Is Angular Velocity a Vector Quantity: 7 Important Facts


The point of compass for the angular velocity is straight way perpendicular to the plane of the rotation. Let us discuss whether angular velocity is vector or not ahead.

The physical property of angular velocity is a vector expression and it is a time division at which a matter revolves about an axle. The angular velocity takes up to the entire matter that travelled towards circular path. The S.I. unit for the angular velocity is radians per second (rad/s).

Angular velocity is a characteristic of an object or mentioned frame and the value of the angular velocity are not depending upon the location where it is calculated. Let us discuss further in this article how the angular velocity is defined as the vector expression.

Why is angular velocity a vector?

Angular velocity is also termed as rotational velocity. Let us explain why angular velocity expresses as a vector.

The angular velocity is a vector because both properties direction and magnitude are present in it. When a body is moving in a circular motion the angular velocity (ω) is expressed as vector expression. The angular velocity is equal to the angular displacement divided by the change in time.

Vector quantity can be defined as when a physical quantity has both properties of direction and magnitude. Some examples of vector expression are velocity, displacement, torque, acceleration, momentum, and many more.

How is angular velocity a vector?

For the case of the moving matter the value of the angular velocity remain unchanged.  Let us describe how angular velocity is vector.

The angular velocity derives from a vector content when a body is moving in a uniform circular motion in a fixed time period from one area to another area at that time. The angular velocity will be equal to angular displacement which is separated by time.

The expression can be written as,

ω = ΔΘ/Δt

Image – Vector expression
Image Credit – Wikipedia

The drawing is about vector addition and scalar multiplication a vector v (blue) is added to another vector w (red, upper illustration). Below, w is stretched by a factor of 2, yielding the sum v + 2w.

Magnitude of angular velocity

Angular velocity is the motion of change of angular displacement. Lets us talk about the magnitude of the angular velocity.

The magnitude of the angular velocity can be estimated using ω = V/r in this equation. The expression for the magnitude of angular velocity is expressing the relation between the linear speed, angular speed, and radius of the path which shaped is circular.

Image – Magnitude of the vector for angular velocity
Image Credit – Wikipedia

In the expression for the magnitude of the angular velocity, V is equivalent to the physical property of linear speed. The angular velocity of the particle at P with respect to the origin O is determined by the perpendicular component of the velocity vector v.

Mathematical form of magnitude of the angular velocity

Let, consider a body is moved in a path which will be in round shaped.  Let us explore the ways to find angular velocity in circular motion.

  • The radius of the path is r and the angular displacement is θ. The angle is, Θ = Arc/Radius
  • So, linear speed is, v = s/t. Where, s is the shortest displacement by a moving matter of arc and angle.
  • Now, the linear displacement is, v = S/r.
  • The linear speed is, V = θ × r/t; V = r × θ/t; V = rω.

After rearranging the mathematical expression for the magnitude of the angular velocity can be written as, ω = V/r.

Direction of angular velocity

The direction of angular velocity can be estimated with the help of right-hand rule. Let us discuss the direction of the angular velocity.

The direction for the angular velocity is together with the axle of revolution and notified away from a fixed notified for a moving object which is rotated in the clockwise direction and on the part of a fixed notified for a matter rotate at the notified of the compass of anti-clockwise.

The rule of the right hand is if someone pointed their finger at the point of the compass the positive charge is traveled, the middle finger in the point of the compass of the field of the magnet, and the thumb in the point of the compass of the magnetic force pushing on the traveled charge.

Is angular velocity a free vector?

Free vector is a classification of vector whose starting position and ending position is continue with same condition. Let see whether the angular velocity is free vector or not.

The angular velocity is a free vector as the value of the angular velocity will remain unchanged for a moving rigid body in a certain time period.

The term free vector means a vector that is not a dot or a line, and something which can travel without any circumference around the area via it has an unchanged magnitude and also an unchanged point of the compass.

Is angular velocity an axial vector?

Axial vectors are vectors that act along the axle of turning. Lets us elaborate on whether angular velocity is an axial vector or not.

Angular velocity is an axial vector as the point of the compass of the angular velocity is towards the axle for this reason the physical property of angular velocity is considered as an axial vector expression.

An axial vector can be explained as; a vector that is not changing its sign during the changes of the coordinate system to a fresh system by a repercussion in the root. Some examples of the axial vector are torque, angular momentum, and many more.

Problem statements with solution:-1

A man traveled from Durgapur to Kolkata via his car. When the car is moved at the same time the wheel of the car is also rotated. The radius of the wheel is 30 inches. When the wheel rotates it’s completed 8 revolutions per second to complete the distance on the road.

Now determine the amount of angular velocity for the wheel of the car.

Solution:-

The expression for the angular velocity can be written as ω = ΔΘ/Δt

According as the car has fulfilled eight revolutions per second, need to multiply 2π according as a full revolution 360° equal 2π

Substituting the values in the equation and we get

ω = 16 π radians/ 1 second

ω = 16 π radians per second.

In the formula an angular rotation Δθ takes place in a fixed time Δt and ω expresses as angular velocity.

So, the angular velocity amount for the wheel is 16 π radians per second.

Problem statements with solution:-2

A toy is moved in a circular path. When the toy has moved the diameter of the path will be 25 meters with a velocity of 46 meters per second.

Now calculate the amount of angular velocity for the toy.

Solution:-

Given, Diameter of the path (D) = 25 meter

Radius (r) = D/2 = 25/2 = 14.5 meter

Velocity (V) = 46 meter per second

The equation for the angular velocity is, ω = V/r

Putting the values in the equation, ω = 45/14.5

ω = 3.10 radians per second

Where,

  • ω is denoted as angular velocity
  • V is denoted as linear velocity
  • r is denoted as radius of the circular path

So, the amount of angular velocity for the toy is 3.10 radians per second.

Problem statements with solution:-3

A man is travelling a circular path. When the man is moved from one place to another place, in that case, the radius of the path will be 25 meters with a velocity of 10 meters per second.

Now calculate the amount of angular velocity for the man.

Solution:-

Given data are,

Radius (r) = 25 meter

Velocity (V) = 10 meter per second

The formula for the angular velocity can be expressed as, V = rω

Here, ω is expressed as the angular velocity of the moving matter. Δθ is expressed as angular displacement of the moving matter and Δt is expressed as the taken time period of the moving matter.

Installation the values in the formula, we can write, ω = 25/10

ω = 2.5 radians per second.

The value of the man’s angular velocity is 2.5 radians per second.

Conclusion

The article concludes that magnitude and direction both are present in the physical property of angular velocity and it is a part of the vector. The dimensional formula for the angular velocity is, M0L0T-1 where, M is the mass, L is the length, and T is denoted as the time taken by the matter.

Indrani Banerjee

Hi..I am Indrani Banerjee. I completed my bachelor's degree in mechanical engineering. I am an enthusiastic person and I am a person who is positive about every aspect of life. I like to read Books and listen to music. Let's connect through LinkedIn-https://www.linkedin.com/in/indrani-banerjee-2487b4214

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