Chapter 10, Problem 27A

Dan and Sue cycle at the same speed. The tires on Dan’s bike are larger in diameter than those on Sue’s bike. Which wheels, if either, have the greater rotational speed?

To determine

The wheel which have greater rotational speed if D and S cycle at the same speed.

Answer to Problem 27A

  ${\omega }_{\text{Sue}}>{\omega }_{\text{Due}}$

Explanation of Solution

Introduction:

A linear speed $\left(v\right)$ is written in terms of the rotational speed $\left(\omega \right)$ as,

  $v=r\omega$

Where, $r=$ distance from the axis

Since the diameter of the tires of D’s bike is greater than that of the S’s bike, thus

  $r_D>r_S$

Now, both the D and S cycle at same speed, thus the rotational speed of the tires of the D’s and S’s cycles will be,

  $\begin{array}{l}\omega \propto \frac{1}{r}\\ \Rightarrow {\omega }_S>{\omega }_D\end{array}$

Conclusion:

Hence, the S’s cycle will have the greater rotational speed.