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

Answer (1)

Given,

d=0.5mm, l=0.25mm

Pitch = 0.5mm,

Number of circular scale divisions = 100

Young's modulus, $\displaystyle Y = \frac{4MLg}{\pi l d ^ 2}$

The maximum possible error in Y due to l and d are,

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

$\displaystyle \begin{aligned}Least\ count &=\frac{Pitch}{Number\ of \ circular\ scale\ divisions}\\&=\frac{0.5}{100} mm=0.005mm\end{aligned}$

Error contribution of l = $\displaystyle \frac{\Delta l}{l} =\frac{0.005}{0.25} =\frac{1}{50}$

Error contribution of d = $\displaystyle \frac{2\Delta d}{d} =\frac{2\times 0.005}{0.5} =\frac{1}{50}$

$i.e,$ The errors in the measurements of d and l are the same