Heat and Thermodynamics 3 Question 21
21. Two metallic spheres $S _1$ and $S _2$ are made of the same material and have got identical surface finish. The mass of $S _1$ is thrice that of $S _2$. Both the spheres are heated to the same high temperature and placed in the same room having lower temperature but are thermally insulated from each other. The ratio of the initial rate of cooling of $S _1$ to that of $S _2$ is
(a) $\frac{1}{3}$
(b) $\frac{1}{\sqrt{3}}$
(c) $\frac{\sqrt{3}}{1}$
(d) $\left(\frac{1}{3}\right)^{1 / 3}$
$(1995,2 M)$
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Solution:
- The rate at which energy radiates from the object is
$\frac{\Delta Q}{\Delta t}=e \sigma A T^{4}$
Since, $\quad \Delta Q=m c \Delta T$, we get
$$ \frac{\Delta T}{\Delta t}=\frac{e \sigma A T^{4}}{m c} $$
Also, since $m=\frac{4}{3} \pi r^{3} \rho$ for a sphere, we get
$$ A=4 \pi r^{2}=4 \pi\left(\frac{3 m}{4 \pi \rho}\right)^{2 / 3} $$
Hence, $\quad \frac{\Delta T}{\Delta t}=\frac{e \sigma T^{4}}{m c}\left[4 \pi\left(\frac{3 m}{4 \pi \rho}\right)^{2 / 3}\right]$
$$ =K\left(\frac{1}{m}\right)^{1 / 3} $$
For the given two bodies
$$ \frac{(\Delta T / \Delta t) _1}{(\Delta T / \Delta t) _2}=\left(\frac{m _2}{m _1}\right)^{1 / 3}=\left(\frac{1}{3}\right)^{1 / 3} $$
