Circular Motion

As we all know , motion is the change in position of an object with respect to its surroundings . Circular Motion is the change in position of an object with respect to its surroundings in a circular path . When an object moves along the circumference of a circle , we call it as Circular Motion . If the body moves in a circular path with a uniform speed , we call it as "**Uniform Circular Motion**"
Please Note : Circular Motion and Rotational Motion are different phenomenon .
Circular Motion has two components
1) Angular
2) Linear

The basic difference between Angular and linear is that in linear dimension , quantities like displacement , velocity and acceleration are measured with respect to its length ; while these quantities are measured with respect to its angle in angular dimension .

## Displacement

### Linear Displacement

Linear Displacement is the distance covered by the object along the circumference of a circle (Arc Length) .

```
  Linear Displacement(s)= 2πr

```

### Angular Displacement

Angular Displacement is defined as the angle subtended by the radius vector with an object performing Circular Motion , at the centre of the circle , in given time . It is denoted by θ .

```
   Angular Displacement (θ) = s/r

```

Angular Displacement is the change in angle while linear displacement is the change in length .

## Velocity

### Linear Velocity

It is the rate of change of distance travelled .(circumference).

```
  Linear Velocity (v) = Displacement / Time

```

.

### Angular Velocity

Angular Velocity is defined as the rate of change of limiting angular displacement .

```
 Angular Velocity (ω) = θ / t

```

Thus , linear velocity can be called as the speed of the object while angular velocity can be called as the rotational speed of the object .

## Acceleration

### Linear Acceleration

Linear Acceleration is the rate of change of linear velocity .

```
  Linear Acceleration (a) = Velocity / Time = Distance / (Time)2

```

### Angular Acceleration

Angular acceleration is the rate of change of angular velocity with respect to time .

```
  Angular Acceleration (α) = ω / t

```

### Relation

a = αr

The direction of radial acceleration is along radius while angular displacement , velocity and acceleration are perpendicular to the plane of circular motion .

## Unit Vectors along Radii and Tangents of a Circle

er= icosθ + jsinθ

et = - isinθ + jcosθ

## Uniform Circular Motion

Uniform Circular Motion is defined as the motion of an object performing circular motion with constant speed .
An object performing circular motion has a linear velocity which remains constant in magnitude but its direction changes along the circular path . Thus , we can say that there is a change in velocity of the object . We can say this because velocity is a vector quantity given by magnitude and direction ; thus even if only the direction changes , the velocity also changes .

As there is a change in velocity ,there is an acceleration in circular motion . This acceleration changes only the direction of the velocity vector and has no effect on the magnitude of the velocity vector (i.e. speed remains constant) . This acceleration is called as tangential acceleration (since tangent gives direction of an object on a circular path at a certain instant) . The tangential acceleration acts along the radius of the circle of motion , towards the centre of the circle . Thus , it is also called as radial acceleration or centripetal acceleration .
It is given by :-

```
  Radial Acceleration (ar) = v2/r = rw2

```

In UCM , speed , angular velocity , radial acceleration , linear and angular momentum remain constant .

## Non-Uniform Circular Motion

A = sqrt (at2+ ar2) [ For UCM , tangential acceleration is zero ]

= r sqrt (w4 + α2)

## Centripetal Force

When a linear force acts on a body , it moves in a line , but this force cannot create circular motion . We need a force that acts in the inward direction to produce Circular Motion . The Force which acts in the inward direction to produce Circular Motion is called as Centripetal Force . Like centripetal acceleration , Centripetal Force , too acts in the inward direction i.e. towards the centre of the circle along the radius .
We get the formula for Centripetal Force by just multiplying the mass of the body by it's centripetal acceleration .

```
  Centripetal Force = mv2/r = mrw2

```

Centripetal force is a real force , since the nature and origin of the force is known .
Centripetal force can be observed in different forms

 Electron around Neutron
 Car around circular track
 Stone tied and whirled around

## Centrifugal Force

Centrifugal force is a pseudo force that acts in the outward direction while performing circular motion . The most common example of centrifugal force is the inertia experienced by an object while performing circular motion . In the case of uniform circular motion , only inertia of direction is merienced , since it is moving with uniform speed but the direction is changing .

## Application of Circular Motion

### **Conical Pendulum**

Time Period = 2π sqrt (l/g)

Tension = mg sqrt [ 1 +(r/h)2]

Important : h2 + r2 = l2  is to be used if needed

### **Banking of Roads**

Suppose a vehicle is moving with a very high speed on a road . We all know that if the vehicle takes a turn at a high speed , it will skid away from the road . In this case , the vehicle skids away from the road due to an outward centrifugal force . Thus , the roads are inclined slightly at a particular angle so as to minimize the centrifugal force and optimize the centripetal force .

For circular unbanked road , vmax = sqrt (μrg)

vmax = sqrt (Rg [us+ tanθ / 1 - ustanθ])

## Vertical Circular Motion

 Mid Point
 Lowest Point

v = sqrt (u2 - 2gh) = sqrt [u2 - 2gr(1 - cosθ)]

For sending an object from height 'h' to a circular loop of radius 'r' , then the height should be h = 5r/2

## Kinematical Equations for Circular Motion

1. ω = ωo + αt
2. θ = ωot + (1/2)αt2
3. ω2 = ωo2 + αθ

Angular displacement in nth second

θn = ωo + (α/2)(2n-1)