Question

A sphere and a cube of same material and same total surface area are placed in the same evacuated space turn by turn after they are heated to the same temperature. Find the ratio of their initial rates of cooling in the enclosure –

• A. $\sqrt{\frac{\pi}{6}}:1$
• B. $\sqrt{\frac{\pi}{3}}:1$
• C. $\frac{\pi}{\sqrt{6}}:1$
• D. $\frac{\pi}{\sqrt{3}}:1$

Answer 1

Answer: (A)

$\sqrt{\frac{\pi}{6}}:1$

Answer 2

S_S = S_C Ž m_S > m_C ; $\frac{dQ}{dt}$ = $\frac{4\sigma AT_{0}^{3}}{ms}$ DT

Given 6a² = 4pr² or $\frac{a}{r}$=$\sqrt{\frac{4\pi}{6}}$=$\sqrt{\frac{2\pi}{3}}$

$\frac{dQ}{dt}$ µ $\frac{A}{m}$ ; $\frac{\left( \frac{dQ}{dt} \right)_{S}}{\left( \frac{dQ}{dt} \right)_{C}} = \frac{\frac{4\pi r^{2}}{\frac{4}{3}\pi r^{3}\rho}}{\frac{6a^{2}}{a^{3}\rho}}$= $\frac{3/r}{6/a} = \frac{a}{2r}$

= $\frac{1}{2}$ $\sqrt{\frac{2\pi}{3}}$ = $\sqrt{\frac{\pi}{6}}$: 1