Problem 78 - Master Resource Book in JEE Main Physics

Question

The initial temperature of a liquid is \((80.0 \pm 0.1)^{\circ} \mathrm{C}\). After it has been cooled, its temperature is \((10.0 \pm 0.1)^{\circ} \mathrm{C}\). The fall in temperature in degree centigrade is (a) \(70.0\) (b) \(70.0 \pm 0.3\) (c) \(70.0 \pm 0.2\) (d) \(70.0 \pm 0.1\)

Step-by-Step Solution

Answer

The fall in temperature is (c) \(70.0 \pm 0.2\) degrees Celsius.

Step 1: Identify Initial and Final Temperatures

First, note the initial temperature of the liquid as \( T_i = 80.0 \, ^\circ \text{C} \) with an uncertainty of \( \pm 0.1 \, ^\circ \text{C} \). The final temperature after cooling is \( T_f = 10.0 \, ^\circ \text{C} \) also with an uncertainty of \( \pm 0.1 \, ^\circ \text{C} \).

Step 2: Calculate the Temperature Difference

The temperature difference \( \Delta T \) is calculated as \( T_i - T_f = 80.0 \, ^\circ \text{C} - 10.0 \, ^\circ \text{C} = 70.0 \, ^\circ \text{C} \).

Step 3: Calculate the Uncertainty in the Temperature Difference

The uncertainty in the temperature difference is calculated by adding the uncertainties of the initial and final temperatures: \( \pm 0.1 \, ^\circ \text{C} + \pm 0.1 \, ^\circ \text{C} = \pm 0.2 \, ^\circ \text{C} \).

Step 4: State the Final Answer with Uncertainty

Considering the calculated temperature difference and the uncertainty, the fall in temperature is \( 70.0 \, ^\circ \text{C} \pm 0.2 \, ^\circ \text{C} \).

Key Concepts

Uncertainty Calculation

When measuring temperature, or any quantity, it's crucial to understand the concept of uncertainty. This tells us how much doubt exists around a measurement. In our exercise, the temperature measurements come with an uncertainty of \( \pm 0.1 \ ^\circ \text{C} \). This represents the possible error range in both the initial and final temperature values. For students, it's essential to recognize that uncertainty is unavoidable but can be minimized with precise instruments and careful measurement techniques.

To calculate the overall uncertainty when comparing two measurements, like in a subtraction (which is our case with initial and final temperatures), we typically sum up their individual uncertainties. Hence, if a temperature initially reads \( 80.0 \pm 0.1 \ ^\circ \text{C} \) and changes to \( 10.0 \pm 0.1 \ ^\circ \text{C} \), the combined uncertainty in the temperature difference is \( \pm 0.1 + \pm 0.1 = \pm 0.2 \ ^\circ \text{C} \).

Understanding how to combine uncertainties gives us a clearer picture of our confidence level in the result.

Error Propagation

The concept of error propagation is all about understanding how uncertainties in our measurements impact the final result. When calculating derived quantities, like the difference in temperature in our exercise, we must determine how the uncertainties in each measurement component affect the final outcome.

In mathematical operations involving measurements, errors propagate differently:

• For addition or subtraction, like calculating a temperature difference, combine absolute uncertainties.
• For multiplication or division, combine relative uncertainties.

In our example, because the final result required subtraction of two temperature readings, we add their absolute uncertainties: \( \pm 0.1 \ ^\circ \text{C} \) from each measurement, resulting in an overall uncertainty of \( \pm 0.2 \ ^\circ \text{C} \).

Accurate error propagation helps ensure that the final reported value truly reflects any variability in measurements.

Temperature Difference

The temperature difference is a simple concept but one that forms a core understanding of many scientific measurements. It's essential to comprehend not just how to calculate this difference, but what it represents and why it's important.

Temperature difference, denoted by \( \Delta T \), represents the change in temperature from an initial state to a final state. In our exercise, this was calculated as \( T_i - T_f = 80.0 \ ^\circ \text{C} - 10.0 \ ^\circ \text{C} = 70.0 \ ^\circ \text{C} \).

This change could result from several processes, like heating, cooling, or reacting substances. Knowing the temperature difference allows scientists and engineers to understand system dynamics, efficiency, and material properties.

By combining this understanding with uncertainty calculation and error propagation, one ensures not only the computation of values but also their reliability and accuracy, which are vital in any scientific analysis.