How Fast Does a Car Tire Spin?

Calculate the angular velocity of a 0.300 m radius car tire when the car travels at $15.0 m/s$ (about $54 km/h$ ). See Figure 6.5.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Strategy
Because the linear speed of the tire rim is the same as the speed of the car, we have $v = 15.0 m/s .$ The radius of the tire is given to be $r = 0.300 m .$ Knowing $v$ and $r$ , we can use the second relationship in $v = rω, ω = \frac{v}{r}$ to calculate the angular velocity.
Solution
To calculate the angular velocity, we will use the following relationship:
$ω = \frac{v}{r} .$
6.10

Substituting the knowns,

$ω = \frac{15.0 m/s}{0.300 m} = 50 . 0 rad/s.$
6.11
Discussion
When we cancel units in the above calculation, we get 50.0/s. But the angular velocity must have units of rad/s. Because radians are actually unitless (radians are defined as a ratio of distance), we can simply insert them into the answer for the angular velocity. Also note that if an earth mover with much larger tires, say 1.20 m in radius, were moving at the same speed of 15.0 m/s, its tires would rotate more slowly. They would have an angular velocity
$ω = (15.0 m/s) / (1.20 m) = 12 . 5 rad/s.$
6.12

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