Question

A metallic sphere having radius 0.08 m and mass m = 10 kg is heated to a temperature of ${227}^{\circ }$C and suspended inside a box whose walls are at a temperature of ${27}^{\circ }$C. The maximum rate at which its temperature will fall is :
(Take e = 1, Stefan's constant; and specific heat of the metal s = 90 cal/kg/deg J = 4.2 Joules/Calorie)

• A. ${.055}^{\circ }$C/sec
• B. ${.276}^{\circ }$ C/sec
• C. ${.044}^{\circ }$ C/sec
• D. ${0.03}^{\circ }$ C/sec

Answer 1

The correct option is B ${.276}^{\circ }$ C/sec

Given: A metallic sphere having radius $0.08m$ and mass $m=10kg$ is heated to a temperature of ${227}^{\circ }C$ and suspended inside a box whose walls are at a temperature of ${27}^{\circ }C.$

To find the maximum rate at which its temperature will fall

Solution:

As per the given criteria,

radius of the metallic sphere, $r=0.08m$

and mass of the metallic sphere, $m=10kg$

Final temperature of the sphere, $T={227}^{\circ }C=227+273=500K$

Initial temperature of the sphere, $T_o={27}^{\circ }C=27+273=300K$

We know,

Rate of cooling $=\frac{e\sigma A(T^4-{T_o}^4)}{mS}$

where $A$ is area of the sphere

$⟹=\frac{1\times 5.67\times {10}^{-8}\times 4\pi r^2\times ({\left(500\right)}^4-{\left(300\right)}^4)}{10\times 90}⟹=\frac{5.67\times {10}^{-8}\times 4\times 3.14\times (0.08{)}^2\times (625\times {10}^8-81\times {10}^8)}{900}⟹=\frac{0.456\times {10}^{-8}(544\times {10}^8)}{900}⟹=\frac{248.1}{900}⟹=0.276℃/s$

is the maximum rate at which temperature will fall