Question

A rod of uniform cross sectional area $A$ and length $L$ has a weight $W$. It is suspended vertically from a fixed support vertically from a fixed support. If Young's modulus for rod is $Y$, then elongation produced in rod is

• A. $\frac{WL}{YA}$
• B. $\frac{WL}{2YA}$
• C. $\frac{WL}{4YA}$
• D. $\frac{3WL}{4YA}$

Answer 1

Answer: (B)

$\frac{WL}{2YA}$

$\text{Each cross-section of rod is experiencing different amoun of stress. Hence,}$

$\begin{array}{c}\text{Let us consider a small element of differential length}dx\\ \text{at a distance}x\text{from free end of the rod.The stress}\end{array}$

$\begin{array}{c}\text{at the position of this element is produced by the weight}\\ \text{of the rod of length}x\text{lying below it i.e}\left(W/L\right)x\end{array}$

$\sigma =\frac{\left(W/L\right)x}{A}=\frac{Wx}{AL}$

$\text{elongaluon}d\delta \text{is produced}\text{in differential}$$\text{element}dx=\frac{\sigma dx}{Y}=\frac{Wwxdx}{AYL}$

$\begin{array}{cc}\delta & =\int d\delta \\ & ={\int }_0^L\frac{Wxdx}{AYL}=\frac{W}{YAL}{\left[\frac{x^2}{2}\right]}_0^L=\frac{\left(W\right)L}{2AY}\end{array}$