# In order to determine the Young's Modulus of a wire of radius 0.2cm (measured using a scale of least count =0.001cm)

Question:

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

1. $0.14 \%$

2. $0.9 \%$

3. $9 \%$

4. $1.4 \%$

Correct Option: , 4

Solution:

$\mathrm{Y}=\frac{\text { Stress }}{\text { Strain }}=\frac{\mathrm{FL}}{\mathrm{Al}}=\frac{\mathrm{mg} \cdot \mathrm{L}}{\pi \mathrm{R}^{2} \cdot \ell}$

$\frac{\Delta \mathrm{Y}}{\mathrm{Y}}=\frac{\Delta \mathrm{m}}{\mathrm{m}}+\frac{\Delta \mathrm{L}}{\mathrm{L}}+2 \cdot \frac{\Delta \mathrm{R}}{\mathrm{R}}+\frac{\Delta \ell}{\ell}$

$\frac{\Delta \mathrm{Y}}{\mathrm{Y}} \times 100=100\left[\frac{1}{1000}+\frac{1}{1000}+2\left(\frac{0.001}{0.2}\right)+\frac{0.001}{0.5}\right]$

$=\frac{1}{10}+\frac{1}{10}+1+\frac{1}{5}=\frac{14}{10}=1.4 \%$