Question

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).

• A. 2.5 s
• B. 5.0 s
• C. 10.0 s
• D. 20.0 s

Answer 1

Answer: (C)

10.0 s

Answer 2

According to Newton?s law of cooling
$- \frac{d\theta}{dt} = k [\theta - \theta_0]$
Here, $k$ is proportionality constant
$\therefore \, \frac{50 - 49}{5} = k \left[ \frac{50 + 49}{2} - 30\right]$
$\frac{1}{5} = k \left[ \frac{99}{2} - 30 \right]$
$\frac{1}{5} = k \left[ \frac{39}{2} \right]$
$\Rightarrow \:\: k \frac{2 }{5 \times 39}$
Now same body cool from $40?C$ to $39?C$.
$\frac{40 - 39}{t} = \frac{2}{5 \times 39} \left[ \frac{40+39}{2} - 30 \right]$
$\frac{1}{t} = \frac{2}{5 \times 39} \left[ \frac{19}{2} \right]$
$\Rightarrow \:\:\: t = \frac{5 \times 39}{19} = 10.26 \, s = 10.0 s$