Circular Motion and Physical Equation

Consider about a circular path of radius r. A body moves from A to B with uniform speed v in time t. The body subtends an angle 𝜃 at the centre of the circular path as angular displacement and an arch s as linear displacement.

Figure as regard this is drawn below.

Mathematical calculation:-

Change in velocity along X axis = vcos𝜃 - v
Change in velocity along Y axis = vsin𝜃 - 0
When 𝜃 is very small
cos𝜃 = 1 and sin𝜃 = 𝜃
Therefore,
Required change in velocity along X axis = 0
Required change in velocity along Y axis = v𝜃
The rate of change of velocity per unit time along Y axis is said to be centripetal acceleration.
It is generally denoted by a.

The rate of change of angular displacement per unit time is said to angular velocity w.
The SI unit of angular velocity is radian per second.
Angular velocity = w = 𝜃/t
Centripetal Acceleration = a = v𝜃/t
Therefore,
a = vw ------------------ (i)

In one complete rotation of the body in the circular path. Time taken to move the body to one complete rotation is T.

w = 2π/T ----------------- (ii)
v = 2πr/T ----------------- (iii)

Therefore,
From (ii) and (iii)
w/v = 1/r
w = vr

Now putting the value of w in (i)
a = v²/r
 This is the expression of centripetal acceleration.
If the mass of the body is m.
Centripetal Force = F = mv²/r

Required Physical Equation in circular motion are

1. Linear Displacement = l = 𝜃r
2. Linear Velocity = v = wr
3. Centripetal Acceleration = a = v²/r
4. Centripetal Force =  F = mv²/r