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Students $I$, $II$ and $III$ perform an experiment for measuring the acceleration due to gravity $(g)$ using a simple pendulum.
They use different lengths of the pendulum and /or record time for different number of oscillations. The observations are shown in the table.
Least count for length $=0.1 \mathrm{~cm}$
Least count for time $=0.1 \mathrm{~s}$
| Student | Length of the pendulum $(cm)$ | Number of oscillations $(n)$ | Total time for $(n)$ oscillations $(s)$ | Time period $(s)$ |
| $I.$ | $64.0$ | $8$ | $128.0$ | $16.0$ |
| $II.$ | $64.0$ | $4$ | $64.0$ | $16.0$ |
| $III.$ | $20.0$ | $4$ | $36.0$ | $9.0$ |
If $\mathrm{E}_{\mathrm{I}}, \mathrm{E}_{\text {II }}$ and $\mathrm{E}_{\text {III }}$ are the percentage errors in g, i.e., $\left(\frac{\Delta \mathrm{g}}{\mathrm{g}} \times 100\right)$ for students $\mathrm{I}, \mathrm{II}$ and III, respectively,
$E_I=0$
$E_I$ is minimum
$\mathrm{E}_{\mathrm{I}}=\mathrm{E}_{\mathrm{II}}$
$\mathrm{E}_{\text {II }}$ is maximum
Solution
$ g=4 \pi^2\left(\frac{\ell}{\mathrm{T}^2}\right) $
$ \frac{\Delta \mathrm{g}}{\mathrm{g}}=\frac{\Delta \ell}{\ell}+2 \frac{\Delta \mathrm{T}}{\mathrm{T}} $
$ \Rightarrow \mathrm{E}=\frac{\Delta \ell}{\ell}+2 \frac{\Delta \mathrm{t}}{\mathrm{t}}, \text { greater the value of } \mathrm{t}, \text { lesser the error }$
Hence, fractional error in the Ist observation is minimum
