Question

The extension in a uniform rod of length $l_1$, mass $m$, cross-section radius $r$ and young's modulus $Y$ when it is suspended at one of its end is : (Consider area of cross section remains same)

• A. $\frac{mgl}{\pi r^{2}Y}$
• B. $\frac{mgl}{2\pi r^{2}Y}$
• C. $\frac{2mgl}{\pi r^{2}Y}$
• D. $\frac{mgl}{4\pi r^{2}Y}$

Answer 1

The correct option is B $\frac{mgl}{2\pi r^{2}Y}$

Tension at point $P$ in the rod is $T=\left[\frac{mg}{l}\right]x$
$\left[\frac{mg}{l}\right]=$ Tension per unit length [uniform-rod]

From Young's modulus formula will be
$(Y=\frac{F/A}{\Delta l/l})\left[\Delta l\right]=\frac{Fl}{AY}$
Extension in the element $dx$ due to tension at $P$
$dl=\frac{Tdx}{AY}$
Put value of $T$
$dl=\left[\frac{mg}{AYl}\right]xdx$
Total extension in the rod
$\Delta l=\underset{0}{\overset{l}{\int }}\left[\frac{mg}{AYl}\right]xdx\Delta l=\frac{mgl}{2AY}=\frac{mgl}{2\pi r^{2}Y}$
Hence, option (B) is correct,