Uniform Circular Motion 2

Relation Between Angular Displacement and Linear Displacement–

Since, Angle = arc/radius

Anglular Displacement = arc P₁P₂/radius

θ = s/r

Angular Velocity – Rate of change of angular displacement of a body with respect to time is known as angular displacement. It is represented by ω.

Average Angular Velocity – It is defined as the ratio of total angular displacement to total time taken.

ω_avg = Total Angular Displacement/Total Time Taken

ω_avg = Δθ/Δ t

Instantaneous Angular Velocity – Angular velocity of a body at some particular instant of time is known as instantaneous angular velocity.

Or

Average angular velocity evaluated for very short duration of time is known as instantaneous angular velocity.

ω = Lim ω_avg = Δθ/Δt

ω = dθ/dt

Relation Between Angular Velocity and Linear Velocity We know that angular velocity

We know that  angular velocity

ω = dθ/dt

Putting, θ = s/r

ω = d (s/r)/dt

or, ω = 1 ds/r dt

or, ω = v/r

or, v = rω

Time Period of Uniform Circular Motion – Total time taken by the particle performing uniform circular motion to complete one full circular path is known as time period.

]In one time period total angle rotated by the particle is 2p and time period is T. Hence angular velocity
ω = 2p/T or, T = 2p/ω   

Frequency - Number of revolutions made by the particle moving on circular path in one second is known as frequency.
f = 1/T = ω/2p

Centripetal Acceleration – When a body performs uniform circular motion its speed remains constant but velocity continuously changes due to change of direction. Hence a body is continuously accelerated and the acceleration experienced by the body is known as centripetal acceleration (that is the acceleration directed towards the center)

 Note: Here p is pie (π)