Question

If $T=2\pi \sqrt{\frac{l}{g}}$, then relative errors in T and l are in the ratio

• A. $1/2$
• B. $2$
• C. $1/2\pi$
• D. none of these

Answer 1

Answer: (A)

$1/2$

$T=2\pi \sqrt{\frac{l}{g}}$

$\frac{dT}{dl}=\frac{\pi }{\sqrt{lg}}$

Approximate error in T $=dT=\left(\frac{dT}{dl}\right)\Delta l$

$=\frac{\pi }{\sqrt{lg}}\Delta l$

Relative error in T $=\frac{dT}{T}=\frac{1}{2l}\Delta l$

Approximate error in l $=dl=\left(\frac{dl}{dT}\right)\Delta T$

$=\frac{\sqrt{lg}}{\pi }\Delta T$

Relative error in l i$=\frac{dl}{l}=\frac{1}{\pi }\sqrt{\frac{g}{l}}\Delta T$

Required ratio $=\frac{\frac{1}{2l}\Delta l}{\frac{1}{\pi }\sqrt{\frac{g}{l}}\Delta T}$

$=\frac{\pi }{2\sqrt{lg}}\frac{\Delta l}{\Delta T}$

$=\frac{1}{2}$