Question

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 = $0.1cm$, Least count for time = $0.1s$

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 $E_1,E_{II}and{E}_{III}$ are the percentage errors in g, i.e. $(\frac{\Delta g}{g}\times 100)$ for students I, II and III, respectively

• A. $E_I=0$
• B. $E_I$ is minimum
• C. $E_I=E_{II}$
• D. $E_{II}$ is maximum

Answer 1

Answer: (B)

$T=2\pi \sqrt{\frac{1}{g}}$
or $\frac{t}{n}=2\pi \sqrt{\frac{1}{g}}$
$\therefore g=\frac{(4{\pi }^2)(n^2)l}{t^2}$
$\%$ error in $g=\frac{\Delta g}{g}\times 100=(\frac{\Delta l}{l}+\frac{2\Delta t}{t})\times 100$
$E_I=(\frac{0.1}{64}+\frac{2\times 0.1}{128})\times 100=0.3125\%$
$E_{II}=(\frac{0.1}{64}+\frac{2\times 0.1}{64})\times 100=0.46875\%$
$E_{III}=(\frac{0.1}{20}+\frac{2\times 0.1}{36})\times 100=1.055\%$
Hence $E_I$ is minimum.