Angular Equation of Motion: 3 Important Concepts


The angular equation of motion, the set of equations illustrates the rotating system’s behavior in terms of its motions as a function of time. The article exhaustively discusses about the angular equations of motion of the rotating system.

The set of three angular equations of motion explains a rotating system as a set of its mathematical functions in dynamic variables. 

First angular equation of motion[latex]\omega = \omega _{0} + \alpha t[/latex]Angular velocity as a function of time
Second angular equation of motion[latex]\Delta \theta = \omega _{0}t + \frac{1}{2} \alpha t^{2}[/latex]Angular Displacement as a function of time
Third angular equation of motion [latex]\omega ^{2} = \omega _{0}^{2} + 2\alpha \Delta \theta[/latex] Angular velocity as a function of displacement

As you see, the variables in all three equations are generally spatial coordinates and time; but also include momentum components. If you identify the dynamics of a system, you can find out these three sets of equations, which are solutions for the differential equations that characterize the motion of the system.

Such a description of motion is classified into two forms: dynamics and kinematics. In the dynamics motion, the force, momenta, energy of objects come into the picture. In comparison, the kinematics motion is only concerned about the variables derived from the positions of objects and time.

In this article, we first determine the set of equations that show connections between variables; and then use these connections to analyze the rotating body’s angular motion. The analysis reports we obtained from angular equations of motion are the foundation for rotational kinematics.

As we discussed the angular acceleration is constant in previous article, the set of angular equation motion can be achieved from the definitions of kinematics quantities of rotating body like displacement [latex]\theta[/latex], initial velocity [latex]\omega_{0}[/latex], final velocity [latex]\omega[/latex] and time t.

Angular Equation of Motion of Rotating Body
Angular Equation of Motion
of Rotating Body

The angular equations are normally recognized as the physical laws and then apply definitions of these kinematic physical quantities. Hence, we can obtain the solutions of these equations by estimating the initial values, which determine the values of the constants.

Read more about our previous article on Angular Velocity of Rotating Body.

Analogy of Angular Motion

There are analogs of all linear motion quantities such as distance, velocity, and acceleration in angular motion, which makes the angular motion more comfortable to work with after learning about linear motion. 

Let’s write the equation of linear velocity as,

[latex]v = \frac{ds}{dt}[/latex]………… (1)

The angular motion is the rotating body’s motion around a fixed axis equal to the angle moved over at the axis by a line drawn to the body. 

As definitions of angular motion, lets substitute displacement [latex]\theta[/latex] instead of ds, and symbol for angular velocity [latex]\omega[/latex], instead linear velocity v, we obtained

[latex]\omega = \frac{d\theta }{dt}[/latex]………..(2)

That means the body’s angular velocity is the angle that the rotating body sweeps per unit of time. 

Analogy of Angular Motion with Linear Motion
Analogy of Angular Motion
with Linear Motion
(source: science abc)

Using the circular polar coordinates, which define a vector from the axis to its position, we can represent a displacement of the rotating body. Like the angular velocity equation, we can determine the position using such a different set of coordinates. Instead of using x,y coordinates, the angular displacement can be written in terms of radius r, which is its distance from the origin.

Analogy of Angular Motion
Analogy of Angular Motion

[latex]\theta[/latex] is the angle between the displacement vector and an axis through the origin, usually measured anti-clockwise from the x-axis and generally expressed in radians – that convert linear motion to angular motion easier.

We can simplify the determination of more angular equations of motion, similar to the linear equations of motion – to describe various applications in physics and engineering where the system has the constant angular acceleration.

First Kinematic Equation of Angular Motion

The first angular equation of motion is the equation of angular velocity with the function of time. It explains the relation among the quantities like [latex]\omega[/latex], [latex]\alpha[/latex] and t.

The first kinematics equation of a rotating body illustrates the correlation between its angular velocity and angular acceleration and time. In simple words, it shows how the rotating body accelerates when its angular velocity changes with time. 

The angular velocity is constant in a uniform circular motion (UCM) but not in rotational motion. Therefore, The angular acceleration results due to the change in its angular velocity with time. 

 [latex]\alpha = \frac{d\omega }{dt}[/latex] ……………(3)

The faster the variation in velocity, the higher its acceleration. If the body rotates clockwise, then its velocity is positive. Otherwise, it is negative. If [latex]\omega[/latex] increases, then [latex]\alpha[/latex] is positive. If it decreases, then [latex]\alpha[/latex] is negative. That means, if the rotating body slows down, then its acceleration is negative, whereas if it speeds up, its acceleration is positive.

Let’s derive the first angular equation of motion relating [latex]\omega[/latex] and [latex]\alpha[/latex].

We recollect the common kinematics equation for linear motion as:

[latex]v = u + at[/latex] …………… (4)

Relation between 
Linear and Angular Velocity
Relation between
Linear and Angular Velocity

As per [latex]v= r\omega[/latex], we have a relationship that shows linear acceleration a and angular acceleration [latex]\alpha[/latex] is constant. i.e.,

[latex]a = r\alpha[/latex]

Substituting values of v and a into equation (4), we get

[latex]r\omega = r\omega_{0} + r\alpha t[/latex]

By canceling radius r, we yield

[latex]\omega = \omega _{0} + \alpha t[/latex] for constant acceleration [latex]\alpha[/latex]  ……………………..(A)

Notice that the above equation is similar to its linear version, besides its angular analogs. We can determine more other situations with a uniform set of angular equations of motion following constant angular acceleration.

Second Kinematic Equation of Angular Motion

The second angular equation of motion is the equation of angular displacement with the function of time. It explains the relation among the quantities like [latex]\theta[/latex], [latex]\alpha[/latex] and t.

The second kinematics equation of the rotating body illustrates the relationship between its angular displacement and angular acceleration and time. In simple words, it shows how the rotating body accelerates when its angular displacement changes with time. 

We have obtained the first angular equation of motion (A), which we will employ to solve more rotational kinematics problems.

Let’s derive the second angular equation of motion by rearranging equation (2) to

[latex]\omega dt = d\theta[/latex]

Since the angular acceleration constant, integrating both sides from its initial to final values, we get

[latex]\int_{t_{i}}^{t_{f}}(\omega _{0} + \alpha t)dt = \int_{\theta _{i}}^{\theta _{f}}d\theta[/latex]

[latex]\int_{t_{i}}^{t_{f}}\omega _{0}dt + \int_{t_{i}}^{t_{f}} \alpha tdt = \int_{\theta _{i}}^{\theta _{f}}d\theta[/latex]

[latex]\omega_{0} t + \frac{1}{2}\alpha t^{2} = \theta _{f} – \theta i[/latex]

[latex]\Delta \theta = \omega_{0} t + \frac{1}{2}\alpha t^{2}[/latex] ………………..(B)

The equation (B) provides us the angular position of the rotating body for given initial forms and angular acceleration of the body in a given time.

Third Kinematic Equation of Angular Motion

The third angular equation of motion is the angular velocity equation with the function of its angular displacement. It explains the relation among the quantities like [latex]\omega[/latex], [latex]\theta[/latex], and t.

The second kinematics equation of the rotating body illustrates the relationship between its angular velocity and angular displacement and time. In simple words, it shows how the rotating body changes its velocity along with its displacement in unit time. 

Let’s find the third angular equation of motion that is independent of time t by solving equation (A) for t,

[latex]t = \frac{\omega -\omega _{0}}{\alpha }[/latex]

Substituting value of t into equation (B), we get

[latex]\Delta \theta = \omega_{0} (\frac{\omega -\omega _{0}}{\alpha }) + \frac{1}{2}\alpha (\frac{\omega -\omega _{0}}{\alpha })[/latex]

= [latex]\frac{\omega \omega _{0}}{\alpha } – \frac{\omega _{0}^{2}}{\alpha } +\frac{1}{2}\frac{\omega ^{2}}{\alpha }- \frac{\omega \omega _{0}}{\alpha }+\frac{1}{2}\frac{\omega _{0}^{2}}{\alpha }[/latex]

= [latex]\frac{1}{2}\frac{\omega ^{2}}{\alpha } – \frac{1}{2}\frac{\omega _{0}^{2}}{\alpha }[/latex]

[latex]\Delta \theta = \frac{\omega ^{2}-\omega _{0}^{2}}{2\alpha }[/latex]

Rearranging above equation for [latex]\omega[/latex], we get

[latex]\omega ^{2} = \omega _{0}^{2} + 2\alpha \Delta \theta[/latex] ………….(C)

The equation (2) through equation (C) illustrates the fixed-axis rotation for constant acceleration

Kinematics Equations of Motion
Kinematics Equations of Motion

Manish Naik

Hello, I'm Manish Naik completed my MSc Physics with Solid-State Electronics as a specialization. I have three years of experience in Article Writing on Physics subject. Writing, which aimed to provide accurate information to all readers, from beginners and experts. In my leisure time, I love to spend my time in nature or visiting historical places. I am honoured to be part of LambdaGeeks. Looking forward to connecting you through LinkedIn - https://www.linkedin.com/in/manish-ashok-naik/ Also, for Maharashtra travel guide and heritage conservation articles, visit my website Wandering Maharashtra - https://wanderingmaharashtra.com/travel-blogs/

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