Classical Physics - Circular Motion

Circular Motion

Circular motion may be divided into two types (a) motion in a horizontal circle (b) motion in a vertical circle. The basic differed being that in case of vertical circle acceleration due to gravity plays a role at every point and changes the speed at every point. We will deal with them separately. 

Motion in a horizontal circle: - here acceleration acts continuously even when speed is uniform, therefore, impel equation of motion as used in translation motion cannot be applied. We define new variables and equations to describe motion. 
    
Angular displacement (θ) change in angular position (initial to final)is called angular displacement.
    
Angular velocity (ω) time rate of change of angular displacement is called angular velocity. 

Thus ω = dθ / dt
    
Angular acceleration (a) time rate of change of angular velocity is called angular acceleration. 

α =dω / dt = d²θ / dt²

if the angular acceleration is constant then 

ω = ωo + αt

ωav = 2π / T

Where T is time period to completer circle.

θ = ωot + 1 / 2 αt2
 
ω² – ω²o + 2 αθ

Relation between v and ax a and α

V = rω where r is radius of the circular path. 

αt = rα where at is tangential acceleration as explained in 

Centripetal or radial acceleration ar = v² / r = rω²

Net acceleration a_net = √a²t + a²r tan β = ar / at

Centripetal force F = mv² / r = mrω². It is a pseudo of orca. It acts towards the centre. 

Centrifugal force the initial reaction required to take into account the acceleration of frame of reference is called centrifugal force. It is equal to – mrω²

Motion in a vertical circle when a body moves in a vertical circle then at the highest point 

mv² / r ≥ mg v_min = √rg at highest point 

v_min at any point P can be determined using the fact that the body has come down by a vertical distance 

AX = AO + OX = r + r cos θ (form)

Thus v²_min p = v²_min A + 2 gh

= rg + 2g (r + r cosθ)

= 3 rg + 2rg cps θ

v_minp = √3rg + 2rg cos θ

v_min at lowed point or bottom is obtained by using θ = 0

That is, v_min bottom = √5rg

Tension, if the string is used or normal reaction at any point P is obtained as 

T or N = mv² / r + mg cos θ

Where v is velocity at that point.  

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