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How to Derive Period From Angular Velocity

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In this post, I will help you understand how to derive the formula for the period from angular velocity.

Definition of Angular Velocity

Angular velocity is the ratio of the angle turned through to the elapsed time. We can also define angular velocity as the change in the angle per second. The unit of angular velocity is radians per second (rad/s or rads-1).

From velocity, we know that velocity, v = displacement (s) / time (t)

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v=s/t

and s is the displacement which represents the length of the arc. In a circle, we have an arc s, the circle of the radius r, and the angle θ.

Now, if we allow θ to represent v and plugin r instead of t, we will get

 θ = s/r

When we cross multiply the above expression to find s, we will end up with

s = θr

We can now go ahead to divide the above equation by time t to obtain

s/t = θr/t

We can clearly see that s/t (which is the ratio of displacement to time) is defined as the velocity of an object (v). Thus, we can now define the above expression as

v = s/t = θr/t

v is the linear velocity of the body

The above expression is the same as

v = (θ/t) x r

But θ/t is equivalent to  ω (which is the angular velocity)

Hence,

 v=ωr

The above formula is the relationship between linear velocity and the angular velocity in radians.

To arrive at the formula for period T, we can say that

T=2π/ω [where 2π represents the circurmference of a circle which is 2 x 1800 = 3600]

But ω = 2πf

Which implies that

T = 2π/2πf = 1/f

Thus, the period is defined as

T = 1/f [where f is the frequency]

Hence, this is the method we apply to understand how to derive the formula for the period from angular velocity.

You may also like to read:

How to Calculate Angular Velocity in Radians per Second

Also, How to Calculate Centripetal Acceleration

How to Calculate Centripetal Force

Reference

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