Students I, II and III perform an experiment for measuring the acceleration due to gravity ( $g$ ) u

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 $=0.1 \mathrm{~cm}$, Least count for time $=0.1 \mathrm{~s}$ \ If $E_{1}$ and $E_{\mathrm{II}}$ and $E_{\mathrm{III}}$ are the percentage errors in g, i.e. $\left(\frac{\Delta g}{g} 	imes 100ight)$ for students I, II and III respectively.\

Professor PC Thomas Classes · Answer

(b) : The time period $T$ of the simple pendulum is related to the acceleration due to gravity $g$ as\ $T=2 \pi \sqrt{\frac{l}{g}}$ or $g=\frac{4 \pi^{2} l}{T^{2}}=\frac{4 \pi^{2} l}{t^{2}} n^{2}$\ $\Rightarrow \frac{\Delta g}{g}=\frac{\Delta l}{l}+2 \frac{\Delta t}{t}$\ For student I\ $\left(100 	imes \frac{\Delta g}{g}ight)_{\mathrm{I}}=E_{\mathrm{I}}=\left(\frac{0.1}{64.0}+\frac{2 	imes 0.1}{128.0}ight) 	imes 100=\frac{20}{64}$\ For student II\ $\left(100 	imes \frac{\Delta g}{g}ight)_{	ext {II }}=E_{	ext {II }}=\left(\frac{0.1}{64.0}+2 	imes \frac{0.1}{64.0}ight) 	imes 100=\frac{30}{64}$\ For student III $$ \begin{aligned} & \left(100 	imes \frac{\Delta g}{g}ight)_{\mathrm{III}}=E_{\mathrm{III}} \ & \quad=\left(\frac{0.1}{20.0}+2 	imes \frac{0.1}{36.0}ight) 	imes 100=\frac{19}{18} . \end{aligned} $$ Values of $E_{\mathrm{I}}, E_{\mathrm{II}}$ and $E_{\mathrm{III}}$ suggest that only alternative (b) is correct.\