Question

A metal sphere cools from 80degree Celsius to 60 degree Celsius in 6 minute.how much time with it take to cool from 60degree Celsius t0 40degree Celsius if the room temperature is 30degree Celsius

Answer 1

The Metal sphere takes 24 minutes to cool from 60°C to 40°C

According to Newton's law of cooling,

The rate of cooling of a body is directly proportional to the temperature difference between body and surroundings.

Let X be the temperature of surrounding,

Y be the temperature of the body before cooling

Z be the temperature of the body after cooling,

Then By Newton's law of cooling

$\frac{Y - Z}{t} \: \propto \: Y - X$

Note : Cooling is a continuous process so the term temperature difference changes continuously, so to solve problems we take the the temperature difference as average temperature of body as it cools and the surroundings.

Given,

A metal sphere cools from 80°C to 60°C in 6 minutes.

Temperature of the surroundings is 30°C

So, Average temperature of cooling =( 80 + 60) ÷ 2 = 70°C

By Newton's law of cooling,

$\implies \frac{80 - 60}{6} = k(70 - 30) \\ \\ \implies \: \frac{20}{6} = 40k \\ \\ \implies \: \frac{1}{12} = k$

Now, The body cools from 60° C to 40°C

Average temperature of cooling = (60+40)÷ 2 = 50°C

Temperature of the surroundings is 30°C

According to Newton's law of cooling,

$\frac{60 - 40}{t} = k(50 - 40) \\ \\ \frac{20}{t} = \frac{1}{12} \times 10 \\ \\ t = \frac{240}{10} \\ \\ t = 24$

Therefore, It takes 24 minutes to cool down from 60° to 40°C