Question 225: Properties of Matter

Question

Two persons pull a wire towards themselves. Each person exerts a force of $$200 \mathrm{~N}$$ on the wire. Young's modulus of the material of wire is $$1 	imes 10^{11} \mathrm{~N} \mathrm{~m}^{-2}$$. Original length of the wire is $$2 \mathrm{~m}$$ and the area of cross section is $$2 \mathrm{~cm}^2$$. The wire will extend in length by _________ $$\mu \mathrm{m}$$.

Accepted Answer

To determine the extension in the length of the wire, we can use the formula derived from Young's modulus:
$$ 	ext{Young's Modulus (Y)} = \frac{	ext{Stress}}{	ext{Strain}} $$
Where:
Stress ($$\sigma$$) is given by:
$$ \sigma = \frac{F}{A} $$
and Strain ($$\epsilon$$) is:
$$ \epsilon = \frac{\Delta L}{L} $$
Here,
$$F$$ is the force exerted,
$$A$$ is the cross-sectional area,
$$\Delta L$$ is the change in length,
$$L$$ is the original length.
We are given:
$$F = 200 \mathrm{~N}$$ (each person pulls with this force, but the total force in the wire should be considered as the tension experienced by one side, so it still remains 200 N),
$$A = 2 \mathrm{~cm}^2 = 2 	imes 10^{-4} \mathrm{~m}^2$$,
$$L = 2 \mathrm{~m}$$,
$$Y = 1 	imes 10^{11} \mathrm{~N} \mathrm{~m}^{-2}$$.
First, calculate the stress ($$\sigma$$):
$$ \sigma = \frac{200 \mathrm{~N}}{2 	imes 10^{-4} \mathrm{~m}^2} = 10^6 \mathrm{~N} \mathrm{~m}^{-2} $$
Using YoungÃ¢ÂÂs modulus formula:
$$ Y = \frac{	ext{Stress}}{	ext{Strain}} \implies 	ext{Strain} = \frac{	ext{Stress}}{Y} $$
Thus, the strain ($$\epsilon$$) is:
$$ \epsilon = \frac{10^6 \mathrm{~N} \mathrm{~m}^{-2}}{1 	imes 10^{11} \mathrm{~N} \mathrm{~m}^{-2}} = 10^{-5} $$
Now, relate the strain to the change in length:
$$ \epsilon = \frac{\Delta L}{L} \implies \Delta L = \epsilon 	imes L = 10^{-5} 	imes 2 \mathrm{~m} = 2 	imes 10^{-5} \mathrm{~m} $$
Convert this change in length to micrometers:
$$ 1 \mathrm{~m} = 10^6 \mu \mathrm{m} $$
$$ 2 	imes 10^{-5} \mathrm{~m} = 2 	imes 10^{-5} 	imes 10^6 \mu \mathrm{m} = 20 \mu \mathrm{m} $$
Therefore, the wire will extend in length by $$20 \mu \mathrm{m}$$.

Properties of Matter

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