Angular Velocity and Linear Velocity

We know that angular displacement is the angle traced by a particle under the 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 = Rθ or θ = \[\frac{S}{r}\] …..(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:

                             ω = \[\frac{\Delta \theta }{\Delta t}\] …(3)                               

In this article, we will derive the relation between linear velocity and angular velocity. So, let’s get started.


Relation Between Linear Velocity and Angular Velocity

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

                           |v| = \[\frac{\Delta (S)}{\Delta t}\] , and S = rθ

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

                        v = \[\frac{\Delta r \theta}{\Delta t}\]

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

                     v = \[\frac{r \Delta \theta}{\Delta t}\] (As \[\frac{\Delta \theta}{\Delta t}\] = ω)

⇒ |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= \[\frac{v}{\omega }\]

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


Difference between Linear Velocity and Angular Velocity


Linear Velocity

Angular Velocity

Definition

We define the linear velocity as the rate of change of the linear displacement.

We define the angular velocity as the rate of change of angular displacement.

Key Differences

  • In a circular motion, the linear velocity of a particle is along the circumference of a circle. 

  • It is different at different points on the circle.

  • The angular velocity of a particle is along the axis of a circle.

  • It remains the same throughout the circular motion.

Symbol

The symbol for the linear velocity is v.

The symbol for angular velocity is a Greek letter, i.e., ω (pronounced as omega).

Formula

v  = \[\frac{\Delta S}{\Delta t}\] = \[\frac{2 \pi r}{t}\]

ω = \[\frac{\Delta \theta}{\Delta t}\] = \[\frac{2 \pi r}{t}\]

Quantity

It is a vector quantity that implies it has both magnitude & direction.

It is an axial vector. Its because the direction of the displacement of the particle is along the axis of the circle.

Measuring Unit

We measure the linear velocity in m/s.

We measure the angular velocity in both degrees and radians. 

A degree is a dimensionless unit.

Radian is an SI unit.

Real-life Example

If your office is at 150 km distance from your home, then your car will acquire a linear velocity.


Do you know?

The correct form of the linear and angular velocity relation is v = rω, not v\[^{\rightarrow }\] = rω\[^{\rightarrow }\].

It’s because v acts on the circumference of the circle and ω along the axis. For vectors of two quantities to be equal, their directions must be the same. Here, we can see that the directions of v and ω are not equal; that’s why the correct way of writing the relation is:

                                           v = rω

FAQs (Frequently Asked Questions)

Question 1: A Car Tire Revolves at 180 rpm. Find its Angular Velocity in rad/sec, Rounded to Two Decimal Places?

Answer: 1 revolution = 2П radians

180 revolutions/minute = (180 x 6.28) rev/min

= (180 x 6.28)/60 = 18.84 rev/sec (in two decimal places)

Question 2: Are Angular Speed and Angular Velocity the Same?

Answer: The answer to this question is evasive because these two quantities have similarities & differences.


Similarity

Both are measured in radian per second.


Difference

  • Angular speed (v) is a scalar quantity.

  • Angular velocity (ω) is a vector quantity

Question 3: If the Angular Speed of the Truck Wheel Increases from 1300 rpm to 1800 rpm in 20 Seconds. Find the Time Taken to Move 40°.

Answer: Here, Initial frequency, f₁ = 1300rpm = 1300/60 rps = 22 rps

               Final frequency, f₂ = 1800rpm = 1800/60 rps = 30 rps

               Angular velocity, ω₁  = 2Пf = 6.28 x 22 = 138 s⁻¹

                           ω₂ = 6.28 x 30 = 188 s⁻¹     

Time taken to move 40⁰ (0.70 rad) is:

t = θ/ω = 0.70/188 = 3.72 milliseconds

Question 4: What is the Angle Between the Velocity Vector and the Acceleration Vector?

Answer: The angle between the velocity & the acceleration vector varies depending on the following types of motion. 

  • If the motion is circular, the angle is 90⁰.

  • In the linear motion, the angle may be 0⁰ (if the speed increases) or 180⁰ (if the speed decreases).