Question

A steel wire has length $l=2\text{m}$, radius $r=1\text{mm}$ and Young's modulus of elasticity $Y=2\times {10}^{11}{\text{Nm}}^{-2}.$ A sphere of mass $1\text{kg}$ is attached to one end of the wire and is whirled in a vertical circle with an angular speed of $2\text{rev/s}$. When the sphere is at the lowest point of the vertical circle, the elongation of the wire is nearly
[Assume , $g=10{\text{ms}}^{-2})$ ]

• A. $1030\mu \text{m}$
• B. $2000\mu \text{m}$
• C. $103\mu \text{m}$
• D. $10.3\mu \text{m}$

Answer 1

The correct option is A $1030\mu \text{m}$

At the bottom most point of the verticle circle ,
From the FBD we can say that,
$T-mg=m{\omega }^{2}l$
From the data given in the question ,
$T=1\times 10+1\times (2\times 2\pi {)}^2\times 2\because \omega =2\text{rev/s}=2\times 2\pi \text{rad/s}$
$\Rightarrow T\approx 326\text{N}........\left(1\right)$


From the definition of modulus of elasticity,
Young's modulus of elasticity $Y=\frac{F/A}{\Delta l/l}$ or $Y=\frac{Fl}{A\Delta l}$
Due to tension force the steel wire experiences a change in length
$\Delta l=\frac{Tl}{YA}$
From the data given in the question,
$\Delta l=\frac{326\times 2}{2\times {10}^{11}\times \pi \times {10}^{-6}}[A=\pi r^2]$
$\Delta l=1030\mu \text{m}$
Thus, option (a) is the correct answer.