First slide

Stress and strain

Question

The length of a metal wire is l₁ when the tension in it is T₁ and is l₂ when the tension is T₂. The natural length of the wire is

Moderate

A

$\frac{I_{1} + l_{2}}{2}$
B

$\sqrt{I_{1} I_{2}}$
C

$\frac{I_{1} T_{2} - I_{2} T_{1}}{T_{2} - T_{1}}$
D

$\frac{I_{1} T_{2} + I_{2} T_{1}}{T_{2} + T_{1}}$

Solution

As, Young's modulus of elasticity, $Y = \frac{Stress}{Strain}$
$\Rightarrow Y = \frac{T_{1} \times L}{A \times (I_{1} - L)}$ .... (i)
where, L is the original length of the wire and also
$Y = \frac{T_{2} \times L}{A \times (I_{2} - L)} \dots . (ii)$
From Eqs. (i) and (ii), we get
$\frac{T_{1} L}{A (I_{1} - L)} = \frac{T_{2} L}{A (I_{2} - L)} \Rightarrow L = \frac{T_{2} I_{1} - T_{1} I_{2}}{T_{2} - T_{1}}$

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