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

A body, initially at temperature \( \theta \) slightly exceeding the surrounding temperature \( \theta_s \), cools. The cooling rate, \( R \), is graphed against the body's temperature \( \theta \). % Step 1: Understanding the Law of Cooling Newton's law of cooling states the temperature change rate \( \frac{d\theta}{dt} \) is proportional to the difference between the body's temperature and the surroundings' \( \theta_s \): \[\frac{d\theta}{dt} = -k(\theta - \theta_s)\] where: - \( k \) is the proportionality constant. - \( \theta_s \) is the ambient temperature. - \( \theta \) is the body's temperature. % Step 2: Interpreting the Cooling Rate As the body cools, \( (\theta - \theta_s) \) decreases. Initially, when \( \theta \) is much greater than \( \theta_s \), the cooling rate \( R \) is high. As the body's temperature nears \( \theta_s \), the cooling rate diminishes because the temperature difference shrinks. This results in a curve with an initially steep negative slope, gradually flattening as the temperature difference lessens. The cooling rate thus declines as the body's temperature approaches the surroundings' temperature. % Step 3: Conclusion Consequently, the relationship between \( R \) and \( \theta \) is a curve descending with a negative slope, aligning with option (C).