Q. A body cools from 50°C to 49°C in 5 seconds. How long will it take to cool from 40°C to 39°C? (Assume temperature of surrounding to be 30°C and Newton’s law of cooling is valid).

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A body cools from 50°C to 49°C in 5 seconds. How long will it take to cool from 40°C to 39°C? (Assume temperature of surrounding to be 30°C and Newton’s law of cooling is valid).

Solution:

According to Newton’s law of cooling

$ – frac{dtheta}{dt} = k [theta – theta_0]$

Here, $k$ is proportionality constant

$therefore , frac{50 – 49}{5} = k left[ frac{50 + 49}{2} – 30right]$

$frac{1}{5} = k left[ frac{99}{2} – 30 right]$

$frac{1}{5} = k left[ frac{39}{2} right] $

$ Rightarrow :: k frac{2 }{5 times 39}$

Now same body cool from $40°C$ to $39°C$.

$ frac{40 – 39}{t} = frac{2}{5 times 39} left[ frac{40+39}{2} – 30 right]$

$frac{1}{t} = frac{2}{5 times 39} left[ frac{19}{2} right]$

$Rightarrow ::: t = frac{5 times 39}{19} = 10.26 , s = 10.0 s $