Answer to Question #105914 in Molecular Physics | Thermodynamics for sridhar

Question

Two metallic spheres P and Q made of same material have same smoothness but the weight of P is 8 times that of Q. If the two are heated to same temperature and left to cool then the ratio of rate of cooling of Q to that of P is
ans 2:1

Answer 1

The rate at which energy radiates from the object is

$$\frac{\Delta Q}{\Delta t}=e\sigma AT^4$$

$$ΔQ=mcΔT$$

$$\frac{\Delta T}{\Delta t}=\frac{e\sigma AT^4}{mc}$$

$$m=\frac{4\pi\rho r^3}{3}\to A=4\pi r^2=4\pi\left(\frac{3m}{4\pi\rho}\right)^{\frac{2}{3}}$$

$$\frac{\Delta T}{\Delta t}=K\left(\frac{1}{m}\right)^{\frac{1}{3}}$$

The ratio of rate of cooling of Q to that of P is  

$$\left(\frac{8}{1}\right)^{\frac{1}{3}}=2$$