Question 226: Units , Dimensions and Error Analysis

Question

A tiny metallic rectangular sheet has length and breadth of 5 mm and 2.5 mm , respectively. Using a specially designed screw gauge which has pitch of 0.75 mm and 15 divisions in the circular scale, you are asked to find the area of the sheet. In this measurement, the maximum fractional error will be $\frac{x}{100}$ where $x$ is _______ .

Accepted Answer

Given, L = 5 mm, B = 2.5 mm
We know, least count of a screw gauge,
$$L.C. = {	ext{Pitch length} \over 	ext{No. of division on circular scale}}$$
$$ \Rightarrow L.C. = {{0.75} \over {15}} = 0.05\,mm$$
We know, $$A = LB$$
By taking $ln$ on both sides,
$$ \Rightarrow \ln A = \ln L + \ln B$$
by differentiating both sides,
$$ \Rightarrow {{dA} \over A} = {{dL} \over L} + {{dB} \over B}$$
Here, $$dL = dB = 0.05\,mm$$ (L.C.)
So fractional error,
$${{dA} \over A} = {{0.05} \over 5} + {{0.05} \over {2.5}}$$
$$ = {1 \over {100}} + {2 \over {100}} = {3 \over {100}} = {x \over {100}}$$ (given)
Hence, $$x = 3$$.

Units , Dimensions and Error Analysis

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