Problem 189 - Master Resource Book in JEE Main Physics

Question

A liquid in a beaker has temperature \(\theta(t)\) at time \(t\) and \(\theta_{0}\) is temperature of surroundings, then according to Newton's law of cooling the correct graph between \(\log _{e}\left(\theta-\theta_{0}\right)\) and \(t\) is \(\quad\) IAIEEE 2012! (a) (b) (d) (c)

Step-by-Step Solution

Answer

The graph is a straight line with a negative slope.

Step 1: Understand Newton's Law of Cooling

Newton's Law of Cooling is given by the differential equation \( \frac{d\theta}{dt} = -k(\theta-\theta_{0}) \), where \( \theta \) is the temperature of the object, \( \theta_{0} \) is the ambient temperature, and \( k \) is a positive constant. This describes how the temperature difference \( \theta - \theta_{0} \) changes over time \( t \).

Step 2: Solve the Differential Equation

Separate variables and integrate: \( \int \frac{1}{\theta-\theta_{0}} \, d\theta = -k \int \, dt \). Upon integrating both sides, we get: \( \ln|\theta-\theta_{0}| = -kt + C \), where \( C \) is the constant of integration.

Step 3: Express the Log Equation

Rewriting the equation from before, we have: \( \log_{e} |\theta-\theta_{0}| = -kt + C \). This describes a linear relationship between \( \log_{e}(\theta-\theta_{0}) \) and \( t \), where the slope is \(-k\) and the y-intercept is \(C\).

Step 4: Determine the Graph Type

Since \( \log_{e}(\theta-\theta_{0}) \) against \( t \) forms a straight line with a negative slope, the correct graph should show a linear relationship, sloping downwards.

Key Concepts

Differential Equations

In the world of mathematics and physics, differential equations are fundamental tools used to describe how things change. They involve equations that relate a function to its derivatives. In simpler words, they show how a particular quantity varies with another, often over time.

When dealing with Newton's Law of Cooling, the differential equation is \[\frac{d\theta}{dt} = -k(\theta-\theta_{0})\]Here, \(\theta\) represents the temperature of the object at time \(t\), \(\theta_{0}\) is the temperature of the surroundings, and \(k\) is a constant that varies depending on the material.

• Separation of Variables: This is a technique often used to solve differential equations. It involves rearranging the equation so that each variable and its derivative or integral term are on opposite sides.
• Integration: Once the equation is separated, integration helps in finding a function that satisfies the original differential equation.

In Newton's Law of Cooling, this process leads to a clear relationship between temperature and time, describing how the temperature changes as time progresses.

Temperature Decay

Temperature decay refers to the gradual decrease in temperature over time as observed in objects cooling down. According to Newton's Law of Cooling, an object's temperature, \(\theta\), will decrease until it matches the ambient temperature, \(\theta_0\).

This law suggests a predictable pattern:

• Exponential Decay: The difference in temperature \(\theta - \theta_0\) decreases exponentially with time. This means that the change is rapid initially but slows down over time.
• Constant Rate: The rate of heat loss is proportional to the temperature difference. The greater the difference, the faster the cooling.

Ultimately, understanding temperature decay helps in predicting how long an object will take to reach its surrounding temperature. It applies to various real-world scenarios, such as cooling down coffee or understanding how fast a hot piece of metal will cool in air.

Graph Interpretation in Physics

Graph interpretation is a crucial skill in physics and is all about understanding the story that a graph tells. By analyzing graphs, one can derive significant insights into physical phenomena. In the context of Newton’s Law of Cooling, we interpret the graph of \(\log_{e}(\theta-\theta_0)\) versus time \(t\).

Key points to remember:

• Linear Relationship: In this exercise, we establish that \(\log_{e}(\theta-\theta_0)\) against \(t\) forms a straight line, indicating a linear relationship. This suggests that as time increases, the logarithm of the temperature difference decreases.
• Negative Slope: The slope of the line, \(-k\), is negative, representing the decline in temperature difference over time. Higher values of \(k\) mean faster cooling.
• Y-intercept: The constant \(C\) is where the line crosses the y-axis. This value can provide insights into initial conditions, such as the initial temperature difference at time \(t = 0\).

By understanding these aspects, students can interpret the physical meaning behind the equations and graphs, which are essential tools for analyzing temperature changes and other dynamic processes in physics.