Did you know that a tornado is a prime example of angular motion? Tornadoes are caused by systems, known as supercells, that generate severe thunderstorms. In these systems, air revolves around a horizontal axis and due to the difference in wind speeds, strong cold winds located higher in the atmosphere and weaker warm winds, collide. This collision lifts the rotating air upward resulting in it rotating about a vertical axis, thus creating a tornado. Hopefully, you have not witnessed this real-life example firsthand but rather can use it as a starting point in understanding angular motion. Therefore, let this article introduce the concept of angular kinematics through definitions and examples to help expand your mind on angular kinematics.
Kinematics
However, before we begin to discuss angular kinematics, let us quickly define the word kinematics. Kinematics focuses on the motion of an object and does not include the forces that cause it to move. Kinematics consists of four variables as follows:
- Velocity, \(v\), which is measured in \(\frac{\mathrm{m}}{\mathrm{s}}\).
- Acceleration, \(a\), which is measured in \(\frac{\mathrm{m}}{\mathrm{s}^2}\).
- Displacement, \(x\), which is measured in \(\mathrm{m}\).
- Time, \(t\), measured in \(\mathrm{s}\).
Note that these variables describe linear motion
Angular Kinematic Equation Definition
We will define and discuss angular kinematics and its associated kinematic variables.
Angular kinematics is the study of angular motion without including external forces.
Now, when we discuss angular kinematics, the four kinematic variables are expressed in terms of angular motion variables as follows:
- Angular velocity, \(\omega\) measured in \(\frac{\mathrm{rad}}{\mathrm{s}}\).
- Angular acceleration, \(\alpha\) measured in \(\frac{\mathrm{rad}}{\mathrm{s}^2}\).
- Angular displacement, \(\theta\) measured in \(\mathrm{rad}\).
- time, \(t\) measured in \(\mathrm{s}\).
Note that these variables describe angular motion
Radians
A radian is the angle from the center of a circle where the arclength, the length along the edge of the circle, is one radius long. Essentially, we can think of a radian as the ratio between the arclength and the radius, \( Radian=\frac{Arclength}{Radius} \).
Fig. 2 - An illustration explaining what a radian is.
A radian per second is the change in an object's orientation, due to rotation, with respect to time.
Relationship Between Linear Motion and Angular Motion Variables
Before delving into angular kinematics, we must make sure to understand that kinematic variables written in the form of linear and angular motion are equivalent to each other. This fact can be seen when looking at the relationships between the variables in the table below.
| Variable | Linear | Linear SI Units | Angular | Angular SI Units | Relationship |
| Acceleration | \(a\) | \(\frac{\mathrm{m}}{\mathrm{s}^2}\) | \(\alpha\) | \(\frac{\mathrm{rad}}{\mathrm{s}^2}\) | \(a = \alpha r\), \(\alpha = \frac{a}{r} \) |
| Velocity | \(v\) | \(\frac{\mathrm{m}}{\mathrm{s}}\) | \(\omega\) | \(\frac{\mathrm{rad}}{\mathrm{s}}\) | \(v = \omega r\), \(\omega = \frac{v}{r} \) |
| displacement | \(x\) | \(\mathrm{m}\) | \(\theta\) | \(\mathrm{rad}\) | \(x = \theta r\\ \),\(\theta = \frac{x}{r} \) |
| time | \(t\) | \(\mathrm{s}\) | \(t\) | \(\mathrm{s}\) | \(t=t\) |
Note that \(r\) represents the radius and time is the same in both linear and angular motion.
Angular Kinematic Equations
For angular motion, there are three kinematic equations, each of which is missing a kinematic variable. The angular kinematic equations are as follows.
\begin{align}\omega &= \omega_{o} + \alpha{t} \\\Delta{\theta} &= \omega_o{t}+\frac{1}{2}{\alpha}t^2 \\\omega^2 &= {\omega_{o}}^2 +2{\alpha}\Delta{\theta}\end{align}
where \(\omega\) is final angular acceleration, \(\omega_0\) is the initial angular velocity, \(\alpha\) is angular acceleration, \(t\) is time, and \(\Delta \theta\) is angular displacement.
Angular Acceleration Kinematic Equations
We can convert the angular velocity and angular velocity squared equations into angular acceleration equations. Rearranging the two results in two explicit angular kinematics equations for angular acceleration.
Angular Velocity Equation
\begin{align}\omega &= \omega_{o} + \alpha{t} \\ \alpha{t} &= \omega-\omega_{o} \\ \alpha &= \frac{\omega-\omega_{o}}{t}\end{align}
Angular Velocity Squared Equation
\begin{align}\omega^2 &= {\omega_{o}}^2 +2{\alpha}\Delta{\theta} \\2{\alpha}\Delta{\theta} &= \omega^2-{\omega_{o}}^2 \\\alpha &= \frac{\omega^2-\omega_{o}^2}{2\Delta{\theta}}\end{align}
Hence, we now have two angular kinematic equations for angular acceleration.
Angular Motion and Rotational Motion
Angular motion and rotational motion are the same. Both terms describe the movement of an object around a fixed point or axis. Hence, when solving angular motion and rotational motion problems, the same kinematic equations are used.
Application of Angular Kinematic Equations
While we may not realize it, we experience angular kinematics in our daily lives. Two prime examples are ceiling fans and Ferris wheels. Both use the concept of rotational motion, i.e. motion in circular paths as they rotate around their central fixed point. Consequently, we can apply the angular kinematic equations and calculate variables such as angular velocity or angular acceleration for both.
Angular Kinematic Examples
To solve angular kinematic problems, one can use the three angular kinematic equations and apply them to a variety of problems. As we have defined kinematics and discussed its relation to angular motion, let us work through some examples to gain a better understanding of angular kinematics. Note that before solving a problem, we must always remember these simple steps:
- Read the problem and identify all variables given within the problem.
- Determine what the problem is asking and what formulas are needed.
- Apply the necessary formulas and solve the problem.
- Draw a picture if necessary to provide a visual aid
Example 1
Let us apply the angular kinematic equations to a rotating fan.
A fan, initially turn off, is turned on. If the fan blade begins to rotate with an angular velocity of \(6.5\,\frac{\mathrm{rad}}{\mathrm{s}}\), what is the angular acceleration of the fan after \(3.5\,\mathrm{s}\)? What is the angular displacement of the fan?
Based on the problem, we are given both the initial and final angular velocity of the fan blade as well as the time. Therefore, we can apply our equation, \(\omega = \omega_0 + \alpha t\), to solve the first part of this problem. Using that equation to solve for angular acceleration results in the following calculations seen below.
First, we must rearrange our equation to solve for angular acceleration.
\begin{align*} \omega &= \omega_0 + \alpha t \\ \alpha t &= \omega - \omega_0 \\ \alpha &= \frac{\omega - \omega_0}{t} \end{align*}
We can now insert our variables and solve as follows:
\begin{align*} \alpha &= \frac{\omega - \omega_0}{t} \\ \alpha &= \frac{6.5\,\frac{\mathrm{rad}}{\mathrm{s}}-0\,\frac{\mathrm{rad}}{\mathrm{s}}}{3.5\,\mathrm{s}} \\ \alpha &= 1.9\,\frac{\mathrm{rad}}{\mathrm{s}^2} \end{align*}
The angular acceleration of the fan blade after \(3.5\,\mathrm{s}\) is \(1.9\,\frac{\mathrm{rad}}{\mathrm{s}^2}\). As we have solved the first part of the problem, we now have to solve for the angular displacement of the fan blade. After looking at the angular motion kinematic equations, we should identify the change in angular displacement equation,\( \Delta{\theta} =\omega_o{t}+\frac{1}{2}at \), as the equation we need to solve for the fan blades' angular displacement. Using this equation and inserting our variables, our calculations will be as follows:
\begin{align*} \theta &= \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2 \\ \theta-\theta_0 &= \omega_0 t + \frac{1}{2}\alpha t^2 \\ \Delta \theta &= \omega_0 t + \frac{1}{2}\alpha t^2 \\ \Delta \theta &= \left(0\,\frac{\mathrm{rad}}{\mathrm{s}}\right)\left(3.5\,\mathrm{s}\right) + \frac{1}{2}\left(1.86\,\frac{\mathrm{rad}}{\mathrm{s}^2}\right) \left(3.5\,\mathrm{s}\right)^2 \\ \Delta \theta &= 11\,\mathrm{rad} \end{align*}
The angular displacement of the fan blade is \(11\,\mathrm{rad}\).
Example 2
Let us apply the angular kinematic equations to a stationary exercise bike.
The tire on a stationary exercise bike has an initial angular velocity of \(5.0\,\frac{\mathrm{rad}}{\mathrm{s}}\). If its angular acceleration is given as \(1.5\,\frac{\mathrm{rad}}{\mathrm{s}^2}\) and its angular displacement is given as \(10.2\,\mathrm{rad}\), what is the final angular velocity of the tire? How long does it take for the tire to reach this velocity?
Based on the problem, we are given both the initial angular velocity of the tire as well as its angular acceleration and angular displacement. Therefore, if we look at all three angular motion kinematic equations, we can identify the angular velocity squared equation, \(\omega^2 = \omega_0^2 + 2 \alpha_x(\theta - \theta_0)\), as the equation needed to solve the first part of this problem. Using that equation to solve for final angular velocity results in the following calculations seen below.
\begin{align}\omega^2 &= \omega_0^2 + 2 \alpha_x(\theta - \theta_0) \\\omega^2 &= \left(5.0\,\frac{\mathrm{rad}}{\mathrm{s}^2}\right)^2 + 2 \left(1.5\,\frac{\mathrm{rad}}{\mathrm{s}^2}\right)(10.2\,\mathrm{rad}) \\\omega^2 &= 25\,\frac{\mathrm{rad}^2}{\mathrm{s}^2} + 30.6\,\frac{\mathrm{rad}^2}{\mathrm{s}^2} \\\omega^2 &= 55.6\,\frac{\mathrm{rad}^2}{\mathrm{s}^2} \\\omega &= 7.5\,\frac{\mathrm{rad}}{\mathrm{s}}\end{align}
The final angular velocity of the tire is \(7.46\,\frac{\mathrm{rad}}{\mathrm{s}}\). As we have solved the first part of the problem, we now have to for time. After looking at the angular motion kinematic equations, we should identify the angular velocity equation, \( \omega = \omega_0 + \alpha t \), as the equation we need to solve for the time it takes the stationary bike tire to reach its final velocity. Using this equation and inserting our variables, our calculations will be as follows:
\begin{align}\omega &= \omega_0 + \alpha t \\t &= \frac{\omega - \omega_0}{\alpha} \\t &= \frac{7.5\,\frac{\mathrm{rad}}{\mathrm{s}} - 5.0\,\frac{\mathrm{rad}}{\mathrm{s}}}{1.5\,\frac{\mathrm{rad}}{\mathrm{s}^2}} \\t &= 1.7\,\mathrm{s}\end{align}
The tire reaches its final angular velocity after \(1.7\,\mathrm{s}\)
Angular Kinematic Equations - Key takeaways
- Kinematics focuses on the motion of an object without including forces.
- Kinematic equations can be written for both linear and angular motion
- Linear motion variables and angular motion variables are equivalents of one another
- Angular kinematics is the study of angular motion without including external forces.
- Angular kinematics includes four variables: angular velocity, angular acceleration, angular displacement, and time
- The relationship between angular kinematic variables is expressed through the four angular kinematic equations
- Angular motion is sometimes referred to as rotational motion, therefore, they have the same kinematic equations
- Applying angular motion kinematic equations to solve examples is a good way to gain familiarity with the equations
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