Question 41: Properties of Matter

Question

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 :

Accepted Answer

To determine the percentage error in Young's modulus, we need to first understand the propagation of errors in the given formula.
Given the equation:
$$ \mathrm{Y}=49000 \frac{\mathrm{M}}{\mathrm{l}} \frac{\mathrm{dyne}}{\mathrm{cm}^2} $$
where:
$$M$$ is the mass (with its value given as $$500 \mathrm{~g}$$)
$$l$$ is the extension (with its value given as $$2 \mathrm{~cm}$$)
The errors in the measurements are determined by the smallest scale divisions on the graph paper, which are:
$$5 \mathrm{~g}$$ for the load axis
$$0.02 \mathrm{~cm}$$ for the extension axis
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.
The relative error in $$M$$ is:
$$ \frac{\Delta M}{M} = \frac{5 \mathrm{~g}}{500 \mathrm{~g}} = 0.01 $$
The relative error in $$l$$ is:
$$ \frac{\Delta l}{l} = \frac{0.02 \mathrm{~cm}}{2 \mathrm{~cm}} = 0.01 $$
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 $$l$$:
$$ \frac{\Delta Y}{Y} = \frac{\Delta M}{M} + \frac{\Delta l}{l} = 0.01 + 0.01 = 0.02 $$
To express this as a percentage, we multiply by 100:
$$ 	ext{Percentage error in } Y = 0.02 	imes 100 = 2\% $$
Thus, the percentage error in Young's modulus $$Y$$ is:
Option A: 2%

Answer

2%

Answer

0.02%

Answer

0.5%

Answer

0.2%

Properties of Matter

Choose the correct answer: