question_answer

A block of metal is heated to a temperature much higher than the room temperature and allowed to cool in a room free from air currents. Which of the following curves correctly represents the rate of cooling [Manipal MEE 1995]

A)                                 

B)  

C)                                 

D)  

Correct Answer: B

Solution :

                   According to Newton's law of cooling Rate of cooling µ Temperature difference Þ \[-\frac{d\theta }{dt}\propto (\theta -{{\theta }_{0}})\]Þ\[-\frac{d\theta }{dt}\]=\[\alpha \ (\theta -{{\theta }_{0}})\] (a= constant) Þ \[\int\limits_{{{\theta }_{i}}}^{\theta }{\frac{d\theta }{(\theta -{{\theta }_{0}})}=-\alpha \int\limits_{0}^{t}{dt}}\] Þ \[\theta ={{\theta }_{0}}+({{\theta }_{i}}-{{\theta }_{0}}){{e}^{-\alpha \,t}}\] This relation tells us that, temperature of the body varies exponentially with time from \[{{\theta }_{i}}\] to \[{{\theta }_{0}}\] Hence graph (b) is correct.