Angular Velocity and its Dimensional Formula

Angular velocity is defined as the disclossion of the physical measurements in terms of length, time and mass. Simply put, angular velocity is nothing but the rate of change of angular displacement (i.e., ⍵=  θ/T). 

Dimensions – These are mathematical measurements for a particular quantity or measure. These give a simplified understanding of the quantity we are measuring. We use many objects in our everyday life, all of which have dimensions of their own. 

Dimensional formula – Each quantity can be expressed by means of power (fundamental units) to each unit for derived quantity. The expression can be simply written as,

A= MpLqTr

Where A, p, q and r are constants, and their values differ with each measure. 

Dimensional formula 

In terms of dimensions, a dimensional formula is an equation that expresses the relationship between fundamental and derived units (equation). The letters L, M, and T are used to represent the three basic dimensions of length, mass, and time in mechanics.

All physical quantities can be stated in terms of the fundamental (base) units of length, mass, and time, multiplied by some factor (exponent).

The dimension of the amount in that base is the exponent of a base quantity that enters into the expression.

The units of fundamental quantities are expressed as follows to determine the dimensions of physical quantities:

• L stands for length 
• M for mass, and 
• T for time

Example: The area is equal to the product of two lengths. As a result, [A] = [L2]. That is, an area has two dimensions of length and zero dimensions of mass and time. In the same way, the volume is the product of three lengths. As a result, [V] = [L3].

Angular velocity derivation

The speed at which an item rotates about another point is known as angular velocity. Learn more about the definition of angular velocity, as well as the three different types of angular velocity formulae and some examples.

Linear velocity

We will go over linear velocity first before moving on to angular velocity. An object’s motion, which is in a straight line, can be studied as linear velocity. It is the pace at which an object’s location changes over time.

Your speed as you drive down the road is one of the most common examples of linear velocity. Your speedometer displays your speed in miles per hour or rate. This is the rate at which your location changes about time; in other words, your linear velocity is your speed.

Linear velocity formula v = s / t, where v = linear velocity, s = distance travelled, and t = time it takes to travel distance. 

Radians

Before we get to angular velocity, there’s one more thing to go over: radians. We employ the radian measure of an angle when dealing with angular velocity. Thus it’s necessary to be familiar with it. The radian measure is technically defined as the length of the arc subtended by the angle, divided by the radius of the circle the angle is a part of (where subtended implies opposite of the angle and extending from one point on the circle to the other), both marked off by the angle.

Angular velocity

Because it only applies to things travelling in a circular route, angular velocity is less prevalent than linear velocity. The angular displacement of an item for time is its angular velocity. The central angle corresponding to the object’s position on the circle changes as it travels along a circular path. The rate of change of this angle for time is denoted by w, which is the angular velocity.

A Ferris wheel, an example for angular velocity, may rotate pi / 6 radians every minute. As a result, the angular velocity of the Ferris wheel would be pi / 6 radians per minute.

Dimensions of angular velocity

M0L0T–1

The dimensions of ⍵ =  θ/T can be written as 

θ = positive angle – which has no units, so it is dimensionless. So its dimensions are M0L0T0.

T = stands for time – which has only one dimension (T). The power to time T is 1 because it is singular, so the dimensions of time T are M0L0T1.

Coming to the dimensions of angular velocity, the formula, as you already know, is ⍵ =  θ/T. So the dimensions can be written as 

M0L0T0M0L0T1 = 1T1 = T-1

So, the dimensions of angular velocity can be written as M0L0T-1. As mass and length are absent, their power is ‘0’, and the dimension time is in the denominator, which can be written as T-1 mathematically.

Angular velocity example: An example of angular velocity is a roulette ball on a roulette wheel, a race car on a circular circuit, or a Ferris wheel. In addition, the angular velocity of an object is its angular displacement for time.

Another most important example is the earth’s angular velocity, which is 1.99 x 10-7 rad/sec. 

 Angular velocity in real-life examples

1. When a film tape has rotated, the velocity of the tape reeling out is calculated by angular velocity.
2. We know that all planets in the solar system rotate in their axis around the sun; the velocity of each planet is calculated by means of angular velocity. The angle of inclination and the time taken for each rotation or spin are measured, and the velocity is derived.
3. The speed of the cycle is measured with the use of angular velocity.
4. The speed of the giant wheel might seem to be slow when seen from the common public view, but the speed of the giant wheel is high when sitting in one of the compartments. The speed/velocity is calculated by angular velocity.
5. Lastly, the most commonly used electrical device is CD players. The rotation and the speed of the spin are pre-calculated based on the angular velocity.

Conclusion

The pace at which an item rotates or circles around an axis, or the rate at which the angular displacement between two bodies varies, is called angular velocity. This displacement is depicted in the diagram by the angle formed by a line on one body and a line on the other.

 The dimensions of angular velocity can be written as M0L0T-1.