the material of the sphere varies as $\alpha T^3$ per unit mass with the temperature $T$ of the sphere, where $\alpha $ is a constant, then the time taken for the sphere to cool down to temperature $2T_0$ will be ( $\sigma $ is Stefan Boltzmann constant)
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 $T_0$. The initial temperature of the sphere is $3T_0$. If the specific heat of the material of the sphere varies as $\alpha T^3$ per unit mass with the temperature $T$ of the sphere, where $\alpha $ is a constant, then the time taken for the sphere to cool down to temperature $2T_0$ will be ( $\sigma $ is Stefan Boltzmann constant)
- [JEE MAIN 2014]
- A
$\frac{{M\alpha }}{{4\pi {R^2}\sigma }}\,\ln \left( {\frac{3}{2}} \right)$
- B
$\frac{{M\alpha }}{{4\pi {R^2}\sigma }}\,\ln \left( {\frac{16}{3}} \right)$
- C
$\frac{{M\alpha }}{{16\pi {R^2}\sigma }}\,\ln \left( {\frac{16}{3}} \right)$
- D
$\frac{{M\alpha }}{{16\pi {R^2}\sigma }}\,\ln \left( {\frac{3}{2}} \right)$
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- [NEET 2021]