Angular Velocity Formula

The angular velocity applies to the entire object that moves along a circular path. Also, in this topic, we will discover the definition, angular velocity formula its derivation and solved example.

Linear Velocity

Linear velocity applies to an object or particle that is moving in a straight line. Also, it refers to the rate of change of an object’s position with respect to time. Besides, its most common example is your car speed when you are driving down on the road. Moreover, the speedometer tells you your speed in kilometer per hour (km/h) and this is your linear velocity.

Angular Velocity

It is less common than linear velocity since it only concerns objects that are moving along a circular path. For example, a roulette ball on a roulette wheel, a race car on a circular path, and a Ferris wheel are all examples of angular velocity.

Moreover, the angular velocity of the object is the object’s angular displacement with respect to time. In addition, when an object travels along a circular path, the central angle equivalent to the object’s position on the circle is changing. Besides, the angular velocity which we represent by using letter w is the rate of change of this angle with respect to time.

Example, suppose a Ferris wheel is rotating pi / 6 (\(\pi\) / 6) radians every minute. As a result, the Ferris wheel’s angular velocity would be pi / 6 (\(\pi\) / 6) radians per minute.

Angular Velocity Formula

There are three formulas that we can use to find the angular velocity of an object.

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 = \(\frac{\theta} {t}\)

Derivation of the formula

w = refers to the angular velocity
\(\theta\) = refers to the position angle
t = refers to the time

2nd option

In the second method, we recognize that \(\theta\) (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)

Derivation

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

Derivation

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

Solved Example on Angular Velocity Formula

Example 1

Suppose a race car is traveling in a circular path or track, and it travels 1 lap or $$2\pi$$ radians in 4 minutes. Then calculate the angular velocity of the car.

Solution:

Now let’s put the values in the first formula to get the answer

w = θ / t
w = 2pi / 4 = pi / 2

So, the angular velocity of the race car is pi / 2 radians per minute.

Example 2

Now consider another race car traveling along a circular track at 110 kilometers per hour, and the radius of the track is 0.2 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 = 110 / 0.2 = 550 radians per hour.
So, the angular velocity of the car is 550 radians per hour.