Question

you are coasting along on your bike (that has a 0.66 m diameter tire). you notice a twig in your spoke (at 1/2 the radius of the tire) hits the bike frame once per second. what is your speed in m/s?
a 0.66
b 1.03
c 2.07
d 4.14

Explanation:

Step1: Find the radius of the tire

The diameter of the tire \(d = 0.66\space m\), so the radius \(r=\frac{d}{2}=\frac{0.66}{2}= 0.33\space m\). The distance of the twig from the center of the tire \(r_{twig}=\frac{r}{2}=\frac{0.33}{2}=0.165\space m\)

Step2: Determine the angular frequency \(\omega\)

The twig hits the bike frame once per second, so the frequency \(f = 1\space Hz\). Using the formula \(\omega=2\pi f\), we get \(\omega = 2\pi\times1=2\pi\space rad/s\)

Step3: Calculate the linear speed \(v\) of the bike

The linear speed \(v=\omega r\). Since the speed of the bike is related to the rotational motion of the tire (the center of the tire moves with the speed of the bike and for a rolling without - slipping tire, the speed of the center of the tire \(v\) is related to the angular speed \(\omega\) of the tire's rotation). The angular speed of the twig is the same as the angular speed of the tire. Using \(v=\omega r\) (where \(r\) is the radius of the tire), \(v = 2\pi\times0.33\)

$$v=2\times3.14\times0.33=2.07\space m/s$$

Answer:

C. 2.07