Animated Solution for Physics - Physics and Measurement: In a simple pendulum, experiment for determination of acceleration due to gravity ($g$), time taken for 20 oscillations is measured by using a watch of 1 second least count. The mean value of time taken comes out to be 30 s. The length of pendulum is measured by using a meter scale of least count 1 mm and the value obtained 55.0 cm. The percentage error in the determination of $g$ is close to

Imagine you are standing in a physics laboratory, tasked with determining the acceleration due to gravity, $g$, using nothing but a simple pendulum, a meter scale, and a stopwatch. It sounds like a straightforward experiment, but the true test of a physicist lies in understanding the limitations of their tools. Every measurement carries a tiny seed of uncertainty, and our goal is to find out how these tiny seeds grow into a larger error in our final calculation.

The heartbeat of this experiment is the formula that connects the time period of the pendulum to its length and the acceleration due to gravity:

Here, $l$ is the length of the pendulum, and $T$ is the time period for one complete oscillation. The terms $4$ and ${\pi }^2$ are pure mathematical constants. They are perfect and carry absolutely zero error. The uncertainty in our calculated value of $g$ comes entirely from our human measurements of $l$ and $T$.

To find the maximum possible percentage error in $g$, we must use the principles of error propagation. When quantities are multiplied or divided, their relative errors add up. Furthermore, if a quantity is raised to a power, that power becomes a multiplier for its relative error.

Applying these rules to our master equation, we get the relative error equation:

Notice the crucial factor of $2$ in front of the time error. Because $T$ is squared in the denominator, any mistake we make in measuring time will have double the impact on our final result! This is a classic trap, and it highlights why measuring time accurately is the most critical part of this experiment.

Now, let's look at the raw data provided in the problem.

1. Length Measurement: The length of the pendulum is measured as $l=55.0\text{ cm}$. The meter scale used has a least count of $1\text{ mm}$. The least count is the smallest value an instrument can measure, and it represents the maximum absolute error possible in a single reading. Therefore, our absolute error in length is:

2. Time Measurement: The time taken for $20$ oscillations is $t=30\text{ s}$. The stopwatch has a least count of $1\text{ s}$, meaning our absolute error in the total time is $\Delta t=1\text{ s}$.

You might wonder, shouldn't we find the error in the time period $T$ for a single oscillation? Mathematically, the relative error in the total time $t$ is exactly equal to the relative error in the time period $T$. Since $T=\frac{t}{n}$ and $\Delta T=\frac{\Delta t}{n}$, the ratio $\frac{\Delta T}{T}$ simplifies perfectly to $\frac{\Delta t}{t}$. We can plug our total time values directly into the formula!

Let's substitute our known values into the relative error equation:

To convert this relative error into a percentage error, we simply multiply the entire expression by $100$:

Now, we execute the final arithmetic. $\frac{10}{55}$ simplifies to $\frac{2}{11}$, which is approximately $0.18\%$. The second term, $\frac{20}{3}$, is approximately $6.67\%$.

Looking at our options, $6.85\%$ is closest to $6.8\%$.

This problem beautifully illustrates a core principle of experimental physics: the quantity with the highest power in your formula demands the most precise measuring instrument. Because time was squared, its relatively small absolute error of $1\text{ s}$ ballooned into a massive $6.67\%$ error, completely dominating the tiny $0.18\%$ error from the length measurement.