Circular motion

In this topic we will discuss about what is circular motion, angular displacement,angular velocity, angular acceleration, relation between angular velocity- linear velocity ,and angular acceleration-linear acceleration . Also we will discuss about centripetal acceleration.

  Circular motion-

When during the motion body moves on a circular path then the such kind of motion of the body is called circular motion .

Some important topics of circular motion:-

1. Angular displacement –  When a body moves on a circular path then the angle traced out by the radius vector at the axis of the circular path in a given time is called angular displacement .

Suppose a body is moving on a circular path of radius ‘r’ , in anticlockwise direction in the plane of paper , with center ‘O’ . let the position of the object changes from P to Q in time t . let <POQ = ϴ  , as shown in figure .

Since,  angle = arc/radius

So we can write   ϴ=PQ/r .

Angular displacement is a vector quantity . Its unit is radian and it has no dimension.

2. Angular velocity – When a body moves on a circular path then angular velocity is defined as the rate of change of its angular displacement . it is denoted by ‘ω’ (omega) .

Its unit is radian/second , and its dimension is [M⁰L⁰T^-1]. It is a vector quantity.

Suppose a point object moving along a circular path of radius r and centre O . let object moves from P to Q in time dt . and <POQ = dϴ .

Then angular velocity  ω= dϴ/dt .

Relation between linear velocity and angular velocity –

Suppose an object is moving with uniform angular velocity ω and linear velocity v  on a circular path of radius r with centre O .

 

Object is at P at time t and after time Δt it reaches at Q . <AOP=Δϴ The length of PQ=Δl

Therefore v= Δl/Δt   or, Δl=vΔt

But , angular velocity ω=Δϴ/Δt  or, Δϴ=ωΔt ,

As we know angle = arc/redius

So, Δϴ=Δl/r  or ωΔt=vΔt/r or  v=ωr ;

 

Angular acceleration –

 In a circular motion angular acceleration is defined as the time rate of change of its angular velocity .

it is denoted by α . Its unit is rad. S^-2 . And its dimension is [M⁰ L⁰ T^-2].

Relation between linear velocity and angular acceleration – as we know v=ωr

So  angular acceleration α= dω/dt =d(v/r)/dt = dv/rdt= a/r ( a= angular acceleration a= dv/dt)

So , a= αr ;

Centripetal acceleration –

In a uniform circular motion , the velocity vector of the object is changing with time . it indicates that the uniform circular motion is the example of circular motion.

Acceleration acting on the object undergoing uniform circular motion is called centripetal acceleration . It always acts along the radius and towards the center.

Suppose a particle of mass m moving with a constant speed v and uniform angular velocity ω around a circular path of radius r .

Let at time t the point is at p and after time Δt it reaches at Q .

Here OP=r₁ and OQ = r₂ and , <POQ = Δϴ

Here, angular velocity  ω=Δϴ/Δt ;

Let v₁ and v₂ are the velocity at position P and Q respectively . as shown in figure (a) .

Here magnitude of PA and QB are equal which is equal to v.

To find the change in velocity in time interval Δt . we draw p’A’ and P’B’ respectively the velocity vector v₁ and v₂ as shown in above  figure (b) .

Here A’B’ = Δv ,

From figure (b)

Δϴ=A’B’/P’A’ = Δv/v

ω Δt= Δv/v ;

ω v = Δv/Δt ;

(v/r)v = a_c

a_c = v²/r =ω²r