Question

Two spheres of radii $R_1$and $R_2$ have densities ${\rho }_1$ and ${\rho }_2$ and specific heat $C_1$ and $C_2$. If they are heated to the same temperature, the ratio of their rates of cooling will be

• A. $\frac{R_2{\rho }_2{C}_2}{R_1{\rho }_1{C}_1}$
• B. $\frac{R_1{\rho }_2{C}_2}{R_2{\rho }_1{C}_1}$
• C. $\frac{R_2{\rho }_1{C}_2}{R_1{\rho }_2{C}_1}$
• D. $\frac{R_2{\rho }_2{C}_1}{R_1{\rho }_1{C}_2}$

Answer 1

The correct option is A $\frac{R_2{\rho }_2{C}_2}{R_1{\rho }_1{C}_1}$

The power at which the body radiates heat is directly proportional to area:
$P=KA=K{R}^2$
and power dissipated is given by:
$P=mC\frac{d\theta }{dt}=\frac{4}{3}\pi R^3\rho C\frac{d\theta }{dt}$
i.e. we get:
$\frac{d\theta }{dt}=G\frac{1}{R\rho C}=Rate$ Rate of cooling: G is an arbitrary constant
i.e.
$\frac{Rat{e}_1}{Rat{e}_2}=\frac{R_2{\rho }_2{C}_2}{R_1{\rho }_1{C}_1}$
Hence option A is correct.