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
Diffcult
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(b) According to Newton's law of cooling
Rate of cooling $\propto$ Temperature difference
Rate of cooling $\propto$ Temperature difference
==> $ - \frac{{d\theta }}{{dt}} \propto (\theta - {\theta _0})$
==>$ - \frac{{d\theta }}{{dt}}$=$\alpha \;(\theta - {\theta _0})$ ($\alpha$= constant)
==> $\int\limits_{{\theta _i}}^\theta {\frac{{d\theta }}{{(\theta - {\theta _0})}} = - \alpha \int\limits_0^t {dt} } $
==> $\theta = {\theta _0} + ({\theta _i} - {\theta _0}){e^{ - \alpha \,t}}$
This relation tells us that, temperature of the body varies exponentially with time from ${\theta _i}$ to ${\theta _0}$
Hence graph $(b)$ is correct.
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