Angular and Linear Velocities

- Trigonometry Angles

This article will look at objects that rotate around a circle and introduce two forms of velocity, which is the rate at which objects move in relation to time: angular and linear velocities. The initial article of this series on trigonometry defined an angle as one ray, the terminal side, rotating away from another ray, the initial side. In the second article, two points of those rays were fixed onto a unit circle. It turns out that this rotation of a point around a circle shows up quite often in the real world.

Angular Velocity

Objects that move, or rotate, in a circular pattern are part of everyday life. Turbines spin at high speeds to generate electricity, helicopter blades spin to produce lift, and even a person standing on the earth has a velocity as the earth spins on its axis and revolves around the sun. This type of velocity is known as angular velocity, the rate at which the angle, formed by an initial starting point and current location of an object, changes over time. The key words in that definition being rate, over, and time; or in other words, a fraction.

\[ \frac{rate}{time} \]

The last article established that one rotation around a unit circle is \( 2\pi \) radians. In terms of rotation, an object moves \( 2\pi \) radians for every 1 revolution.

\[ \frac{2\pi}{1\ \text{rev}} \]

Therefore, angular velocity \( \omega \) can be described mathmatically as the angle \( \alpha \) in radians that an object moves through over time \( t \).

\[ \omega = \frac{\alpha}{t} \]

With this information, it is now easy to convert between revolutions over time and radians over time. To see this in action, imagine a wind turbine which has blades spinning at 20 revolutions per minute.

Wind turbine spinning

Now, simply convert using the ratio that allows the desired units to cancel.

\[ \require{cancel} \omega = \frac{20\ \text{rev}}{1\ \text{min}} = \frac{20\ \cancel{\text{rev}}}{1\ \text{min}} * \frac{2\pi\ \text{rad}}{1\ \cancel{\text{rev}}} = \frac{40\pi\ \text{rad}}{1\ \text{min}} \approx 125.7\ \text{rad/min} \]

Converting Between Units of Time

The previous example showed revolutions in terms of minutes, but what if there is a need to see these in different units of time, such as seconds or hours? The answer is as simple as adding another conversion ratio to the equation. Taking the previous example, convert the measurement in terms of seconds. In this case, convert between minutes and seconds. This ratio can be shown in two ways:

\[ \frac{1\ \text{min}}{60\ \text{sec}}\text{ or }\frac{60\ \text{sec}}{1\ \text{min}} \]

Which conversion ratio will give the desired result of converting from minutes to seconds in the previous example? For those paying attention to the previous articles in this series, the answer is simple: the one that allows the minutes to cancel out. Revisiting the last equation, the conversion to seconds might look like this:

\[ \require{cancel} \omega = \frac{20\ \text{rev}}{1\ \text{min}} = \frac{20\ \cancel{\text{rev}}}{1\ \cancel{\text{min}}} * \frac{2\pi\ \text{rad}}{1\ \cancel{\text{rev}}} * \frac{1\ \cancel{\text{min}}}{60\ \text{sec}} = \frac{40\pi\ \text{rad}}{60\ \text{sec}} \approx 2.1\ \text{rad/sec} \]

Linear Velocity

Linear velocity is much the same as angular velocity, except that linear velocity is concert with units of distance that change over time, rather than the measure of angles that change over time. With regards to a circle, this distance is represented by the arc length. The last article established that an arc length \( s \) can be determined by multiplying the angle in radians by the radius of the circle, \( s = \alpha r \). Therefore, linear velocity \( v \) can be determined as follows:

\[ v = \frac{s}{t} \]

Returning to the wind turbine example, suppose that the length of a blade is \( 116\ \text{ft} \), the length of the blades for common wind turbines.

Wind turbine linear velocity

Since the length of the blade is also equal to the radius, the circumference of the circle created by blade rotation can be determined as \( 2\pi(116)\ \text{ft} = 232\pi\ \text{ft} \). With this information, an equation be created to determine the linear velocity.

\[ v = \frac{20\ \text{rev}}{1\ \text{min}} = \frac{20\ \cancel{\text{rev}}}{1\ \text{min}} * \frac{232\pi\ \text{ft}}{1\ \cancel{\text{rev}}} = \frac{4640\pi\ \text{ft}}{1\ \text{min}} \approx 14576.99\ \text{ft/min} \]

The Linear/Angular Velocity Relationship

One of the most interesting aspects of math is how concepts build on each other. This article has so far defined angular velocity as \( \omega = \frac{\alpha}{t} \), where \( \alpha \) is the angle, and linear velocity as \( v = \frac{s}{t} \), where \( s \) is the arc length. Combine this with the definition of an arc length \( s = \alpha r \) and a relationship emerges:

\[ v = \frac{s}{t} = \frac{\alpha r}{t} = r * \frac{\alpha}{t} = r\omega \]

Breaking down the above, start with the linear velocity \( v \) as \( \frac{s}{t} \). Next, replace \( s \), the arc length, with its definition \( \alpha r \). This step produces the angular velocity \( \frac{\alpha}{t} \) times \( r \), the radius. Simplifying the angular velocity produces the equation for linear velocity in terms of angular velocity:

\[ v = r\omega \]

In the example of the wind turbine, the angular velocity was determined to be \( \frac{40\pi\ \text{rad}}{1\ \text{min}} \) with a radius \( r = 116\ \text{ft} \). Use the above linear velocity formula in terms of angular velocity and compare the results to the previously found linear velocity:

\[ v = \frac{40\pi\ \text{rad}}{1\ \text{min}} * 116\ \text{ft} = \frac{4640\pi\ \text{ft}}{1\ \min} \approx 14576.99\ \text{ft/min} \]

The resulting linear velocity is the same as when it was derived from the relationship to the circumference of the bounding circle. Note also that the radian units seem to "disappear", this is because the radian unit is just a real number.

Conclusion

This article defines angular velocity, \( \frac{\text{angle}}{\text{time}} \), and linear velocity, \( \frac{\text{arc length}}{\text{time}} \). It also provides a real world example to show where these types of velocity can be found in everyday life, as well as how to employ them. The next article in this series will be defining the trigonomic functions, until then!