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Combining the definition of angular speed with the arc length equation, we can find a relationship between angular and linear speeds. The angular speed equation can be solved for giving Substituting this into the arc length equation gives:
Substituting this into the linear speed equation gives:
As a point moves along a circle of radius its angular speed , is the angular rotation per unit time,
The linear speed , of the point can be found as the distance traveled, arc length per unit time,
When the angular speed is measured in radians per unit time, linear speed and angular speed are related by the equation
This equation states that the angular speed in radians, representing the amount of rotation occurring in a unit of time, can be multiplied by the radius to calculate the total arc length traveled in a unit of time, which is the definition of linear speed.
Given the amount of angle rotation and the time elapsed, calculate the angular speed.
A water wheel, shown in [link] , completes 1 rotation every 5 seconds. Find the angular speed in radians per second.
The wheel completes 1 rotation, or passes through an angle of radians in 5 seconds, so the angular speed would be radians per second.
An old vinyl record is played on a turntable rotating clockwise at a rate of 45 rotations per minute. Find the angular speed in radians per second.
rad/s
Given the radius of a circle, an angle of rotation, and a length of elapsed time, determine the linear speed.
A bicycle has wheels 28 inches in diameter. A tachometer determines the wheels are rotating at 180 RPM (revolutions per minute). Find the speed the bicycle is traveling down the road.
Here, we have an angular speed and need to find the corresponding linear speed, since the linear speed of the outside of the tires is the speed at which the bicycle travels down the road.
We begin by converting from rotations per minute to radians per minute. It can be helpful to utilize the units to make this conversion:
Using the formula from above along with the radius of the wheels, we can find the linear speed:
Remember that radians are a unitless measure, so it is not necessary to include them.
Finally, we may wish to convert this linear speed into a more familiar measurement, like miles per hour.
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