Question

Temperature of a body \( \theta \) is slightly more than the temperature of the surrounding \( \theta_s \). Its rate of cooling (R) versus temperature of the body (\( \theta \)) is plotted. Its shape would be:

• A.

  

• B.

  

• C.

  

• D.

  

Hint:

The rate of cooling decreases as the temperature difference between the body and its surroundings decreases. This leads to a curve that flattens over time.

Answer 1

The Correct Option is C

We are given the situation of a body \( \theta \) with a temperature slightly higher than the surrounding temperature \( \theta_s \). The rate of cooling, \( R \), is plotted against the temperature of the body \( \theta \). % Step 1: Understanding the Law of Cooling According to Newton's law of cooling, the rate of change of temperature \( \theta \) with respect to time is proportional to the difference between the body's temperature and the surrounding temperature \( \theta_s \). This can be written as: \[ \frac{d\theta}{dt} = -k(\theta - \theta_s) \] where: - \( k \) is the constant of proportionality, - \( \theta_s \) is the surrounding temperature, - \( \theta \) is the temperature of the body. % Step 2: Interpreting the Cooling Rate As the body cools, the difference \( (\theta - \theta_s) \) becomes smaller. Initially, when \( \theta \) is much higher than \( \theta_s \), the rate of cooling \( R \) is large. As the temperature of the body approaches \( \theta_s \), the rate of cooling decreases, since the difference between the body’s temperature and the surrounding temperature is smaller. This behavior leads to a curve that initially has a steep negative slope but gradually flattens out as the temperature difference reduces. This implies the rate of cooling decreases as the body’s temperature approaches that of the surroundings. % Step 3: Conclusion The relationship between \( R \) and \( \theta \) would, therefore, follow a curve that decreases with a negative slope, which corresponds to option (C).