JEE MAIN - Physics (2025 - 28th January Morning Shift - No. 24)
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 _______ .
คำตอบ
3
คำอธิบาย
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Given, L = 5 mm, B = 2.5 mm
We know, least count of a screw gauge,
$$L.C. = {{Pitch\,length} \over {No.\,of\,division\,on\,circular\,scale}}$$
$$ \Rightarrow L.C. = {{0.75} \over {15}} = 0.05\,mm$$
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, $$x = 3$$.