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Question:

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 T0. The initial temperature of the sphere is 3T0. If the specific heat of the material of the sphere varies as αT³ 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 2T0 will be (σ is Stefan Boltzmann constant)

Mα4πR²σln(32)

Mα4πR²σln(16/3)

Mα16πR²σln(32)

Mα16πR²σln(16/3)

Solution:

Answer is A. For a small change in temperature, heat loss is dQ = MSdT = MαT³dT
Differentiating w.r.t dt above, we get
dQ/dt = -4πR²σ(T⁴ - T0⁴) = MαT³dT/dt
Assuming T0 is negligible compared to T, we get
-4πR²σT⁴ = MαT³dT/dt
dT/T = -4πR²σ/Mα dt
Integrating both sides from 3T0 to 2T0 and 0 to t respectively, we get
∫₃ᵀ₀²ᵀ₀ dT/T = -4πR²σ/Mα ∫₀ᵗ dt
ln(2/3) = -4πR²σt/Mα
t = Mαln(3/2)/4πR²σ = Mαln(32)/4πR²σ