Angular Velocity and Linear Velocity

For the students of science subjects, it is very much important to know the terms exactly. This is to say, it is important for the science students to know the definitions of the term because it makes the meaning very clear and also in a concise manner. Also, many times the terms seem the same and therefore the students get confused about the meaning, in such cases, if the students have the definitions at hand, they can easily differentiate between the two. And hence Vedantu provides the same to the students of science for free.

A quick overview of the terms Angular Velocity, and Linear Velocity.

Angular Velocity: A relation by which the position of one line with the other line is established, is called angular position. And the rate at which the angular position of a rotating body changes is called angular velocity.

Linear Velocity: In simple terms, Linear Velocity refers to the velocity of an object in a straight line. That is to say, it shows the time taken by an object for moving from point A to point B.

Difference between the angular velocity and Linear Velocity.

Advantages of having the Definitions, along with the relations, of the linear and angular Velocity

• To understand science better it is very important for the students to have a clear understanding of the words and the terms used in science, which is to say the students must have a good scientific vocabulary. And the best way of having that effective vocabulary is by having the definitions and the meaning of the terms available. And the same goes for the definition of linear Velocity and Angular Velocity.

• The concept of linear velocity and angular velocity does seem completely different by learning the definitions of it. But still, there are some relations between the two as well, and hence Vedantu provides the same to the students for completely free of cost.

•  Also, sometimes students may find the concept of linear velocity and angular velocity a little difficult to understand, hence Vedantu provides the explanation of the same in a simple and lucid language. That is to say, along with the definition, the explanation is also given.

• Once the students have understood the concept of linear velocity and angular velocity in a good manner then other things become pretty easy. Such as the definition of linear and angular velocity can be remembered easily because you already know what exactly it means. Also, finding the difference between the both becomes quite easy, as well as its relations.

Why is angular displacement considered as an axial vector?

We know that angular displacement is the angle traced by a particle under circular motion. Since the direction of the displacement is along the axis, that’s why angular displacement is an axial vector.   

A particle in a circular motion exhibits two types of displacements; these are:

• Along the circumference: Linear Displacement (s).

• Making an angle θ: Angular Displacement.

If ‘r’ is the radius of the circle, then the relation between angular and linear displacement is:  

\[S=R\theta \] or \[\theta =\frac{s}{r}\] . . . . . . .  (1)

Linear velocity is defined as the rate of change of linear displacement. For a particle P, it is given by:

                     \[V=\frac{\Delta s}{\Delta t}\].....(2)

Angular velocity of the particle is the rate of change of angular displacement, i.e., how fast an angle is changing. It is given by:

    \[w=\frac{\Delta \theta }{\Delta t}\].....(3)

Relation Between Linear Velocity and Angular Velocity

We know that the linear velocity of particle P is given by: 

\[\mid v \mid =\frac{\Delta (s)}{\Delta t}\].....(2), and S = rθ

Now, putting the value of eq (1) in (3), we get:

                     \[v=\frac{\Delta r\theta }{\Delta t}\].....(2)

We know that the radius ‘r’  is a constant value, so we get:

                 \[v=\frac{\Delta r\theta }{\Delta t}\].....(2) (As \[\frac{\Delta \theta }{\Delta t}\] = w)

⇒ |v| = rω, which is the ‘relation between linear and angular velocity.

Here, v depends on r and ω in the following manner:

ω = How fast an object is rotating, and

r  = The linear velocity of the particle at the center will be zero. As the particle starts moving away from the center, the linear velocity starts increasing. It is maximum at the circumference of the circle.

The tangent to the circle gives the direction of the linear velocity.  

Point to Remember

• Putting the value of θ in radians in the equation: S = rθ makes the calculation easier.

• Similar to the displacement, the angular velocity is also an axial vector quantity. We can find its direction by using the Right-hand Thumb Rule. This rule says:

Curl your fingers in a counterclockwise direction, and the thumb pointing outwards (along the axis) is the direction of the angular velocity. Similarly, if you curl your fingers in a clockwise direction, then the thumb pointing inwards gives the direction of ω.

Angular Velocity and Linear Velocity Relation

For a body in a uniform circular motion, the relation between linear and angular velocity is:

                          v =  rω 

This equation states that the linear velocity (v) is directly proportional to the distance of the particle from the center of the circular path and its angular velocity. 

• The linear velocity is different at different points on the circle. 

• It is zero at the center.

• It is minimum in between the center and any point on the circumference.

• It is maximum at the circumference of the circle. However, angular velocity remains the same at all points on the circular path.

Angular Velocity to Linear Velocity

From our knowledge of the circular motion, we can infer that the magnitude of the linear velocity of a particle moving in a circle links to the angular velocity of the particle by the following relation:

                         \[r=\frac{v}{w}\]

At any moment, this relationship applies to every particle that has a rigid body.