Question

The wheel in figure has eight equally spaced spokes and a radius of $30cm$ It is mounted on a fixed axle and is spinning at $2.5rev/s.$ You want to shoot a $20-cm$-long arrow parallel to this axle and through the wheel without hitting any of the spokes. Assume that the arrow and the spokes are very thin. What minimum speed must the arrow have?

• A. $4.0m/s$
• B. $3.50m/s$
• C. $2.5m/s$
• D. $5.0m/s$

Answer 1

The correct option is A $4.0m/s$

Step 1:Find time taken by the spoke to reach the adjacent spoke.
Given, angular velocity of wheel,
$\omega =2.5rev/s$
$\omega =2.5\left(2\pi \right)rad/s$
For arrow to pass through the spinning rim, It has to pass between any two spokes of the wheel.
The angle between spokes,
$\theta =\frac{2\pi }{8}=\frac{\pi }{4}rad$
Time taken by spoke to reach to the position of adjacent spoke,
$t=\frac{\theta }{\omega }=\frac{\pi /4}{2.5\left(2\pi \right)}rad=\frac{1}{20}sec$

Step 2:Find the minimum speed of the arrow.
Given, Length of arrow $=20cm=0.2m$ for arrow to pass through the spokes of wheel, it should take $<\frac{1}{20}sec$ to pass through the wheel
Minimum speed of the arrow,
$v=\frac{l}{t}=\frac{0.2}{\frac{1}{20}}$
$v=4m/s$
Final answer:$\left(d\right)$