Question

A wheel has six equally spaced thin spokes of length $20\text{cm}$. It is rotating about an axis $\text{xx'}$ which is perpendicular to its plane and passing through its center with $5\text{rev/sec}$. What should be the minimum velocity of an arrow of length $30\text{cm}$ to pass the wheel without hitting any spoke. (Arrow is projected parallel to the axis).

• A. $9\text{m/s}$
• B. $4\text{m/s}$
• C. $5\text{m/s}$
• D. $4.5\text{m/s}$

Answer 1

The correct option is A $9\text{m/s}$

Angle between successive spokes is,
$\Delta \theta =\frac{2\pi }{6}=\frac{\pi }{3}$
Angular speed of wheel $\omega =5\text{rps}=(5\times 2\pi )\text{rad/s}$
$\therefore \omega =10\pi \text{rad/s}$
Time taken by one spoke to pass is,
$t=\frac{\Delta \theta }{\omega }=\frac{\left(\frac{\pi }{3}\right)}{10\pi }$
$\Rightarrow t=\frac{1}{30}s$
During this time interval, the arrow must pass through.
$x=v\times t$
$\Rightarrow x=\text{length of arrow = 30 cm = 0.3 m}$
$\Rightarrow v=\frac{0.30}{\left(1/30\right)}=9\text{m/s}$
So, minimum velocity of arrow will be $9\text{m/s}$
Thus, option $\left(a\right)$ is the correct answer.