Angular Velocity Concept

What is angular velocity. Relation between period, frequency and angular velocity. Subject expression and solved examples.

Angular Velocity

The angle at which radius vector of an object doing circular motion is scanning in 1 second is called "angular velocity".

The unit for angular velocity is rad/s (radian/second)

1 rad/s velocity is the velocity of the radius vector that scan 1 radian angle in 1 second.

The Radian Concept

An object that does one complete turn on a circular path is scan 360 degree angle. Also 360 degree is equals 2π radian. Let π = 3,14.

2π radyan = 360°

2•3.14 radyan = 360

6.28 radyan = 360

1 rad = 360/6.28

1 rad = 57.32

One radian is 57.32 degrees. Accordingly, an object moving at 1 rad/s velocity is scanning 57.32 degrees angle per second. Angular velocity is mostly represented by the symbol w.

ω = 2πf

f: frequency

frequency is the inverse of the period.

f = 1

If we put 1/T instead of f,

ω =

T: period

Period: The time elapsed for one full turn of an object which does circular motion is called the period.

In a broader sense, the period is the time elapsed for one complete motion of an object which recurrent continuous the same motion.


If an object is doing its one full turn at 10 second, the turn of 360 degrees of this object is takes at 10 second.

This object is scan 2π/10 rad angle in 10 second. The angular velocity of the object is, 2π/10 rad.

 = 0.63 rad

Because of 2π rad = 360° 

360° = 36°


According to the equality at above, the velocity of the object is 36 degrees/second (36°/s).


In a generator, the shaft to which the wire coils are connected turns at velocity of 20 revolutions per second.

A) How many seconds does it take one full turn of the generator shaft?

B) What is the frequency of the generator?

C) What is the angular velocity of the generator?

D) How many seconds does it takes the 135 degrees angle scan of the generator shaft?



The time piece that the generator shaft does one full turn is the period of the generator.

The generator shaft which does 20 full turning in 1 second, is does "x" turn in 1 second.

Is found "x" from ratio.

1 s           20 rev

T s           1 rev

T = 1

T = 0.05 s

The generator's period is 0,05 second.


The generator's frequency  is the number of revs per second of the shaft.

If the shaft does 20 revs per second, the generator's frequency is 20 Hz.

f = 20 Hz


The angular velocity of the object is found from equality in the below.

ω = 2•π•f

ω = 2• 3.14 • 20

ω = 125.6 rad/s


Scan time for 135 degree angle

ω =2•π•f

ω = 2•20π

ω = 40π

If we write 180° instead of π, we find the angular velocity in degree unit.

ω = 40 • 180

ω = 7200 derece/s

The object is scan 7200 degrees angle in 1 second.

For 135 degrees angle,

1 s           7200 revs

x              135 revs

x = 135/7200

x = 0.01875 s

The object is scan 135° angle in 0, 01875 second.

Can we find the period from this result.

Of course,

0.1875 s          135°

t                     360°

t = 360 • 0.1875

t = 0.05 s

We found this result in the A option too.


The frequency of an object that does circular motion is 60 Hz.

Find the angular velocity of this object.


ω = 2•π•f

ω = 2•3.14 • 60

ω = 376.8 rad/s


The period of an object that does circular motion is 0.02 s.

Find the angular velocity of this object.


The frequency and the period are opposite to each other.

T = 1/f

f = 1/T

f = 1/0.02

f = 50 Hz

ω = 2πf

ω = 2•3.14•50

ω = 314 rad/s




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