Linear and Angular Speeds, Area of Sectors, and Length of Arcs

This section covers:

Around the time you’re learning about radian measures, you may have to work with Linear and Angular Speeds, and also Area of Sectors and Lengths of Arcs.

We discussed radians and how they related to degrees of central angles here in the Angles and the Unit Circle section.   Before we talk about linear and angular speeds, let’s go over how radians relate back to the circumference of a circle and also revolutions of a circle.

Note that θ is the central angle of rotation in radians, r is the radius of the circle, s is the arc length, or intercepted arc (part of the circumference) of the circle, and a revolution (or rotation) is when an object has gone all the way around a circle (or the circle returns back to where it started).  These units can all relate back to the circumference of a circle:  2πr, where r is the radius.   (Remember in a Unit Circle, the circumference is just 2π, since r = 1).

Note that the words radian and radius are related, since there are 2πr (radians) in a revolution, and 2πr (radius) is the measurement of the circumference.

One of the most important concepts is that the length of an intercepted arc is the radius times the radian measure of that arc’s central angle.  To see this, set up a proportion comparing this arc to the whole circumference:

arc-proportion

From here you can see that  \(s=r\theta \).

Here are some formulas:

linear-and-angular-speed-formulas

We’ll use these formulas in some of the linear/angular speed problems below.

Remember that you typically use radius with linear speed, and radians with angular speed in 2πr.

Linear Speed

Linear speed is the speed at which a point on the outside of the object travels in its circular path around the center of that object.  The units can be any usual speed units, such as miles per hour, meters per second, and so on.

We remember that Distance = Rate x Time, or Rate (Speed) =  \(\frac{{\text{Distance}}}{{\text{Time}}}\).   We’ll first talk about how fast an object along the circumference of a circle is changing.

Think of a car that drives around in a circle on a track with arc length (the actual length of the curvy part – part of the circumference) s.  The formula for the speed around a circle, or the linear speed is   \(v=\frac{s}{t}\),  where s is the arc length and t is the time.

Here’s a type of problem that you might have.  Note that we have to use Unit Multipliers (Dimensional Analysis) when the units don’t match.

LInear Speed Example

Angular Speed

Angular speed is the rate at which the object turns, described in units like revolutions per minute, degrees per second, radians per hour, and so on.

Angular speed has to do with how fast the central angle of a circle is changing, as opposed to the circumference of the circle.

Again, think of a car that drives around in a circle on a track with central angle  θ.   The formula for the speed around a circle in terms of this angle, or the angular speed is  \(\omega =\frac{\theta }{t}\),  where θ is in radians, and t is the time.

Here’s a type of problem that you might have.  Note that we have to use Unit Multipliers (Dimensional Analysis) again when the units don’t match.

linear-speed-example

Here’s an example that shows the difference between finding angular speeds and linear speeds.  Notice with angular speeds, we ignore the radius, since we are just dealing with rotation of an angle.  Also notice that Linear Speed = Radius x Angular Speed, or , or Radius =  \(\frac{{\text{Linear Speed}}}{{\text{Angular Speed}}}\).

Linear And Angular Speed Problems

Here are more problems with linear and angular speeds, and rotations/revolutions:

Linear and Angular Speed Problems 2

Area of Sectors

Let’s get the area of a sector of a circle based on the radius and a central angle in radians.     We know from geometry that a sector of a circle is is like a pizza slice; it’s a region bounded by a central angle and its intercepted arc.

Picture of Circle

You may have seen in geometry how to get the area of a sector based on radius and degrees of the sector:     \(\frac{{\text{area of the sector}}}{{\text{area of the whole circle}}}=\frac{{\text{degrees of the sector}}}{{360{}^\circ }}\).

Let’s use the fact that the area of a circle is  \(\pi \,{{r}^{2}}\),  the arc length of a sector is  \(r\theta \),  and the arc length of a whole circle is  \(2\pi r\).  Now we can solve for the area of a sector given a radius and central angle:  \(\frac{{\text{area of the sector }(A)}}{{\pi\,{{r}^{2}}}}=\frac{{r\theta }}{{2\pi }};\,\,\,\,\,\,\,\,\,A=\frac{1}{2}{{r}^{2}}\theta \).

Length of Arcs

You may have seen in geometry how to get the length of an arc based on radius and degrees of the sector: \(\frac{{\text{length of intercepted arc}}}{{\text{ arc length (circumference) of circle}}}=\frac{{\text{degrees of the central angle of the arc}}}{{360{}^\circ }}\).

But we actually already know a simple equation from one of the first equations above:  \(s=r\theta \),  where s = the length of the intercepted arc, r is the radius of the circle, and θ is the intercepted arc.  It’s as simple as that!

So let’s get the area of a sector and length of intercepted arc, given the radius and central angle:

Area of Sector and Length of Arc

Understand these problems, and practice, practice, practice!

On to Graphs of Trig Functions – you’re ready!