Question 228: Properties of Matter

Question

A wire of cross sectional area A, modulus of elasticity $$2 	imes 10^{11} \mathrm{~Nm}^{-2}$$ and length $$2 \mathrm{~m}$$ is stretched between two vertical rigid supports. When a mass of $$2 \mathrm{~kg}$$ is suspended at the middle it sags lower from its original position making angle $$	heta=\frac{1}{100}$$ radian on the points of support. The value of A is ________ $$	imes 10^{-4} \mathrm{~m}^2$$ (consider $$x<<\mathrm{L}$$ ).
(given : $$\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2$$)

Accepted Answer

$$\begin{aligned}
\Delta I_{	ext {spring }} & =\left(\sqrt{L^2+x^2}ight)-L \
& =\frac{x^2}{2 L} \
T=k \Delta I & =\frac{k n^2}{2 L}
\end{aligned}$$
$$\begin{array}{rl}
&\Rightarrow 2 T 	heta=m g \
& 2\left(\frac{Y A}{l}ight) \frac{x^2}{2 l} 	heta=m g \
& A =\frac{m g}{	heta^3 Y} \
& =\frac{20}{10^{-6} 	imes 2 	imes 10^{11}} \
& =10 	imes 10^{-5} \
& =1 	imes 10^{-4}
\end{array}$$

Properties of Matter

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