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  •  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 T0. The initial tempe

 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 T0. The initial temperature of the sphere is 3T0. If  the specific heat of the material of the sphere varies as  \alphaT3 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 2T0 will be (\sigma is Stefan Boltzmann constant)                                                                                    

 

Option: 1

\frac{M\alpha }{4\pi R^{2}\sigma }ln\left ( \frac{3}{2} \right )


Option: 2

\frac{M\alpha }{4\pi R^{2}\sigma }ln\left ( \frac{16}{3} \right )


Option: 3

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


Option: 4

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


Answers (1)

\\ In \ the \ given \ problem, fall \ in \ temperature \ of \ sphere, \\ \\ d T=\left(3 T_{0}-2 T_{0}\right)=T_{0} \\ \\Temperature \ of \ surrounding, \ T_{\text {surr}}=T_{0} \ \\ \\ Initial \ temperature \ of \ sphere, \ T_{\text {initial}}=3 T_{0} \\ \\ Specific\ heat \ of \ the \ material \ of \ the \ sphere \ varies \ as \ c=\alpha T^{3} \ per \ unit \ mass \\ \\ (\alpha=a \ constant) \
\\ Applying \ formula. \\ \\ \frac{d T}{d t}=\frac{\sigma A}{M c J}\left(T^{4}-T_{s u r r}^{4}\right) \\ \\ \Rightarrow \frac{T_{0}}{d t}=\frac{\sigma 4 \pi R^{2}}{M \alpha\left(3 T_{0}\right)^{3} J}\left[\left(3 T_{0}\right)^{4}-\left(T_{0}\right)^{4}\right] \\ \\ \Rightarrow d t=\frac{M \alpha 27 T_{0}^{4} J}{\sigma 4 \pi R^{2} \times 80 T_{0}^{4}} \\ \\ Solving \ we \ get, \text{Time taken for the sphere to cool down temperature} \ \ 2 T_{0}, \\ \ \\ t=\frac{M \alpha}{16 \pi R^{2} \sigma} \ln \left(\frac{16}{3}\right)
 

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Kshitij

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