Question

The wheel in Fig. $\mathbf{10}\mathbf{-}\mathbf{30}\mathbf{}$ Has eight equally spaced spokes and a radius of $\mathbf{30}\mathbf{}\mathbf{cm}$. It is mounted on a fixed axle and is spinning at $\mathbf{2}\mathbf{.}\mathbf{5}\mathbf{}\mathbf{rev}\mathbf{/}\mathbf{s}$. You want to shoot a $\mathbf{20}\mathbf{}\mathbf{cm}$ 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?

Short answer

1. Minimum speed of the arrow is$4.0\frac{\text{m}}{\text{s}}$ .
2. It doesn’t matter where the arrow is aimed between the axle and rim of the wheel.

Step-by-step solution

Listing the given quantities

Radius of wheel is,$\mathrm{R}=30\text{cm}=0.30\text{m}.$

Angular velocity of wheel is,$\mathrm{\omega }=2.5\frac{\text{rev}}{\text{s}}$

Length of an arrow is,$\mathrm{L}=20\text{cm}=0.20\text{m}.$

Determine the concept of velocity and number of revolutions

Find the time in which the arrow should be passed through the wheel using its relation with the number of revolutions and angular velocity of the wheel. Using time, find the minimum speed of the arrow.

Formula:

$\begin{array}{l}\mathrm{t}=\frac{\text{No}.\text{ofrevolutions}}{\mathrm{\omega }}\\ \mathrm{v}=\frac{\mathrm{x}}{\mathrm{t}}\end{array}$

(a) Calculate the minimum speed of arrow

As the arrow is shot, it should go through the space between the spokes. If it is aimed at one of the spokes, it should be aimed in such way that it will pass through the wheel in time equal to

$\mathrm{t}=\frac{\mathrm{No}.\mathrm{of}\mathrm{revolutions}}{\mathrm{\omega }}$

Since the wheel has 8 spokes substitute the values and solve as:

$\begin{array}{c}\mathrm{t}=\frac{\frac{1}{8}\text{\hspace{0.17em}rev}}{2.5\frac{\text{rev}}{\text{s}}}\\ =0.05\text{\hspace{0.17em}s}\end{array}$

Minimum speed of the arrow is obtained as follows:

$\begin{array}{c}{\mathrm{v}}_{\min }=\frac{\mathrm{L}}{\mathrm{t}}\\ =\frac{0.20}{0.05}\\ =4.0\frac{\text{m}}{\text{s}}\end{array}$

Therefore, minimum speed of the arrow is $4.0\frac{\text{m}}{\text{s}}$.

(b) Determine the location of rim and axle

Since minimum speed of the arrow is independent of the position of the arrow, it doesn’t matter where between the axle and rim of the wheel the arrow is aimed.