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)

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), M α 16 π R 2 σ ln 3 2, M α 16 π R 2 σ ln 16 3, M α 4 π R 2 σ ln 3 2, M α 4 π R 2 σ ln 16 3

MARKS App · 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

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