Angular velocity: All you need to know about angular velocity

The angular velocity is the rate of change in the angular displacement of a particle. The angular velocity has two words: angular and velocity. Velocity means displacement upon a time, and angular velocity means angular displacement upon a time. If any particle is making a circular motion, then suppose the particle equals p and center equal to q.

Angular velocity

Now p comes to q during t time. Now it will create an angle in the center which is theta. This angle theta is also known as angular displacement. However, angular displacement travels per unit time, or angle sustains per unit time. However, we know this ratio is angular velocity. Angular velocity’s symbol is omega. Next, omega equals the angle subtended per unit time. Let particles’ direction of motion be anti-clockwise. Angular velocity and angular displacement are the vector quantities. So the direction of angular displacements is the same as angular velocity.

Angular velocity formula

Angular velocity formula is omega equals 2 pi N over 60 rad per second. We know the velocity formula in linear motion is the general formula; velocity is nothing but the distance by time. If your body covers two beats in one second, then we can say that the velocity in the body in linear motion is two minutes per second. Suppose we apply this technique in the angular motion also, so in this case, the angular displacement unit per second time. So at first, if your body is rotating, let N be the body’s rpm. The rpm means revolutions per minute. If we want per second, then we need to divide by 60.

Angular velocity

Because one minute will contain 60 seconds and revolutions per minute converts into per second by dividing by 60, for one revolution, the body rotates one complete revolution; we know that it will cover 360 degrees which is equal to 2 pi radians. If you take 1 minute, it will be N revolution. N revolution means 2 pi into N radians. Next, you take one second that is 2 pi N divided by 60 radians per second. This we got the angular velocity formula, omega equals 2 pi N by 60.

ω= 2πN/60 rad/s

Velocity in linear motion

v= d/t= 2/1 m/s

v=2 m/s, t=1

 

In angular motion

If N=rpm=revolutions per minute

N/60=rps=revolutions per second

1 revolution=360 degree=2π radians

Next, 1 minute= (N) revolutions =2π into N radians

Next, 1 second=2πN/60 rad/s

ω= 2πN/60 rad/s

Angular velocity unit

The unit of angular velocity is radians per second.

Angular velocity symbol

Omega (ω) is the symbol of angular velocity.

Angular velocity to linear velocity

Let, v is linear velocity and ⍵ is angular velocity. At first, a body which is moving on the circular path of radius r, then the position changes from p to q point. Next, that’s why theta is made in the centre. However, theta is angular displacement. We know, angular displacement theta equals l by r. Next, we can write l equals to r into theta. Now, we need to differentiate both sides with respect to time t. So we can write dl/dt equal to r into dθ/dt where r is constant. Next, dl/dt will be v and dθ/dt will be ⍵. However, angular displacement is now angular velocity and rate of change of linear displacement by time is linear velocity. Now, this is an scaler form and in vector form we can write v equals to ⍵ ⨯ r.

Now, v is the linear velocity, ⍵ is the angular velocity and r is the radius vector or position vector.

Then, v= ⍵ ⨯ r.

Angular velocity dimensional formula

The dimensional formula of angular velocity is omega equals angular displacement by time. Now the SI unit of angular velocity is radian per second. Now theta is the dimensionless quantity. If a circle of radius r, initially a particle is p, after some time interval the position of the particle is p. Formula for angular displacement is equal to arc upon radius. The arc length is considered as s. So theta equals to s by r. Nows measures the intern of the meter, and r measures the intern of the meter too. So both meters will be canceled, and the result is 1. Now the dimension of angular displacement is M power zero, L power zero, and T power zero. It is the dimensionless quantity.

So the place of angular displacement is written as 1 upon t, and t is nothing but time, so the SI unit of time is second. Now omega equals 1 upon second is expressed in terms of dimension T. Now T will be the numerator which will be T to the power minus 1. Next, the dimensional formula’s format of angular velocity is M to the power absent L to the power absent and T to the power minus 1.

Angular Velocity = Angular displacement × [Time]-1…(1)

Now, the dimensional formula of Angular displacement = [M^0 L^0 T^0]….(2)

And, the dimensions of time = [M^0 L^0 T^1]……(3)

Now, on substituting equation (2) and (3) in equation (1) we get,

Next, angular Velocity = Angular displacement × [Time]-1

Or, v = [M^0 L^0 T^0] × [M^0 L^0 T^1]-1 = [M^0 L^0 T^-1]

However, the angular velocity is dimensionally represented as [M^0 L^0 T^-1].

Angular velocity of the earth

We know the formula of angular velocity, which is omega equals 2 pi divided by T. We are assuming that it is rotating in a circular path. However, T is nothing but how much time it takes to rotate that is T. However, we know earth takes 24 hours to complete one rotation. But we need to transform the hour to the minute. Therefore we need to convert 24 hours which equals 24, into 3600. Now omega equals to 2 pi upon 86400.

Angular velocity of the earth’s equation

ω=dθ/dt

Here,dθ=2πRadians

dt = 24Hours

Now, The SI unit of angular velocity is rad/sec

so we need to convert our time to seconds

Next, 1 hour = 3600 sec

⇒24 hours

=24×3600

=24×3600 sec

Now, ω=2π/24×3600 rad/sec

⇒ω=7.29×10^−5 rad/sec

Angular velocity and speed

Let’s say we have some object that’s moving in a circular path. So the object is moving in a circular path that looks something like a center clockwise circular path. Now, the object is making five revolutions every second. However, radians are just one way to measure angles. You could do with degrees per second. Now, If we do it with radians, we know that each revolution is 2 pi radians. Next, If we go all the way around a circle, we have gone 2 pi radians. There is 2 pi per revolution, so we can do a little bit of dimensional analysis. Next, we get 5 times 2 pi which gets us 10 pi radians per second. And it works out the dimensional analysis, and it also makes sense. However, If we are doing five revolutions a second, each of these revolutions is 2 pi radians.

So we are doing 10 pi radians/second. So either 5 revs/second or 10 pi radians/second, they are both essentially measuring the same thing. However, this measure of how fast you are orbiting around a central point is called angular velocity. Next, it is angular velocity because if you think about it, it is telling us how fast our angle is changing or the speed of the angle changing. However, angular velocity tends to be treated as angular speed. Next, It is a vector quantity, and it is a little unintuitive that the vector’s actually popping out of the page. However, it is actually a pseudo vector. So it’s a vector quantity, and the direction of the vector is dependent on which way it’s spinning. For example, when it is spinning in a counterclockwise direction, there is a vector; the real angular vector does pop out of the page.

Result

If it is going clockwise, the angular velocity vector will pop into the page. When we are just thinking about a two-dimensional plane, we can really think of an angular velocity as the pseudo-scalar. But we can include that as a scalar quantity, as long as we specify which way it is rotating. So 10 pi radian per second, we could call it angular velocity. And this tends to be denoted by an omega. However, We could say angular velocity is equal to a change in angle over a change in time. So after the calculation, we could tell that omega is equal to speed which we are using v for, divided by the radius.

Angular velocity equation

Q.  Calculate the angular velocity of a particle moving along the straight line given by θ = 3t3 + 6t + 2 when t = 5s.

  1. So, we know θ = 3t3 + 6t + 2 and t= 5 second

      However, ⍵= dθ/dt= 9t^2+6.

    Next, ⍵= 9(5^2)+6 = 231units/second.  

Some frequently asked questions

What is the angular velocity in simple terms?

Angular velocity is the rate of change in the angular displacement of a particle. However, angular velocity has two words: angular and velocity. Velocity means displacement upon time, and angular velocity means angular displacement upon a time.

What is called angular speed?

Angular velocity tends to be treated as angular speed. It is a vector quantity, and it is a little unintuitive that the vector’s actually popping out of the page. However, it is actually a pseudovector.

Is Omega angular velocity?

Yes, omega is the symbol of angular velocity.

What is the symbol of angular acceleration?

 Alpha (α) is the symbol of angular acceleration.

What is omega equal to?

 ω = 2πf, omega is equal to 2 pi f.

What is the formula of angular velocity?

Angular velocity formula is omega equals to 2 pi N over 60 rad per second. However, we know the velocity formula in linear motion is the general formula; velocity is nothing but the distance by time.

What is omega in physics class 11?

Average angular speed = ΔΘ/Δt. However, angular speed, ω = dΘ / dt. V = w r, in which v – linear speed of particle transferring in a circle of radius r.

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