
A rolling wheel has both linear and angular velocity. If the wheel is not slipping, the two velocities are related by geometry, because the circumference of a circle is 2π times the radius r. A wheel rotates through an angle of 2π radians (a full rotation) as it moves forward by a distance of 2πr (one circumference).


The linear velocity is the circumference divided by the time it takes to make one turn, i.e., v = 2πr/t. For one rotation, the quantity 2π/t is the angular velocity ω. The linear velocity is therefore equal to ω multiplied by the radius r. Linear and angular velocity are related to each other by the equation

(7.2)  $$v=\omega r$$
 v  =  linear velocity (m/s)  ω  =  angular velocity (rad/s)  r  =  radius (m) 
 Linear velocity from angular velocity 

The radius r that appears in equation (7.2) means that larger wheels have a higher linear velocity than smaller wheels for a given angular velocity. Early bicycles had very large wheels because the larger wheels would create a higher linear velocity. Unfortunately, they were very difficult to ride and quite unstable! Modern bicycles use gears and chains so that the pedals can turn at a different angular velocity from the wheels.


A car has wheels with a radius of 30 cm. What is the angular velocity of the wheels in both radians per second and revolutions per minute when the car is moving at 30 m/s (67 mph)?
Asked: 
angular velocity ω calculated in two units: rad/s and rpm 
Given: 
radius r = 0.3 m; velocity v = 30 m/s 
Relationships: 
linear velocity v = ωr; one revolution is 2π = 6.28 radians 
Solution: 
In rad/s, ω = v/r = (30 m/s)/(0.3 m) = 100 rad/s. In rpm, ω = (100 rad/s)×(60 s/min)×(1 revolution/6.28 rad) = 955 rpm. 
Answer: 
ω = 100 rad/s = 955 rpm 

A wheel is rolling at a linear velocity of 25 m/s. If the radius of the wheel is decreased, and the linear velocity remains the same, what does this indicate about the angular velocity of the smaller wheel?
 It remains unchanged.
 It increases.
 It decreases.
 There is not enough information to answer.

The answer is b. For a smaller wheel to move at the same linear velocity of a larger wheel, it must rotate faster. It will therefore have a higher angular velocity.
