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

$\left[\right.$ Take $\left.g =10\,m / s ^{2}\right]$

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

c $L =1\,m$ $\Delta L =0.4 \times 10^{-3} m$ $m =1\,kg$ $d =0.4 \times 10^{-3}\,m$ $\frac{ F }{ A }= Y \frac{\Delta L }{ L }$ $Y =\frac{ FL }{ A \Delta L }=\frac{( mg ) \cdot(1)}{\left(\frac{\pi d ^{2}}{4}\right) 0.4 \times 10^{-3}}$ $\Rightarrow \frac{10 \times 4}{\pi\left(0.4 \times 10^{-3}\right)^{2} \times 0.4 \times 10^{-3}}$ $Y =\frac{40}{\pi\left(0.4 \times 10^{-3}\right)^{3}}$ $Y =\frac{40 \times 7}{22 \times 64 \times 10^{-3} \times 10^{-9}}$ $Y =0.199 \times 10^{-12} N / m ^{2}$ $\frac{\Delta Y }{ Y }=\frac{\Delta F }{ F }+\frac{\Delta L }{ L }+\frac{\Delta A }{ A }+\frac{\Delta(\Delta L )}{(\Delta L )}$ $=\frac{0.02}{0.4}+2 \frac{\Delta d }{ d }=\frac{0.2}{4}+2 \times \frac{0.01}{0.4}$ $=\frac{0.1}{2}+\frac{0.1}{2}=0.1$ $\Rightarrow \Delta Y =0.1 \times Y$ $=0.199 \times 10^{11}=1.99 \times 10^{10}$

Question

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

$\left[\right.$ Take $\left.g =10\,m / s ^{2}\right]$

A$25$
B$20$
C$2$
D$8$

JEE MAIN 2022, Diffcult

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