Question

A sphere and a cube made of the same metal have equal volumes, identical surface characteristics and are at the same temperature. If they are allowed to cool in identical surroundings, the ratio of their rates of loss of heat will be

(A) \(\frac{4\pi}3\): 1 

(B) 1 : 1

(C) \((\frac{\pi}6{})^{2/3}\) : 1

(D) \((\frac{\pi}6{})^{1/3}\) : 1

Answer 1

Correct Option is (D) \((\frac{\pi}{6})^{\frac{1}{3}} : 1\)

Given, A sphere and cube have same volume

then, \(\frac{4}{3} \pi r^3 = a^3\)

 \(a = (\frac{4}{3} \pi)^{\frac{1}{3}} r\)

Heat radiated,

\(\theta = \sigma AT^4\)

\(\frac{\theta _1}{\theta_2} = \frac{4\pi^2}{6a^2}\)

\(\frac{\theta _1}{\theta_2} = \frac{4 \pi r^2}{6[(\frac{4}{3}\pi)^\frac{2}{3}r^2]}\)

\(\frac{\theta _1}{\theta_2} = \frac{2 \pi r^2}{3[(\frac{4 \pi}{3})^\frac{2}{3}r^2]}\)

\(\frac{\theta_1}{\theta_2} = (\frac{\pi}{6})^{\frac{1}{3}} :1\)

\(\theta _1 = \) heat radiated by sphere

\(\theta_2 = \) Heat radiated by cube

Answer 2

(D) \((\frac{\pi}6{})^{1/3}\) : 1