Hurricanes are considered the powerhouse of weather phenomena. To fuel their need for fury, they use warm ocean air to absorb warm ocean water. Winds, which come together at the surface of the ocean, then force the warm ocean air to rise. The air eventually cools and forms clouds. This process is continuously repeated, resulting in air and clouds rotating around what is known as the eye of the storm. As this occurs at faster and faster rates, the hurricane generates more and more power to unleash on those closest to it. Now, these chilling, yet majestic, phenomena are prime examples of rotational motion. Therefore, let this article introduces the concept of rotational motion.
Fig. 1 - A hurricane demonstrating rotational motion.
Rotational Motion Definition
Below we will define rotational motion and discuss how it is divided into different types.
Rotational Motion is defined as a type of motion associated with objects that travel in a circular path.
Types of Rotational Motion
Rotational Motion can be divided into three types.
- Motion about a fixed axis: Is also known as pure rotation and describes the rotation of an object around a fixed point. Some examples are the rotating of fan blades or the rotating of hands on an analog clock as both rotate about a central fixed point.
- A combination of rotational and translational motion. This motion describes an object, whose components can rotate about a fixed point, while the object itself travels along a linear path. An example is the rolling of wheels on a car. The wheels have two velocities, one as a result of the rotating wheel and another due to the car's translational motion.
- Rotation about an axis of rotation. This motion describes objects that rotate about an axis while also rotating around another object. An example is Earth orbiting around the sun while it also rotates about its own axis.
Rotational Motion Physics
The physics behind rotational motion is described by a concept known as kinematics. Kinematics is a field within physics that focuses on the motion of an object without referencing the forces causing the motion. Kinematics focuses on variables such as acceleration, velocity, displacement, and time which can be written in terms of linear or rotational motion. When studying rotational motion, we use the concept of rotational kinematics. Rotational kinematics refers to rotational motion and discusses the relationship between rotational motion variables.
Note that velocity, acceleration, and displacement are all vector quantities meaning they have magnitude and direction.
Rotational Motion Variables
The rotational motion variables are:
- angular velocity
- angular acceleration
- angular displacement
- time
Angular Velocity, \(\omega\)
Angular velocity is the change in the angle with respect to time. Its corresponding formula is $$ \omega = \frac{\theta}{t}$$ where angular velocity is measured in radians per second, \(\mathrm{\frac{rad}{s}}\).
The derivative of this equation yields the equation
$$\omega = \frac{\mathrm{d}\theta}{\mathrm{d}t},$$
which is the definition of instantaneous angular velocity.
Angular Acceleration , \(\alpha\)
Angular acceleration is the change in angular velocity with respect to time. Its corresponding formula is $$ \alpha = \frac{\omega}{t} $$ where angular acceleration is measured in radians per second squared, \(\mathrm{\frac{rad}{s^2}}\).
The derivative of this equation yields the equation
$$\alpha = \frac{\mathrm{d}\omega}{\mathrm{d}t},$$
which is the definition of instantaneous angular acceleration.
Angular Displacement, \(\theta\)
Angular displacement is the product of angular velocity and time. Its corresponding formula is $$ \theta = \omega t $$ where angular displacement is measured in radians, \(\mathrm{rad}\).
Time, \(t\)
Time is time. $$ \mathrm{time} = t $$ where time is measured in seconds, \(s\).
Relationship Between Rotational Kinematics and Linear Kinematics
Before diving deeper into rotational kinematics, we must be sure to recognize and understand the relationship between kinematic variables. This can be seen when looking at the variables in the table below.
Variable | Linear | Linear SI units | Angular | Angular SI units | Relationship |
acceleration | $$a$$ | $$\frac{m}{s^2}$$ | $$\alpha$$ | $$\mathrm{\frac{rad}{s^2}}$$ | $$\begin{aligned}a &= \alpha r \\\alpha &= \frac{a}{r}\end{aligned}$$ |
velocity | $$v$$ | $$\frac{m}{s}$$ | \(\omega\) | $$\mathrm{\frac{rad}{s}}$$ | $$\begin{aligned}v &= \omega r \\\omega &= \frac{v}{r}\end{aligned}$$ |
displacement | $$x$$ | $$m$$ | \(\theta\) | $$\mathrm{rad}$$ | $$\begin{aligned}x &= \theta r \\\theta &= \frac{x}{r}\end{aligned}$$ |
time | $$t$$ | $$s$$ | \(t\) | $$\mathrm{s}$$ | $$t = t$$ |
Note that \(r\) represents the radius and time is the same in both linear and angular motion.
As a result, kinematic equations of motion can be written in terms of linear and rotational motion. However, it is important to understand that although equations are written in terms of different variables, they are of the same form because rotational motion is the equivalent counterpart of linear motion.
Remember these kinematic equations only apply when acceleration, for linear motion, and angular acceleration, for rotational motion, are constant.
Rotational Motion Formulas
The relationship between rotational motion and rotational motion variables is expressed through three kinematic equations, each of which is missing a kinematic variable.
$$\omega=\omega_{o} + \alpha{t}$$
$$\Delta{\theta} =\omega_o{t}+\frac{1}{2}{\alpha}t$$
$$\omega^2={\omega_{o}}^2 +2{\alpha}\Delta{\theta}$$
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.
These kinematic equations only apply when angular acceleration is constant.
Rotational Kinematics and Rotational Dynamics
As we have discussed rotational kinematics, it is also important for us to discuss rotational dynamics. Rotational dynamics deals with the motion of an object and the forces causing the object to rotate. In rotational motion, we know this force is torque.
Newton's Second Law for Rotational Motion
Below will we define torque and its corresponding mathematical formula.
Torque
In order to formulate Newton's second law in terms of rotational motion, we must first define torque.
Torque is represented by \(\tau\) and is defined as the amount of force applied to an object that will cause it to rotate about an axis.
The equation for torque can be written in the same form as Newton's second law, \(F=ma\), and is expressed as $$\tau = I \alpha$$
where \(I\) is the moment of inertia and \(\alpha\) is angular acceleration. Torque can be expressed this way as it is the rotational equivalent of force.
Note that the moment of inertia is the measurement of an object's resistance to angular acceleration. Formulas regarding an object's moment inertia will vary depending on the shape of the object.
However, when the system is at rest, it is said to be in rotational equilibrium. Rotational equilibrium is defined as a state in which neither a system's state of motion nor its internal energy state changes with respect to time. Therefore, for a system to be at equilibrium, the sum of all forces acting on the system must be zero. In rotational motion, this means that the sum of all torques acting on a system must equal zero.
$$ \sum \tau = 0 $$
The sum of all torques acting on a system can be zero if the torques are acting in opposite directions thus canceling out.
Torque and Angular Acceleration
The relationship between angular acceleration and torque is expressed when the equation, \( \tau={I}\alpha \) is rearranged to solve for angular acceleration. As a result, the equation becomes\( \alpha=\frac{\tau}{I} \). Thus, we can determine that angular acceleration is proportional to torque and inversely proportional to the moment of inertia.
Rotational Motion Examples
To solve rotational motion examples, the five rotational kinematic equations can be used. As we have defined rotational motion and discussed its relation to kinematics and linear motion, let us work through some examples to gain a better understanding of rotational motion. 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 rotational kinematic equations to a spinning top.
A spinning top, initially at rest, is spun and moves with an angular velocity of \(3.5\,\mathrm{\frac{rad}{s}}\). Calculate the top's angular acceleration after \(1.5\,\mathrm{s}\).
Fig. 2 - A spinning top demonstrating rotational motion.
Based on the problem, we are given the following:
- initial velocity
- final velocity
- time
As a result, we can identify and use the equation, ,\( \omega=\omega_{o} + \alpha{t} \) to solve this problem. Therefore, our calculations are:
$$\begin{aligned}\omega &= \omega_{o} + \alpha{t} \\\omega-\omega_{o} &= \alpha{t} \\\alpha &= \frac{\omega-\omega_{o}}{t} \\\alpha &= \frac{3.5\,\frac{rad}{s}- 0}{1.5\,s} \\\alpha &= 2.33\,\frac{rad}{s}\end{aligned}$$
The angular acceleration of the top is \(2.33\,\mathrm{\frac{rad}{s^2}}\).
Example 2
Next, we will do the same thing for a tornado.
What is the angular acceleration of a tornado, initially at rest, if its angular velocity is given to be \(95\,\mathrm{\frac{rad}{s}}\) after \(7.5\,\mathrm{s}\)? What is the tornado's angular displacement?
Fig. 3 - A tornado demonstrating rotational motion.
Based on the problem, we are given the following:
- initial velocity
- final velocity
- time
As a result, we can identify and use the equation, \( \omega=\omega_{o}+\alpha{t} \), to solve the first part of this problem. Therefore, our calculations are:\begin{align}\omega &= \omega_{o} + \alpha{t} \\\omega-\omega_{o} &= \alpha{t} \\\alpha &= \frac{\omega-\omega_{o}}{t} \\\alpha &= \frac{95\,\mathrm{\frac{rad}{s}} - 0}{7.5\,\mathrm{s}} \\\alpha &= 12.67\,\mathrm{\frac{rad}{s^2}}\end{align}
Now using this calculated angular acceleration value and the equation, \( \Delta{\theta}=\omega_o{t}+\frac{1}{2}{\alpha}t \), we can calculate the tornado's angular displacement as follows:\begin{align}\Delta{\theta} &= \omega_o{t}+\frac{1}{2}{\alpha}t \\\Delta{\theta} &= \left(0\right) \left(7.5\,\mathrm{s}\right) + \frac{1}{2}\left(12.67\,\mathrm{\frac{rad}{s^2}} \right)\left({7.5\,\mathrm{s}}\right)^2 \\\Delta{\theta} &= \frac{1}{2}\left(12.67\,\mathrm{\frac{rad}{s^2}} \right) ({7.5\,\mathrm{s}})^2 \\\Delta{\theta} &= 356.3\,\mathrm{rad}\end{align}
The angular displacement of the tornado is \(356.3\,\mathrm{rad}\).
Example 3
For our last example, we will apply the torque equation to a rotating object.
An object, whose moment of inertia is \( 32\,\mathrm{\frac{kg}{m^2}} \) rotates with an angular acceleration of \( 6.8\,\mathrm{\frac{rad}{s^2}} \). Calculate the amount of torque needed for this object to rotate about an axis.
After reading the problem, we are given:
- angular acceleration
- moment of inertia
Therefore, applying the equation for torque expressed in the form of Newton's second law, our calculations will be as followed:\begin{align}\tau &= {I}\alpha \\\tau &= \left(32\,\mathrm{\frac{kg}{m^2}}\right)\left(6.8\,\mathrm{\frac{rad}{s^2}}\right) \\\tau &= 217.6\,\mathrm{N\,m}\end{align}
The amount of torque needed to rotate the object about an axis is \( 217.6\,\mathrm{N\,m} \).
Rotational Motion - Key takeaways
- Rotational Motion is defined as a type of motion associated with objects that travel in a circular path.
- Types of rotational motion include motion about a fixed axis, motion about an axis in rotation, and a combination of rotational motion and translational motion.
- Rotational kinematics refers to rotational motion and discusses the relationship between rotational motion variables.
- Rotational motion variables include angular acceleration, angular velocity, angular displacement, and time.
- Rotational motion variables and rotational kinematic equations can be written in terms of linear motion.
- Rotational motion is the equivalent counterpart to linear motion.
- Rotational dynamics deals with the motion of an object and the forces causing the object to rotate which is torque.
- Torque is defined as the amount of force applied to an object that will cause it to rotate about an axis and can be written in terms of Newton's Second Law.
- When the sum of all torques acting on a system equals zero, the system is said to be in rotational equilibrium.
References
- Fig. 1 - Eye of the Storm from Outer Space (https://www.pexels.com/photo/eye-of-the-storm-image-from-outer-space-71116/) by pixabay (https://www.pexels.com/@pixabay/) public domain
- Fig. 2 - Multi Color Striped Ceramic Vase (https://www.pexels.com/photo/multi-color-striped-ceramic-vase-972511/) by Markus Spiske (https://www.pexels.com/@markusspiske/) public domain
- Fig. 3 - Tornado on Body of Water during Golden Hour (https://www.pexels.com/photo/tornado-on-body-of-water-during-golden-hour-1119974/) by Johannes Plenio (https://www.pexels.com/@jplenio/) public domain
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