A black coloured solid sphere of radius R and mass M is inside a cavity with a vacuum inside. The walls of the cavity are maintained at temperature T 0 . The initial temperature of the sphere is 3 T 0

Question

A black coloured solid sphere of radius $R$ and mass $M$ is inside a cavity with a vacuum inside. The walls of the cavity are maintained at temperature $T_{0}$ . The initial temperature of the sphere is $3 T_{0}$ . If the specific heat of the material of the sphere varies as $α T^{3}$ per unit mass with the temperature $T$ of the sphere, where $α$ is a constant, then the time taken for the sphere to cool down to temperature $2 T_{0}$ will be

( $σ$ is Stefan Boltzmann constant)

TestHub · Accepted Answer

d θ d t = - σ A e M s θ 4 - θ 0 4 d θ d t = - σ A e M α θ 3 θ 4 - θ 0 4 ∫ θ 3 θ 4 - θ 0 4 d θ = - σ A e M α ∫ d t Let θ 4 - θ 0 4 = x 4 θ 3 d θ = d x ∫ d x 4 x = - σ A e M α t 0 t 1 4 ln x = M σ A e M α t 0 t 1 4 ln θ 4 - θ 0 4 2 T 0 3 T 0 = - σ A e M α t 1 4 ln 2 T 0 4 - T 0 4 3 T 0 4 - T 0 4 = - σ A e M α t t = - α M 4 σ A e ln 15 T 0 4 80 T 0 4 = - α M 4 σ A e ln 3 16 = - α M 4 σ 4 π R 2 e ln 3 16 where e = 1 t = α M 16 π σ R 2 ln 16 3

Physics - Heat & Thermo Question with Solution | TestHub

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