question_answer

A black colored 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 \[3{{T}_{0}}.\]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 \[2{{T}_{0}}\]  will be (\[\sigma \]is Stefan Boltzmann constant)                  [JEE ONLINE 19-04-2014]

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)       \[\frac{M\alpha }{16\pi {{R}^{2}}\sigma }\ln \left( \frac{3}{2} \right)\]

Correct Answer: C

Solution :

[c] In the given problem, fall in temperature of sphere, Temperature of surrounding, Initial temperature of sphere, Specific heat of the material of the sphere varies as, per unit mass ( constant) Applying formula,

Solving we get,

Time taken for the sphere to cool down temperature ,