Question

In the determination of Young's modulus $(Y= \frac {4MLg}{\pi ld^2}) $ by using Searle's method, a wire of length $L = 2m$ and diameter $d = 0.5 \,mm$ is used. For a load $M = 2.5 \,kg$, 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 $0.5\, mm$. The number of divisions on their circular scale is $100$. The contributions to the maximum probable error of the $Y$ measurement is

• 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 I
• 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 I

Answer 1

Answer: (C)

$\Delta d=\Delta l=\frac{0.5}{100} \,mm =0.005 \, mm$
$Y=\frac{4 M L g}{\pi l d^{2}} \Rightarrow\left(\frac{\Delta Y}{Y}\right)_{\max }=\left(\frac{\Delta l}{l}\right)+2\left(\frac{\Delta d}{d}\right)$
$\left(\frac{\Delta l}{l}\right)=\frac{0.5 / 100}{0.25}=0.02$
and $\frac{2 \Delta d}{d}=\frac{(2)(0.5 / 100)}{0.5}=0.02$
or $ \frac{\Delta l}{l}=2 \cdot \frac{\Delta d}{d}$