Question

A wheel has eight equally spaced spokes and a radius of 30 cm. It is mounted on a fixed axle and is spinning at $2.5{\displaystyle \frac{rad}{\sec }}$. 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.
(a) What minimum speed must the arrow have ?
(b) Does it matter where between the axle and rim of the wheel you aim? If so, what is the best location?

Answer 1

Hint: In order to solve this question, you need to find out the angle between the spokes and also you need to find the time taken by one spoke to reach the position of the adjacent spoke. Along with this we also need to find what is the best location between the axle and rim of the wheel you should aim.

Formula used:
Formula required to solve this question is
$t\text{ = }{\displaystyle \frac{\theta }{\omega }}$
Here, $t$ refers to the time taken by one spoke to reach the position of adjacent spoke, $\theta$ refers to the angle between the adjacent spoke and $\omega$ refers to the angular velocity.

Complete step by step answer:

Angular velocity of wheel = $2.5{\displaystyle \frac{rad}{\sec }}$
Length of arrow = 20 cm
For arrow to pass through the spinning rim, it has to pass between any two spokes of wheel
The angle between the spokes = ${\displaystyle \frac{2\pi }{8}}$
First of all, we will convert the angle between the spokes in degrees and then we will convert it into radian (rad).
Angle in degree
${\displaystyle \frac{2\pi }{8}}\text{ = }{45}^{\circ }$
Now, angle in radium (rad)
${45}^{\circ }\text{ = }{\displaystyle \frac{\pi }{4}}\text{ rad}$

(a) Now we will find the time taken by one spoke to reach the position of adjacent spoke.
$t\text{ = }{\displaystyle \frac{\theta }{\omega }}$
$\Rightarrow t\text{ = }{\displaystyle \frac{{\displaystyle \frac{\pi }{4}}}{2.5\times 2\pi }}$
$\Rightarrow t\text{ = }{\displaystyle \frac{1}{20}}\sec$
Therefore to pass the spokes of the wheel, it should take less than ${\displaystyle \frac{1}{20}}\sec$ to pass the wheel. Answer to the first question which states that what is the minimum speed is equal to
$\text{Speed}={\displaystyle \frac{{\displaystyle \frac{20}{100}}}{{\displaystyle \frac{1}{20}}}}\text{ m/sec}$
$\therefore \text{Speed}=4\text{ m/sec}$
The minimum speed is equal to $4\text{ m/sec}$.

(b) It does matter where to in the error to shoot with the minimum speed without hitting spike we need to aim to a spike to acquire maximum time to pass through wheel

Note: Many of the people make mistakes by not making diagrams in these types of but it is important as these types of questions become easier to solve with the help of the diagram as many activities are holding together so it is important to visualize it.Along with this it is very common to forget to convert the angle into radium so it should be taken care.