First slide

Stress and strain

Question

A wire of radius r stretched without tension along a straight line is lightly fixed at A and B as shown in fig. What is the tension in the wire when it is
pulled into the shape A C B ? Assume Young's modulus of the material of the wire to be Y and d < l

Difficult

A

$\frac{{Yd}^{2}}{2 {πr}^{2} l^{2}}$
B

$\frac{{Yπr}^{2} d^{2}}{2 l^{2}}$
C

$\frac{{Yπr}^{2}}{2 l^{2} d^{2}}$
D

$\frac{{Yπr}^{2} l^{2}}{2 d^{2}}$

Solution

From figure, the change in length
$\begin{array}{l} = AC + CB - AB \\ = 2 \sqrt{(l^{2} + d^{2})} - 2 l [∵ AC = CB = \sqrt{(l^{2} + d^{2})}] \end{array}$
Let T be the tension in the wire. Then
Longitudinal stress $= T / ({πr}^{2})$
and Longitudinal strain $= \frac{Change in length}{Original length}$
$\begin{array}{r} = \frac{2 \sqrt{(l^{2} + d^{2})} - 2 l}{2 l} \\ ∴ Y = \frac{T / {πr}^{2}}{[2 \sqrt{(l^{2} + d^{2})} - 2 l] / 2 l}, \end{array}$
$= \frac{Tl}{{πr}^{2} [\sqrt{(l^{2} + d^{2})} - l]}$
or $T = \frac{{Yπr}^{2} [\sqrt{(l^{2} + d^{2})} - l]}{l}$
$\begin{array}{l} = {Yπr}^{2} [(1 + \frac{d^{2}}{2 l^{2}} \dots) - 1] \\ = \frac{{Yπr}^{2} d^{2}}{2 l^{2}} \end{array}$

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