Linear & Angular Speed Lesson

In this lesson, we will learn how to solve word problems that involve linear and angular speed. We have previously learned about the distance formula, used to solve motion word problems: $$d=r \cdot t$$ Where d is the distance traveled, r is the rate of speed, and t is the time traveled. If we solve this equation for r: $$r=\frac{d}{t}$$

Formula for Linear Speed

To find linear speed, we will use a similar formula:
Suppose that some point P moves at a constant speed along a circle with a radius r and a center O. The measure of how fast the position of P is changing is the linear speed. $$v=\frac{s}{t}$$ Where v represents the linear speed, s is the length of the arc traced by point P at time t. To put this in more simple terms, linear speed is the speed at which a point on the outside of an object travels in a circular path around the center of that object.

Formula for Angular Speed

If we look at our image above, we can see as our point P moves along the circle, ray OP rotates around the origin. Our ray OP represents the terminal side of angle POB, and the measure of this angle changes as P moves along the circle. Angular speed is the measure of how fast angle POB is changing. In other words, angular speed represents how fast the central angle is changing.
Most books will use the Greek letter omega ω to represent angular speed: $$ω=\frac{θ}{t}$$ θ is given in radians as the measure of angle POB at time t. ω will be expressed in radians per unit of time.
We previously learned about the formula for arc length on a circle. We found that the length s of the arc intercepted on a circle of radius r by a central angle of measure θ radians was found to be: $$s=rθ$$ If we plug in for s in our formula for linear speed: $$v=\frac{s}{t}$$ $$v=\frac{rθ}{t}$$ $$v=r \cdot \frac{θ}{t}$$ Since ω is given as θ/t, we can replace this in our formula: $$v=rω$$ Let's look at a few examples.
Example #1: Solve each word problem.
The tires of a bicycle have radius 15.0 inches and are turning at the rate of 230 revolutions per minute. How fast is the bicycle traveling in miles per hour?
Let's begin by calculating the angular speed ω. $$ω=\frac{θ}{t}$$ How do we calculate θ? One revolution represents one complete rotation or an angle of 360° or 2$π$ radians. We have 230 revolutions: $$ω=\frac{230 \cdot 2π}{1 \hspace{.2em}\text{min}}=\frac{460π}{1 \hspace{.2em}\text{min}}$$ We could also write this as 460$π$ radians per minute.
Since the problem asks for our time unit to be in hours, let's convert this over:
1 hour = 60 minutes
Set up a unit fraction and multiply: $$\require{cancel}\frac{460 π}{1 \hspace{.2em}\cancel{\text{min}}}\cdot \frac{60 \hspace{.2em}\cancel{\text{min}}}{1 \hspace{.2em}\text{h}}=\frac{27{,}600 π}{1 \hspace{.2em}\text{h}}$$ Now that we have our angular speed as 27,600$π$ radians per hour, let's find v, our linear speed: $$v=rω$$ $$v=15 \hspace{.15em}\text{in}\cdot \frac{27{,}600 π}{1 \hspace{.15em}h}$$ $$v=\frac{414{,}000 π \hspace{.15em}\text{in}}{1 \hspace{.15em}\text{h}}$$ Since the problem asks for miles per hour, we need to convert inches into miles:
63,360 inches =  1 mile $$v=\frac{414{,}000 π \hspace{.15em}\cancel{\text{in}}}{1 \hspace{.15em}\text{h}}\cdot \frac{1 \hspace{.15em}\text{mi}}{63{,}360 \hspace{.15em}\cancel{\text{in}}}$$ $$v=\frac{575 π \hspace{.15em}\text{mi}}{88 \hspace{.15em}\text{h}}$$ To convert this into miles per hour, we will multiply 575 by $π$ and then divide the result by 88:
v ≈ 20.5 miles per hour
Example #2: Solve each word problem.
At a local high school, Max rides his cart around a circular track at 2 revolutions per minute. If the radius of the circular track is 150 meters, how fast is Max's cart traveling in meters per second?
Let's begin by calculating the angular speed ω. $$ω=\frac{θ}{t}$$ How do we calculate θ? One revolution represents one complete rotation or an angle of 360° or 2$π$ radians. We have 2 revolutions: $$ω=\frac{2 \cdot 2π}{1 \hspace{.15em}\text{min}}=\frac{4π}{1 \hspace{.15em}\text{min}}$$ This gives us an angular speed of 4$π$ radians per minute. Let's find v, our linear speed: $$v=rω$$ $$v=150 \hspace{.15em}\text{m}\cdot \frac{4π}{1 \hspace{.15em}\text{min}}=\frac{600π \hspace{.15em}\text{m}}{1 \hspace{.15em}\text{min}}$$ This tells us Max is riding at a rate of 600$π$ meters per minute, but the problem asks for seconds, so we need to perform another unit conversion: $$v=\frac{600π \hspace{.15em}\text{m}}{1 \hspace{.15em}\cancel{\text{min}}}\cdot \frac{1 \hspace{.15em}\cancel{\text{min}}}{60 \hspace{.15em}\text{sec}}=\frac{10π \hspace{.15em}\text{m}}{1 \hspace{.15em}\text{sec}}$$ v ≈ 31.4 meters per second

---

Skills Check: