Angular Velocity Formula

Angular Velocity Formula

As a vector quantity, angular velocity is defined as the rate of change of angular displacement, which identifies the rotational speed and angular speed of an object around a given axis. Angular velocity refers to the amount of change in the particle’s angular displacement over a certain time interval. The angular velocity vector tracks vertically to the plane of rotation, as represented by the right-hand rule.

Every point on an object that is revolving about an axis has the same angular velocity. The tangential velocity of points distant from the axis of rotation is, nevertheless, different from that of points closer to the axis of rotation. Rotational velocity and an angular frequency vector are other names for angular velocity.

ω = θ/ t

Where t is the change in time, ω is the angular velocity, and θ  is the angular displacement.

Linear Velocity

A particle or item travelling straight ahead is said to have linear velocity. It also describes how quickly the location of an object changes over time. The most typical illustration of this is the pace at which you drive along the road. Additionally, your speed is displayed on the speedometer in kilometres per hour (km/h), which is your linear velocity.

Angular Velocity

Due to the fact that it only applies to things that are travelling in a circular motion, it is less frequent than linear velocity. Examples of angular velocity include a roulette ball on a wheel, a race car travelling in a circle, and a Ferris wheel.

In addition, the object’s angular displacement with respect to time is represented by the angular velocity of the object. Additionally, the central angle that corresponds to an object’s position on a circle changes as it moves along a circular path. Additionally, the angular velocity, denoted by the letter w, is the rate at which this angle changes with respect to time.

Angular Velocity Formula

It is the rate at which an object’s position angle changes over time. Consequently, the Angular Velocity Formula is

w = θ/t

Angular Velocity Formula Derivation

1st option

This one comes from its definition. It is the rate of change of the position angle of an object with respect to time. So, in this way the formula is

w = θt

w = refers to the angular velocity
θ = refers to the position angle
t = refers to the time

2nd option

In the second method, we recognize that θ (theta) is given in radians, and the definition of radian measure gives theta = s / r. Also, we can put theta in first angular velocity formula. This will give us

w = (s / r) / t

on further simplification we get

w = s / (rt)

s = refers to the arc length
r = refers to the radius of the circle
t = refers to the time taken

3rd Option

The third formula comes from distinguishing that we can rewrite the second formula as

w = s / (rt)
w = (s / t) (1 / r)

Now recall that s / t is linear velocity. Hence, we can rewrite it as

w = v (1 / r) = v / r

w = is the angular velocity
v = linear velocity
r = is the radius of the circle

Solved Example on Angular Velocity Formula

Example 1: Now consider another race car traveling along a circular track at 120 kilometers per hour, and the radius of the track is 0.4 km. Now, find the angular velocity of the car.

Solution: We can solve this with the help of the third formula
w = v / r
w = 120 / 0.4 = 300 radians per hour.
So, the angular velocity of the car is 300 radians per hour.

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