In Section 1.3, we learned that the radian measure of an angle was equal to the length of the arc on the unit circle associated with that angle. So an arc of length 1 on the unit circle subtends an angle of 1 radian. There will be times when it will also be useful to know the length of arcs on other circles that subtend the same angle.
Figure \(\PageIndex\): Arcs subtended by an angle of 1 radian.
In Figure \(\PageIndex\), the inner circle has a radius of 1, the outer circle has a radius of \(r\), and the angle shown has a measure of \(\theta\) radians. So the arc length on the unit circle subtended by the angle is \(\theta\), and we have used s to represent the arc length on the circle of radius \(r\) subtended by the angle.
Recall that the circumference of a circle of radius \(r\) is \(2\pi r\) while the circumference of the circle of radius 1 is \(2\pi\). Therefore, the ratio of an arc length \(s\) on the circle of radius \(r\) that subtends an angle of \(\theta\) radians to the corresponding arc on the unit circle is \(\dfrac<2\pi r> <2\pi>= r\). So it follows that
On a circle of radius \(r\), the arc length s intercepted by a central angle with radian measure is
It is important to remember that to calculate arc length , we must measure the central angle in radians.
(It is not clear why the letter \(s\) is usually used to represent arc length. One explanation is that the arc “subtends” an angle.)
Using the circles in the beginning activity for this section:
Degree measure is familiar and convenient, so why do we introduce the unit of radian? This is a good question, but one with a subtle answer. As we just saw, the length \(s\) of an arc on a circle of radius \(r\) subtended by angle of \(\theta\) radians is given by \(s = r\theta\), so \(\theta = \dfrac\). As a result, a radian is a ratio of two lengths (the quotient of the length of an arc by a radius of a circle), which makes a radian a dimensionless quantity. Thus, a measurement in radians can just be thought of as a real number. This is convenient for dealing with arc length (and angular velocity as we will soon see), and it will also be useful when we study periodic phenomena in Chapter 2. For this reason radian measure is universally used in mathematics, physics, and engineering as opposed to degrees, because when we use degree measure we al- ways have to take the degree dimension into account in computations. This means that radian measure is actually more natural from a mathematical standpoint than degree measure.
The connection between an arc on a circle and the angle it subtends measured in radians allows us to define quantities related to motion on a circle. Objects traveling along circular paths exhibit two types of velocity: linear and angular velocity. Think of spinning on a merry-go-round. If you drop a pebble off the edge of a moving merry-go-round, the pebble will not drop straight down. Instead, it will continue to move forward with the velocity the merry-go-round had the moment the pebble was released. This is the linear velocity of the pebble. The linear velocity measures how the arc length changes over time.
Consider a point \(P\) moving at a constant velocity along the circumference of a circle of radius \(r\). This is called uniform circular motion. Suppose that P moves a distance of s units in time \(t\). The linear velocity v of the point \(P\) is the distance it traveled divided by the time elapsed. That is, \(v = \dfrac\). The distance s is the arc length and we know that \(s = r\theta\).
Definition: linear velocity
Consider a point \(P\) moving at a constant velocity along the circumference of a circle of radius \(r\). The linear velocity \(v\) of the point \(P\) is given by
where \(\theta\), measured in radians, is the central angle subtended by the arc of length \(s\).
Another way to measure how fast an object is moving at a constant speed on a circular path is called angular velocity. Whereas the linear velocity measures how the arc length changes over time, the angular velocity is a measure of how fast the central angle is changing over time.
Definition: angular velocity
Consider a point P moving with constant velocity along the circumference of a circle of radius r on an arc that corresponds to a central angle of measure \(\theta\) (in radians). The angular velocity \(\omega\) of the point is the radian measure of the angle \(\theta\) divided by the time t it takes to sweep out this angle. That is
The symbol \(\omega\) is the lower case Greek letter “omega.” Also, notice that the angular velocity does not depend on the radius r.
This is a somewhat specialized definition of angular velocity that is slightly different than a common term used to describe how fast a point is revolving around a circle. This term is revolutions per minute or rpm. Sometimes the unit revolutions per second is used. A better way to represent revolutions per minute is to use the “unit fraction” \(\dfrac\). Since 1 revolution is \(2\pi\) radians, we see that if an object min is moving at x revolutions per minute, then
Suppose a circular disk is rotating at a rate of 40 revolutions per minute. We wish to determine the linear velocity v (in feet per second) of a point that is 3 feet from the center of the disk.
2. The result from part (a) gives
3. We now convert feet per minute to feet per second.
Notice that in Exercise 1.18, once we determined the angular velocity, we were able to determine the linear velocity. What we did in this specific case we can do in general. There is a simple formula that directly relates linear velocity to angular velocity. Our formula for linear velocity is \(v =\dfrac \dfrac\). Notice that we can write this is \(v = r\dfrac\). That is, \(v = r\omega\)
Consider a point \(P\) moving with constant (linear) velocity \(v\) along the circumference of a circle of radius \(r\). If the angular velocity is \(\omega\), then
So in Exercise 1.18, once we determined that \(\omega = 80\pi \dfrac\), we could determine v as follows:
\[v = r\omega = (3\space ft)(80\pi\dfrac = 240\pi\dfrac).\]
Notice that since radians are “unit-less”, we can drop them when dealing with equations such as the preceding one.
Example \(\PageIndex\): Linear and Angular Velocity
The LP (long play) or 331 rpm vinyl record is an analog sound storage medium and has been used for a long time to listen to music. An LP is usually 12 inches or 10 inches in diameter. In order to work with our formulas for linear and angular velocity, we need to know the angular velocity in radians per time unit. To do this, we will convert \(33\dfrac\) revolutions per minute to radians per minute. We will use the fact that \(33\dfrac = \dfrac\)
We can now use the formula v D r! to determine the linear velocity of a point on the edge of a 12 inch LP. The radius is 6 inches and so
It might be more convenient to express this as a decimal value in inches per second. So we get
The linear velocity is approximately 20.944 inches per second.
For these problems, we will assume that the Earth is a sphere with a radius of 3959 miles. As the Earth rotates on its axis, a person standing on the Earth will travel in a circle that is perpendicular to the axis.
So the linear velocity is approximately \(1075.1\) feet per second.
In this section, we studied the following important concepts and ideas:
This page titled 1.4: Velocity and Angular Velocity is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Ted Sundstrom & Steven Schlicker (ScholarWorks @Grand Valley State University) via source content that was edited to the style and standards of the LibreTexts platform.
