How To Find Angular Velocity: 5 Answers You Should Know


The motion of an object on a circular path on its fixed axis is known to be the rotational motion. This article is about the velocity generated by this motion. 

To know how to find angular velocity lets me know about it first. When the body moves on a circular path, the velocity that comes into action due to this motion is said to be angular velocity. Examples of angular velocity are the wheels, merry-go-round, and many more. 

We all know about the translation velocity that is caused due to the linear motion of an object. But when the body rotates or moves on a circular path, apart from linear velocity, there is also angular velocity. This velocity tells us about the vector rate of changes of the angle of a moving body. It gives the idea about how rapidly the body is rotating or revolving. In physics, rotational velocity is also a term for angular velocity. 

Image Credit: dnet based on raster version released under GFDL, Angular velocityCC BY-SA 3.0

Orbital angular velocity and spin angular velocity are the two kinds of angular velocity. The orbital angular velocity specifies the rate of angle change of an object about a fixed point. At the same time, the spin angular velocity specifies the rapidity of a rotating object about its center of rotation. 

How to find angular velocity from linear velocity

How to find angular velocity

We know that angular velocity is specified as the rate of angle change of the moving body. So we have:

[latex]\omega = \frac{\Delta \theta }{\Delta t} = \frac{\mathrm{d}\theta }{\mathrm{d} t}[/latex]

From the figure above, we can see that the particle is displacing from point A to B. Here the distance covered, s, by the particle is equal to the arc length of the circular path. 

Using the formula: 

[latex]Arc = Radius \times Angle[/latex]

[latex]s = \theta\cdot r[/latex]

differentiating both sides with respect to t

[latex]\frac{\mathrm{d} s}{\mathrm{d} t}= r \cdot \frac{\mathrm{d} \theta }{\mathrm{d} t}[/latex]

We know that the differentiation of displacement gives us the linear velocity, and differentiation of angle gives angular velocity. On substituting the values, we get the relation between linear and angular velocity becomes:

[latex]v = \omega . r[/latex]

This formula is used to find the angular velocity from the linear velocity of a rotating body. 

For example, a ball is moving on a circular path with a radius 5 m and velocity of 20 m/s then the angular velocity is given as;

[latex]v = \omega . r[/latex]

[latex]20 = \omega . 5[/latex]

[latex]\omega = \frac{20}{5}[/latex]

[latex]\omega = 4 rad \cdot s^{-1}[/latex]

How to find angular velocity in radians per second

In rotational kinematics, the particle moves along the circular path. The angular velocity specifies how fast the object is moving. Hence by calculating the revolutions made by the object in a given time, we can find out its velocity. 

[latex]\omega = \frac{revolutions}{time}[/latex]

We know that for a circular path, 1 complete revolution makes 360°. And 360° equals 2π in radians. Taking time in the standard unit, the unit of omega becomes; 

[latex]\omega = \frac{radians}{second}[/latex]

Suppose a spinning wheel is making 4830revolutions per minute. Then the angular velocity in radians per second would be:

[latex]\omega = \frac{4830 rev}{1 min}[/latex]

1 revolution = 2π radian

4830 revolution = 4830 × 2π 

and 1 min = 60 seconds 

Therefore:

[latex]\omega = \frac{4830 \cdot 2\Pi }{60}[/latex]

[latex]\omega = 161 rad \cdot s^{-1}[/latex]

How to find angular velocity with mass and radius

Just like there is linear momentum for the translational motion, similarly for the rotational motion of an object, there exists angular momentum, L. The formula for angular momentum is given as:

[latex]L = mvr[/latex]

Substituting the value of v from the equation [latex]v = \omega \cdot r[/latex] , we get;

[latex]L = m (\omega \cdot r) r[/latex]

[latex]L = m \omega r^{2}[/latex]

[latex]\omega = \frac{L}{m r^{2}}[/latex]

This equation is used to find the angular velocity with mass and radius. 

How to find angular velocity without time

Just like we have equations of motions for linear motion, in the same way, there are three equations of motion for rotating objects. 

[latex]\omega _{f} = \omega _{i} + \alpha t[/latex]

[latex]\theta = \omega _{i} t + \frac{1}{2} \alpha t^{2}[/latex]

[latex]{\omega _{f}}^2 = {\omega _{i}}^2 + 2 \alpha \theta[/latex]

Here, 

[latex]\omega _{f}[/latex] = final angular velocity

[latex]\omega _{i}[/latex] = initial angular velocity

[latex]\alpha[/latex] = angular acceleration

[latex]\theta[/latex] = angular displacement

t = time taken 

Using this equation of rotational motions, we can find the angular motion without time. 

For calculating the angular velocity without time, the previous formulas can also be used;

[latex]v = \omega . r[/latex]

[latex]L = m \omega r^{2}[/latex]

[latex]\omega = 2 \pi f[/latex]

Suppose we are given that a rotating wheel initially at rest displaces at an angle of 4π radians with an angular acceleration of [latex]2 rad . s^{-2}[/latex] , then the angular velocity with which the wheel is rotating is given as;

[latex]{\omega _{f}}^2 = {\omega _{i}}^2 + 2 \alpha \theta[/latex]

[latex]{\omega_{f}}^{2} = 0 + 2 \times 2 \times 4 \pi[/latex]

[latex]{\omega_{f}}^{2} = 16 \pi[/latex]

[latex]{\omega_{f}} = \sqrt{16 \pi}[/latex]

[latex]{\omega_{f}} = 4 \sqrt{\pi}[/latex]

[latex]\omega_{f} \approx 7 rad \cdot s^{-1}[/latex]

Frequently Asked Questions (FAQs)

What is rotational motion? 

Motion can be classified into various types, one of which is rotational motion.

When a body moves on a circular path about a fixed point, it is said to be in rotational motion. The particles of the rotating body move in the same phase with the same angular speed—for example, electron movement around the nucleus. 

Define angular velocity? 

When a particle moves on a circular path, it attains angular velocity along with linear velocity. 

The rate of angle change of the rotating object is said to be its angular velocity. It is analogous to the linear velocity of the translational motion. It specifies how fast the object is rotating. 

Give an example of angular velocity from everyday life 

The Ferris wheel is a basic example of angular velocity from our day-to-day life. 

The Ferris wheel moves on a circular path about its fixed point. On rotating, its angle keeps changing, and hence the angular velocity comes into action. Together the linear and the angular velocity makes the Ferris wheel move

How are angular velocity and linear velocity related? 

The linear velocity and angular velocity are related by the formula;

[latex]v = \omega . r[/latex]

What is the unit of angular velocity? 

Since the angular velocity is the rate of angle change of a circulatory body, its unit is;

[latex]1 \omega = \frac{1 radians }{1 second}[/latex]

[latex]\omega = rad \cdot s^{-1}[/latex]

How to find the angular velocity of a particle?  

The angular displacement of the moving body leads to the angular speed. 

To find the angular speed, we use the following formulas;

[latex]\omega = \frac{\Delta \theta }{\Delta t} = \frac{\mathrm{d} \theta }{\mathrm{d} t}[/latex]

[latex]v = \omega . r[/latex]

For example, a wheel displaces at an angle of 12π radians in 2s then the angular speed would be:

[latex]\omega = \frac{12 \pi}{2 s}[/latex]

[latex]\omega = 6 rad \cdot s^{-1}[/latex]

Rabiya Khalid

Hi,  I am Rabiya Khalid, currently pursuing my masters in Mathematics. Article writing is my passion and I have been professionally writing for more than a year now. Being a science student, I have a knack for reading and writing about science and everything related to it. If you like what I write you can connect with me on LinkedIn: https://www.linkedin.com/mwlite/in/rabiya-khalid-bba02921a In my free time, I let out my creative side on a canvas. You can check my paintings at: https://www.instagram.com/chronicles_studio/

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