Problem 117 - Master Resource Book in JEE Main Physics

Question

A body cools in a surrounding which is at a constant temperature of \(\theta_{0}\). Assume that it obeys Newton's law of cooling. Its temperature \(\theta\) is plotted against time \(t\). Tangents are drawn to the curve at the points \(P\left(\theta=\theta_{2}\right)\) and \(Q\left(\theta=\theta_{1}\right) .\) These tangents meet the time axis at angles of \(\phi_{2}\) and \(\phi_{1}\), as shown(a) \(\frac{\tan \phi_{2}}{\tan \phi_{1}}=\frac{\theta_{1}-\theta_{0}}{\theta_{2}-\theta_{0}}\) (b) \(\frac{\tan \phi_{2}}{\tan \phi_{1}}=\frac{\theta_{2}-\theta_{0}}{\theta_{1}-\theta_{0}}\) (c) \(\frac{\tan \phi_{1}}{\tan \phi_{2}}=\frac{\theta_{1}}{\theta_{2}}\) (d) \(\frac{\tan \phi_{1}}{\tan \phi_{2}}=\frac{\theta_{2}}{\theta_{1}}\)

Step-by-Step Solution

Answer

(a) \( \frac{\tan \phi_2}{\tan \phi_1} = \frac{\theta_1-\theta_0}{\theta_2-\theta_0} \) is the correct choice.

Step 1: Understanding Newton's Law of Cooling

Newton's law of cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature, \( \theta_0 \). Mathematically, this is \( \frac{d\theta}{dt} = -k(\theta - \theta_0) \), where \( k \) is a positive constant.

Step 2: Slope of the Tangent and its Interpretation

The slope of the tangent at a point on the temperature-time curve represents \( \frac{d\theta}{dt} \). At point \( P(\theta = \theta_2) \), the tangent makes an angle \( \phi_2 \) with the time axis, and \( \tan \phi_2 = \left| \frac{d\theta}{dt} \right| \) at \( P \). At \( Q(\theta = \theta_1) \), similarly, \( \tan \phi_1 = \left| \frac{d\theta}{dt} \right| \) at \( Q \).

Key Concepts

Rate of Temperature Change

The core idea behind Newton's Law of Cooling is centered around the rate of temperature change. Imagine you have a hot coffee cup cooling down in a room. The speed at which it cools, or its rate of temperature change, is not consistent—it depends on how hot the coffee is compared to room temperature. For a very hot cup, the cooling is fast. As it cools, this rate slows down. Mathematically, this is expressed as:

• \( \frac{d\theta}{dt} = -k(\theta - \theta_0) \)
• "\( \theta \)" represents the object's temperature.
• "\( \theta_0 \)" represents the ambient or surrounding temperature.
• "\( k \)" is a constant that affects how fast cooling happens.

This equation tells us that the rate \( \frac{d\theta}{dt} \), or how quickly temperature changes, is directly linked to the temperature difference between the object and its environment. The greater this difference, the faster the object's temperature will change, and vice versa. This proportionality is key in understanding cooling dynamics.

Ambient Temperature

Ambient temperature is essentially the temperature of the surroundings where a cooling process takes place. In our cooling coffee example, it's the temperature of the room. This is the baseline, the reference temperature to which the object's temperature \( \theta \) will move toward.
The concept of ambient temperature \( \theta_0 \) is crucial, as it influences how an object cools:

• If the ambient temperature is high, the object cools more slowly since the temperature difference is smaller.
• If the ambient temperature is lower, the object cools quickly as the difference is greater.

In Newton's Law of Cooling, you often see \( \theta_0 \) playing a vital role. It acts like a gravitational pull towards equilibrium, pulling the temperature of the object closer over time. It's important for understanding how objects stabilize their temperature relative to their surroundings.

Proportional Difference in Temperature

The phrase 'proportional difference in temperature' is about how Newton's Law of Cooling links the rate of temperature change to how far the current temperature \( \theta \) is from the ambient temperature \( \theta_0 \).

In simpler terms, the difference \( (\theta - \theta_0) \) controls how fast or slow the temperature changes:

• A larger difference means faster cooling as the object loses or gains heat quickly.
• A smaller difference results in slower cooling, indicating equilibrium is near.

The proportionality factor \( k \) in the equation \( \frac{d\theta}{dt} = -k(\theta - \theta_0) \) is crucial because it dictates how sensitive the rate of cooling is to this temperature difference. Thus, understanding this factor helps predict cooling rates and time needed to reach near ambient temperature.