Problem 85 - Master Resource Book in JEE Main Physics

Question

The least count of a stop watch is \(0.2 \mathrm{~s}\). The time of 20 oscillations of a pendulum is measured to be \(25 \mathrm{~s}\). The percentage error in the measurement of time will be (a) \(8 \%\) (b) \(1.8 \%\) (c) \(0.8 \%\) (d) \(0.1 \%\)

Step-by-Step Solution

Answer

(c) \(0.8\%\)

Step 1: Identify the Formula

To calculate the percentage error, we use the formula \( \text{Percentage Error} = \left( \frac{\text{Least Count of the Stopwatch}}{\text{Measured Time}} \right) \times 100 \). This formula will allow us to find the error in the measured time as a percentage of the total measured time.

Step 2: Insert Given Values

From the problem, the least count of the stop watch is given as \(0.2\,\text{s}\) and the measured time for 20 oscillations is \(25\,\text{s}\). Substitute these values into the formula: \( \text{Percentage Error} = \left( \frac{0.2}{25} \right) \times 100 \).

Step 3: Calculate the Fraction

Calculate the fraction \( \frac{0.2}{25} \). This division simplifies to \(0.008\).

Step 4: Convert to Percentage

Now, multiply the fraction from the previous step by 100 to convert it to a percentage: \(0.008 \times 100 = 0.8\%\).

Step 5: Choose the Correct Answer

The calculated percentage error is \(0.8\%\). Therefore, the correct answer is (c) \(0.8\%\).

Key Concepts

Least Count

When you're dealing with measurements, accuracy matters, and that's where the concept of 'least count' comes into play. The least count is the smallest measurement that an instrument can precisely measure. It's a crucial part of any measurement device's precision. In simple terms, it tells you how fine-tuned or sensitive your measuring device is.

For instance, if a stopwatch has a least count of 0.2 seconds, it means the smallest time interval it can measure is 0.2 seconds. Anything shorter is beyond its capability. So, when measuring time intervals, like pendulum oscillations, knowing the least count helps determine if the device is adequate for the task.

This concept is important because:

• It influences the precision of your measurement process.
• It defines the smallest change in measurement that can be detected.
• Understanding it prevents underestimating or overestimating the true value of a measurement.

Hence, when proceeding with measurements, always keep the least count in mind to ensure that your results are as accurate as possible!

Pendulum Oscillations

Pendulum oscillations refer to the periodic swinging motion of a pendulum. It's an interesting concept that provides a wealth of information about time and dynamics in physics. The oscillation begins when a pendulum is displaced from its equilibrium position and then released. Gravity pulls the pendulum back, and it swings back and forth.

Each complete cycle of motion from one side to the other and back again is called an oscillation. And several factors can influence the time it takes for a pendulum to complete one oscillation, such as the length of the pendulum and the acceleration due to gravity.

The formula that governs pendulum motion is:
\[ T = 2\pi \sqrt{\frac{L}{g}} \]
where:

• \( T \) is the period (time for one complete oscillation),
• \( L \) is the length of the pendulum, and
• \( g \) is the acceleration due to gravity.

Understanding pendulum oscillations is not only crucial for grasping basic physics but also forms a foundation for understanding more complex systems of motion. A pendulum's predictable patterns make it a useful scientific tool for measuring time.

Measurement Uncertainty

Measurement uncertainty accounts for the doubt that exists about the result of any measurement. This is a natural aspect of scientific inquiry, as no measurement can be perfectly exact. Several factors contribute to uncertainties, such as limitations of instruments, environmental conditions, or human error.

In the example of measuring the time of pendulum oscillations, measurement uncertainty can arise from the stopwatch's least count. It sets a bound on how close the measurement can be to the true value. Even though you might measure something repeatedly, you'll often get slightly different results.

Here's why understanding measurement uncertainty is important:

• It provides a comprehensive picture of the accuracy and reliability of your measurements.
• Helps calculate how much risk there is of an error being present in reported results.
• Enables comparison between competing measurements by understanding possible deviations.

Expressing measurement uncertainty helps you to find percentage error, which represents how much error is present in measured values as a proportion of the actual measurement. This knowledge is crucial in ensuring scientific experiments yield trustworthy and transparent results.