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**Circular Motion**

When a particle moves in a plane such that its distance from a fixed (or moving) point remains constant then its motion is called as the circular motion with respect to that fixed (or moving) point. That fixed point is called centre and the distance between fixed point and particle is called radius.

**For Example:**

The car is moving in a straight line with respect to the man A. But the man B continuously rotate his face to see the car. So with respect to man A

But with respect to man B

Therefore we conclude that with respect to A the motion of car is straight line but for man B it has some angular velocity.

**Kinematics of Circular Motion**

**1. Angular Position **

The angle made by the position vector with given line (reference line) is called angular position Circular motion is a two dimensional motion or motion in a plane.

Suppose a particle P is moving in a circle of radius r and centre O. The position of the particle P at a given instant may be described by the angle q between OP and OX. This angle θ is called the angular position of the particle. As the particle moves on the circle its angular position θ change. Suppose the point rotates an angle Δθ in time Δt.

**2. Angular Displacement **

**➢ Definition **

Angle rotated by a position vector of the moving particle in a given time interval with some reference line is called its **angular displacement**.

Important Points

- It is dimensionless and has proper unit SI unit radian while other units are degree or revolution 2p rad = 360° = 1 rev
- Infinitely small angular displacement is a vector quantity but finite angular displacement is not because the addition of the small angular displacement is commutative while for large is not.
- Direction of small angular displacement is decided by right hand thumb rule. When the fingers are directed along the motion of the point then thumb will represents the direction of angular displacement.
- Angular displacement can be different for different observers

**3. Angular velocity ω**

**➢ Average Angular Velocity**

where θ_{1} and θ_{2} are angular position of the particle at time t_{1} and t_{2} respectively.

**➢ Instantaneous Angular Velocity**

The rate at which the position vector of a particle with respect to the centre rotates, is called as instantaneous angular velocity with respect to the centre.

Important points:

- It is an axial vector with dimensions [T
^{-1}] and SI unit rad/s.- For a rigid body as all points will rotate through same angle in same time, angular velocity is a characteristic of the body as a whole, e.g., angular velocity of all points of earth about its own axis is (2π/24) rad/hr.
- If a body makes `n' rotations in `t' seconds then angular velocity in radian per second will be

If T is the period and ‘f’ the frequency of uniform circular motion.- If θ = a – bt + ct2 then ω = dθ/dt = – b + 2ct

**➢ Relation Between Speed and Angular Velocity**

The rate of change of angular velocity is called the angular acceleration (α). Thus,

The linear distance PP' travelled by the particle in time Δt is

Here, v is the linear speeds of the particle

It is only valid for circular motion.

v = rω is a scalar quantity.

**4. Relative Angular Velocity**

Angular velocity is defined with respect to the point from which the position vector of the moving particle is drawn Here angular velocity of the particle w.r.t. `O' and `A' will be different.

**➢ Definition**

Relative angular velocity of a particle ‘A’ with respect to the other moving particle ‘B’ is the angular velocity of the position vector of ‘A’ with respect to ‘B’. That means it is the rate at which position vector of ‘A’ with respect to ‘B’ rotates at that instant

Important Points:

- If two particles are moving on the same circle or different coplanar concentric circles in same direction with different uniform angular speed ωA and ωB respectively, the rate of change of angle between OA and OB is

So the time taken by one to complete one revolution around O w.r.t. the other- If two particles are moving on two different concentric circles with different velocities then angular velocity of B relative to A as observed by A will depend on their positions and velocities. consider the case when A and B are closest to each other moving in same direction as shown in figure. In this situation

**5. Angular Acceleration α**

**➢ Average Angular Acceleration**

Let ω_{1} and ω_{2} be the instantaneous angular speeds at times t_{1} and t_{2} respectively, then the average angular acceleration α_{av} is defined as

**➢ Instantaneous Angular Acceleration**

It is the limit of average angular acceleration as ∆t approaches zero, i.e.,

Important points:

- It is also an axial vector with dimension [T
^{-2}] and unit rad/s^{2}- If α = 0, circular motion is said to be uniform.
- As

i.e., second derivative of angular displacement w.r.t time gives angular acceleration.- α is a axial vector and direction of α is along ω

if ω increases and opposite to ω if ω decreases.

**Types of Circular Motion**

For better understanding, __we can classify the circular motion in following three__

__types:__

**1. Uniform circular motion** in which v and ω are constant.

In this motion,

a is towards centre, v is tangential and according to the shown figure, ω is perpendicular to paper inwards or in ⊗ direction.

**2. Accelerated circular motion** in which v and ω are increasing. So, a_{t }is in the direction of v and α is in the direction of ω.

In the figure shown, α and ω both are perpendicular to paper inwards. Further,

and

**3. Retarded circular motion** in which v and ω are decreasing. So, a_{t} is in the opposite direction of v and α is in the opposite direction of ω.

In the figure shown, ω is perpendicular to paper inwards in ⊗direction and

α is perpendicular to paper outwards in O direction.

**Note:** In the above figures, θ is the angle between v and a.

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