A wire of cross-sectional area A, modulus of elasticity and length 2 m is stretched between two vertical rigid supports. When a mass of 2 kg is suspended at the middle, it sags lower from its original position, making an angle radian on the points of s

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Solution

Q.
A wire of cross-sectional area A, modulus of elasticity $2 \times 10^{11}$ ${Nm}^{- 2}$ and length 2 m is stretched between two vertical rigid supports. When a mass of 2 kg is suspended at the middle, it sags lower from its original position, making an angle $θ = \frac{1}{100}$ radian on the points of support. The value of A is _____ $\times 10^{- 4}$ $m^{2}$ (consider $x < < L$ ). [2024]

Ans.
(1)

$\tan θ = \frac{x}{L} \approx θ, As θ is small$

$2 T \sin θ = 20 \Rightarrow T θ = 10$

$T \cdot \frac{1}{100} = 10 \Rightarrow T = 1000 N$

Change in length, $Δ L = 2 \sqrt{x^{2} + L^{2}} - 2 L$

$\Rightarrow Δ L = 2 L [1 + \frac{x^{2}}{2 L^{2}} - 1] = \frac{x^{2}}{L}$

$Stress = Y strain \Rightarrow \frac{T}{A} = Y \frac{Δ L}{(2 L)}$

$\frac{T}{A} = Y \cdot \frac{x^{2}}{L (2 L)} = \frac{Y}{2} \cdot (\frac{x^{2}}{L^{2}}) = \frac{Y}{2} \cdot θ^{2}$

$\frac{1000}{A} = \frac{Y θ^{2}}{2} = \frac{2 \times 10^{11}}{2} \times \frac{1}{100 \times 100}$

$\Rightarrow A = 1 \times 10^{- 4} m^{2}$

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