Dynamics of uniform circular motion

The motion of a body around a circular path at a fixed point is called circular motion. Here, a particle moving along a circular path with constant speed is known as uniform circular motion. If the speed of the particle moving in a circle is not constant . This type of motion is known as non uniform circular motion.

Uniform circular motion 

If a particle moves along a circular path with a constant speed (i.e., it covers equal distance along the circumference of the circle in equal intervals of time), then its motion is said to be a uniform circular motion. Here, the velocity changes but the speed is constant.

Angular Displacement  

The angular displacement of a particle moving along a circular path is defined as the angle swept out by its radius vector in the given time interval.

   𝜃 = Arc Length/ Radius 

𝛳  is measured in radian .

It is a dimensionless quantity.

The angular velocity at any instant is known as instantaneous angular velocity. 

Angular Velocity

     The time rate of angular displacement of a particle is called angular velocity. If Δ𝜃 is the angle 

Δt = time

         ⍵=Δ𝛳/Δt

Time period 

A time period is the time taken by a particle for the complete cycle of the wave to pass a given point.

            T=2𝜋/⍵ 

Frequency 

Frequency is the number of rotations complete in unit time.

                   f=1/T

Angular Acceleration 

Angular acceleration is defined as the time rate of change of angular velocity.

𝛂 =d⍵/dt

Relation between linear velocity and angular velocity  , v= r⍵

Relation between linear and angular acceleration , a = r𝛂

 Radial or Centripetal acceleration

Body undergoing uniform circular motion is acted upon by an acceleration which is directed along the radius towards the centre of the circular path. It is called centripetal acceleration.  

Relation between linear velocity and angular velocity

As Δ𝛳=Δs/r

Dividing by Δt

Δ𝛳/Δt = 1/r (Δs/Δt)

For a very small time interval i.e.,Δt→ 0,

d𝛳/dt = 1/r (ds/dt)

w=v/r

v= rw

Now angular acceleration 𝜶 = d𝟂/dt

dv/dt = d(𝜔r)/dt

dv/dt = r d𝜔/dt

a=r𝜶

Also a = v2/r

            =𝟂2r

In uniform circular motion , the kinetic energy remains constant.

Dynamics of circular motion 

When we observe a particle moving in a circle from an inertial frame of reference,we can observe a non-zero force acting on it. This tends to happen as force acting on the particle in an inertial frame is non-zero which causes the particle to move in an accelerated motion. The acceleration of the particle with a magnitude of v2/r is directed towards the centre of circular path.In the formula v2/r,v defines magnitude of velocity of the particle,r defines radius of the circular path followed by the particle. The non-zero force acting on the particle is directed towards the centre with a magnitude F and satisfies the following equation.

      a=F/m

      v2/r =F/m

      F = mv2/r

As this force is directed towards the centre of the circular path,this is called Centripetal force. Hence to keep particle in circular motion,centripetal force f value mv2/r is required.

It is understandable that a substitute of force acting towards the centre is Centripetal force. Centripetal force arises from some other force such as  tension force,gravitational force etc.

Nonuniform circular motion 

If the speed of the particle moving in a circle is not constant , the acceleration has both the radial and the tangential components. This type of motion is known as nonuniform circular motion.

    a= dv/dt

Circular turnings and banking of roads 

When vehicles go through turnings, they travel along a nearly circular arc. There must be some force which will produce the required acceleration. If the vehicle goes in a horizontal circular path, this force is also horizontal. A vehicle of mass M moving at a speed v is making a turn on the circular path of radius r. The external forces acting on the vehicle are 

       (i) weight Mg

       (ii) Normal contact force N and

       (iii)friction f

If the road is horizontal, the normal force N is vertically upward. The only horizontal force that can act towards the centre is the friction f. This is static friction and is self adjustable. The tyres get a tendency to skid outward and the frictional force which opposes this skidding acts towards the centre. Thus, for a safe turn we must have 

       v2/r = f/M

      f = Mv2/r 

Conclusion 

A particle moves along a circular path with a constant speed, then its motion is said to be a uniform circular motion. The angular displacement is the angle swept out by its radius vector in the given time interval. The rate of angular displacement is called angular velocity and rate of angular velocity is called angular acceleration.