If $\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{~L}}{8}}, g$ is a constant and the relative error in T is $k$ times…
If $\mathrm{T}=2 \pi \sqrt{\frac{\mathrm{~L}}{8}}, g$ is a constant and the relative error in T is $k$ times to the percentage error in L then $\frac{1}{k}=$
$2$
$\frac{1}{200}$
$200$
$\frac{1}{2}$
Solution
$T=2 \pi \sqrt{\frac{L}{g}} \Rightarrow \frac{d T}{d L}=\frac{\pi}{\sqrt{L g}}$
Relation error in $T=\frac{d T}{T}=\frac{\pi}{\sqrt{L g}} d L \times \sqrt{\frac{g}{L}} \times \frac{1}{2 \pi}$
$\Rightarrow \frac{d T}{T}=\frac{d L}{2 L}$ ....(i)
Also, $\frac{d T}{T}=k\left(\frac{d L}{L} \times 100\right)$ ....(ii)
So, from (i) and (ii), $k\left(\frac{d L}{L} \times 100\right)=\frac{d L}{2 L}$
$\Rightarrow k=\frac{1}{200} \Rightarrow \frac{1}{k}=200$