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.
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 = r ω, ω = \frac{v}{r} to calculate the angular velocity.
To calculate the angular velocity, we will use the following relationship:
ω = \frac{v}{r} (6.10)
Substituting the knowns,
w = \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
w = (15.0 m/s) / (1.20 m) = 12.5 rad/s. (6.12)