Young's modulus is determined by the equation given by $$\mathrm{Y}=49000 \frac{\mathrm{m}}{\mathrm{l}} \frac{\mathrm{dyne}}{\mathrm{cm}^2}$$ where $M$ is the mass and $l$ is the extension of wire used in the experiment. Now error in Young modules $(Y)$ is estimated by taking data from $M-l$ plot in graph paper. The smallest scale divisions are $5 \mathrm{~g}$ and $0.02 \mathrm{~cm}$ along load axis and extension axis respectively. If the value of $M$ and $l$ are $500 \mathrm{~g}$ and $2 \mathrm{~cm}$ respectively then percentage error of $Y$ is :

<p>To determine the percentage error in Young's modulus, we need to first understand the propagation of errors in the given formula.</p> <p>Given the equation:</p> <p>$$ \mathrm{Y}=49000 \frac{\mathrm{M}}{\mathrm{l}} \frac{\mathrm{dyne}}{\mathrm{cm}^2} $$</p> <p>where:</p> <ul> <li>$M$ is the mass (with its value given as $500 \mathrm{~g}$)</li> <li>$l$ is the extension (with its value given as $2 \mathrm{~cm}$)</li> </ul> <p>The errors in the measurements are determined by the smallest scale divisions on the graph paper, which are:</p> <ul> <li>$5 \mathrm{~g}$ for the load axis</li> <li>$0.02 \mathrm{~cm}$ for the extension axis</li> </ul> <p>To find the percentage error in Young's modulus ($Y$), we need to compute the relative errors in the measurements $M$ and $l$, and then propagate these errors through the given formula.</p> <p>The relative error in $M$ is:</p> <p>$\frac{\Delta M}{M} = \frac{5 \mathrm{~g}}{500 \mathrm{~g}} = 0.01$</p> <p>The relative error in $l$ is:</p> <p>$\frac{\Delta l}{l} = \frac{0.02 \mathrm{~cm}}{2 \mathrm{~cm}} = 0.01$</p> <p>Since $Y$ is proportional to $M$ and inversely proportional to $l$, the overall percentage error in $Y$ is the sum of the percentage errors in $M$ and $l$:</p> <p>$$ \frac{\Delta Y}{Y} = \frac{\Delta M}{M} + \frac{\Delta l}{l} = 0.01 + 0.01 = 0.02 $$</p> <p>To express this as a percentage, we multiply by 100:</p> <p>$\text{Percentage error in } Y = 0.02 \times 100 = 2\%$</p> <p>Thus, the percentage error in Young's modulus $Y$ is:</p> <p><strong>Option A: 2%</strong></p>