Question

A metal block is placed in a room which is at ${10}^{\circ }\text{C}$. It is heated by an electric heater of power $500\text{W}$ till its temperature becomes ${50}^{\circ }\text{C}$. Its initial rate of rise of temperature is ${2.5}^{\circ }\text{C/sec}$. The heater is switched off and now a heater of $100\text{W}$ is required to maintain the temperature of the block at ${50}^{\circ }\text{C}$. (Assume Newton's Law of cooling to be valid). The rate of cooling in of the block at ${50}^{\circ }\text{C}$ after switching off the $100\text{W}$ heater is ${\frac{-1}{n}}^{\circ }\text{C/s}$, find $n$.

Answer 1

$P_{\text{heater}}=P_{\text{given to block}}+P_{\text{loss to surroundings}}$
$500=ms\frac{dT}{dt}+0$
$500=C\times 2.5+0$
$\therefore C=200{\text{J/}}^{\circ }C$ (where C is thermal capacity of the block)
At ${50}^{\circ }\text{C}$, power loss to surroundings = power of heater = $100\text{W}$
$100=-ms\frac{dT}{dt}=-C\frac{dT}{dt}$
$\frac{dT}{dt}=-\frac{100}{200}=-{0.5}^{\circ }\text{C}/s=\frac{1}{2}{}^{\circ }C/s$