"In an experiment to determine the Young's modulus of wire of a length exactly $1 \mathrm{~m}$, the extension in the length of the wire is measured as $0.4 \mathrm{~mm}$ with an uncertainty of $\pm\, 0.02 \mathrm{~mm}$ when a load of $1 \mathrm{~kg}$ is applied. The diameter of the wire is measured as $0.4 \mathrm{~mm}$ with an uncertainty of $\pm \,0.01 \mathrm{~mm}$. The error in the measurement of Young's modulus $(\Delta \mathrm{Y})$ is found to be $x \times 10^{10}\, \mathrm{Nm}^{-2}$. The value of $x$ is _________________. $\left(\right.$take $\mathrm{g}=10 \mathrm{~ms}^{-2}$ )"

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"

${{F/A} \over {l/L}} = Y,\,A = \pi {D^2}$

$${{\Delta Y} \over Y} = {{\Delta F} \over F} + {{2\Delta D} \over D} + {{\Delta l} \over e} + {{\Delta L} \over L}$$

$= 2 \times {{0.01} \over {0.4}} + {{0.02} \over {0.4}}$

$= {{0.04} \over {0.4}} = {1 \over {10}}$

$Y = {{Fl} \over {A\Delta l}}$

$= {{10 \times 1} \over {\pi {{(0.1\,mm)}^2} \times 0.4\,mm}}$

$= 1.988 \times {10^{11}}$

$\approx 2 \times {10^{11}}$

${{\Delta y} \over y} = {1 \over {10}}$

$\Delta y = {y \over {10}} = 2 \times {10^{10}}$

"

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