Use Linear and Angular Speed to Describe Motion on a Circular Path
In addition to finding the area of a sector, we can use angles to describe the speed of a moving object. An object traveling in a circular path has two types of speed. Linear speed is speed along a straight path and can be determined by the distance it moves along (its displacement) in a given time interval. For instance, if a wheel with radius 5 inches rotates once a second, a point on the edge of the wheel moves a distance equal to the circumference, or [latex]10\pi[/latex] inches, every second. So the linear speed of the point is [latex]10\pi[/latex] in./s. The equation for linear speed is as follows where [latex]v[/latex] is linear speed, [latex]s[/latex] is displacement, and [latex]t[/latex]
is time.
Angular speed results from circular motion and can be determined by the angle through which a point rotates in a given time interval. In other words, angular speed is angular rotation per unit time. So, for instance, if a gear makes a full rotation every 4 seconds, we can calculate its angular speed as [latex]\frac{360\text{ degrees}}{4\text{ seconds}}=[/latex] 90 degrees per second. Angular speed can be given in radians per second, rotations per minute, or degrees per hour for example. The equation for angular speed is as follows, where [latex]\omega[/latex] (read as omega) is angular speed, [latex]\theta[/latex] is the angle traversed, and [latex]t[/latex] is time.
angular speed
As a point moves along a circle of radius [latex]r[/latex], its angular speed, [latex]\omega[/latex], is the angular rotation [latex]\theta[/latex] per unit time, [latex]t[/latex].
[latex]\omega =\frac{\theta }{t}[/latex]
Combining the definition of angular speed with the arc length equation, [latex]s=r\theta[/latex], we can find a relationship between angular and linear speeds. The angular speed equation can be solved for [latex]\theta[/latex], giving [latex]\theta =\omega t[/latex]. Substituting this into the arc length equation gives:
Substituting this into the linear speed equation gives:
linear speed
The linear speed. [latex]v[/latex], of the point can be found as the distance traveled, arc length [latex]s[/latex], per unit time, [latex]t[/latex].
[latex]v=\frac{s}{t}[/latex]
When the angular speed is measured in radians per unit time, linear speed and angular speed are related by the equation
[latex]v=r\omega[/latex]
This equation states that the angular speed in radians, [latex]\omega[/latex], representing the amount of rotation occurring in a unit of time, can be multiplied by the radius [latex]r[/latex] to calculate the total arc length traveled in a unit of time, which is the definition of linear speed.
- If necessary, convert the angle measure to radians.
- Divide the angle in radians by the number of time units elapsed: [latex]\omega =\frac{\theta }{t}[/latex].
- The resulting speed will be in radians per time unit.
- Convert the total rotation to radians if necessary.
- Divide the total rotation in radians by the elapsed time to find the angular speed: apply [latex]\omega =\frac{\theta }{t}[/latex].
- Multiply the angular speed by the length of the radius to find the linear speed, expressed in terms of the length unit used for the radius and the time unit used for the elapsed time: apply [latex]v=r\mathrm{\omega}[/latex].Example 11: Finding a Linear Speed