Question

In order to determine the Young`s Modulus of a wire of radius $0.2\text{cm}$ (measured using a scale of least count $=0.001\text{cm}$) and length $1\text{m}$ (measured using a scale of least count $=1\text{mm}$), a weight of mass $1\text{kg}$ (measured using a scale of least count $=1\text{g}$) was hanged to get the elongation of $0.5\text{cm}$ (measured using a scale of least count $0.001\text{cm}$). What will be the fractional error in the value of Young`s Modulus determined by this experiment.

• A. $9$%
• B. $1.4$%
• C. $0.9$%
• D. $0.14$%

Answer 1

The correct option is B $1.4$%

In this case, Young's modulus will be given as:
$$\gamma =\frac{mgL}{A.l}$$
$\frac{\Delta \gamma }{\gamma }=\left(\frac{\Delta m}{m}\right)+\left(\frac{\Delta g}{g}\right)+\left(\frac{\Delta A}{A}\right)+\left(\frac{\Delta l}{l}\right)+\left(\frac{\Delta L}{L}\right)$

$\frac{\Delta \gamma }{\gamma }=\left(\frac{1\text{g}}{1\text{kg}}\right)+0+2\left(\frac{\Delta r}{r}\right)+\left(\frac{\Delta l}{l}\right)+\left(\frac{\Delta L}{L}\right)$

$=\left(\frac{1\text{g}}{1\text{kg}}\right)+2\left(\frac{0.001\text{cm}}{0.2\text{cm}}\right)+\left(\frac{0.001\text{cm}}{0.5\text{cm}}\right)+\left(\frac{0.001\text{m}}{1\text{m}}\right)$

$=\left(\frac{1}{1000}\right)+2\left(\frac{1\times 10}{2\times {10}^3}\right)+(\frac{1}{5}\times \frac{10}{{10}^3})+\left(\frac{1}{{10}^3}\right)$

$=\frac{1}{1000}+\frac{1}{100}+\frac{2}{{10}^3}+\frac{1}{{10}^3}$

$=\frac{1+10+2+1}{1000}=\frac{14}{1000}$
$=1.4$%