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In order to determine the Young's Modulus of a wire of radius $0.2\, cm$ (measured using a scale of least count $=0.001\, cm )$ and length $1 \,m$ (measured using a scale of least count $=1\, mm$ ), a weight of mass $1\, kg$ (measured using a scale of least count $=1 \,g$ ) was hanged to get the elongation of $0.5\, cm$ (measured using a scale of least count $0.001\, cm$ ). What will be the fractional error in the value of Young's Modulus determined by this experiment? (in $\%$)
$0.14$
$0.9$
$9$
$1.4$
Solution
$Y=\frac{\text { Stress }}{\text { Strain }}=\frac{ FL }{ Al }=\frac{ mg . L }{\pi R ^{2} \cdot \ell}$
$\frac{\Delta Y }{ Y }=\frac{\Delta m }{ m }+\frac{\Delta L }{ L }+2 \cdot \frac{\Delta R }{ R }+\frac{\Delta \ell}{\ell}$
$\frac{\Delta Y }{ Y } \times 100=100\left[\frac{1}{1000}+\frac{1}{1000}+2\left(\frac{0.001}{0.2}\right)+\frac{0.001}{0.5}\right]$
$=\frac{1}{10}+\frac{1}{10}+1+\frac{1}{5}=\frac{14}{10}=1.4\, \%$
