Two identical metal bars are heated in two different temperatures and allowed to cool in the same surroundings. Which one of the following figures correctly shows their cooling curves?
Show Hint
Cooling curves for identical objects in the same surroundings can never intersect. At any given temperature, the instantaneous rate of cooling is uniquely determined, so the curves must remain parallel-like, gradually converging at infinity.
Step 1: Understanding the Question:
We need to identify the correct cooling curves for two identical metal bars starting from different initial temperatures and cooling in the same environment.
Step 2: Key Formula or Approach:
According to Newton's Law of Cooling, the rate of cooling of a body is proportional to the difference in temperature between the body and its surroundings:
\[ \frac{dT}{dt} = -k(T - T_0) \]
Integrating this differential equation gives the temperature at any time \( t \):
\[ T(t) = T_0 + (T_i - T_0)e^{-kt} \]
where \( T_i \) is the initial temperature, \( T_0 \) is the surrounding temperature, and \( k \) is the cooling constant.
Step 3: Detailed Explanation:
For two identical bars, the cooling constant \( k \) is identical. Since they are kept in the same surroundings, \( T_0 \) is also identical.
Let the initial temperatures of the two bars be \( T_{i1} \) and \( T_{i2} \), with \( T_{i1} > T_{i2} \).
The temperature profiles for the two bars are:
\[ T_1(t) = T_0 + (T_{i1} - T_0)e^{-kt} \]
\[ T_2(t) = T_0 + (T_{i2} - T_0)e^{-kt} \]
Since \( T_{i1} - T_0 > T_{i2} - T_0 \) and \( e^{-kt} > 0 \), we have \( T_1(t) > T_2(t) \) for all \( t \ge 0 \).
This means the two cooling curves never cross each other and both approach the surrounding temperature \( T_0 \) asymptotically as \( t \to \infty \).
This behavior is correctly depicted only in figure (D).
Step 4: Final Answer:
The correct cooling curves are shown in figure (D).
