Question

Two metal spheres A and B have radii r and 4r respectively. They are heated to 4000 K each and allowed to cool down. The respective ratio of their rates of cooling is :

• A. 1:4
• B. 4:1
• C. 1:256
• D. 1:16

Answer 1

The correct option is B 4:1

The formula for rate of cooling is given by

$mc(-\frac{dT}{dt})=\sigma .A(T^4-T_0^4)$. . . . . .(1)

density, $\rho =\frac{mass}{volume}=\frac{m}{V}$

$m=\rho .\frac{4}{3}\pi r^4$

Area, $A=4\pi r^2$

Equation (1) becomes

$\rho .\frac{4}{3}\pi r^{3}c.(-\frac{dT}{dt})=\sigma .4\pi r^2(T^4-T_0^4)$

$-\frac{dT}{dt}=\frac{3\sigma (T^4-T_0^4)}{\rho cr}$

$H=-\frac{dT}{dt}\propto \frac{1}{r}$

$\frac{H_1}{H_2}=\frac{r_2}{r_1}$

$\frac{H_1}{H_2}=\frac{4r}{r}$

$H_1:H_2=4:1$

The correct option is B.