Question

In the determination of Young's modulus $(Y=\frac{4MLg}{\pi l{d}^2})$ by using the Searle's method, a wire of length $L=2m$ and diameter $d=0.5mm$ is used. For a load $M=2.5kg$, an extension $l$ = 0.25 mm in the length of the wire is observed. Quantities $d$ and $l$ are measured using a screw gauge and a micrometer, respectively. They have the same pitch of $0.5mm$. The number of divisions on their circular scale is $100$. The contributions to the maximum probable error of the $Y$ measurement:

• A. due to the errors in the measurements of $d$ and $l$ are the same.
• B. due to the error in the measurement of $d$ is twice that due to the error in the measurement of $l$.
• C. due to the error in the measurement of $l$ is twice that due to the error in the measurement of $d$.
• D. due to the error in the measurement of $d$ is four times that due to the error in the measurement of $l$.

Answer 1

The correct option is A due to the errors in the measurements of $d$ and $l$ are the same.

$L.C=\frac{0.5}{100}=0.005$

$\frac{\Delta Y}{Y}$ $=$ $\frac{\Delta l}{l}$ + $\frac{2\Delta d}{d}$

$\frac{\Delta l}{l}$ $=$ $\frac{0.005\times {10}^{-3}}{0.25\times {10}^{-3}}$ $=$ $\frac{1}{50}$

$\frac{2\Delta d}{d}$ $=$ $\frac{2\times 0.005\times {10}^{-3}}{0.5\times {10}^{-3}}$ $=$ $\frac{1}{50}$