# Circular Motion | Definition, Equations, Formulas, Units – Motion in a Plane

Circular Motion Definition
Circular motion is the movement of an object in a circular path.

We are giving a detailed and clear sheet on all Physics Notes that are very useful to understand the Basic Physics Concepts.

## Circular Motion | Definition, Equations, Formulas, Types, Units – Motion in a Plane

Circular Motion Types:
1. Uniform Circular Motion Definition:
If the magnitude of the velocity of the particle in circular motion remains constant, then it is called uniform circular motion.

2. Non-uniform Circular Motion Definition:
If the magnitude of the velocity of the body in circular motion is not constant, then it is called non-uniform circular motion.
Note:
Spinning Motion Definition:

A special kind of circular motion where an an object rotates around itself is called as spinning motion.

Variables in Circular Motion
(i) Angular Displacement Definition:
Angular displacement is the angle subtended by the position vector at the centre of the circular path.

Angular Displacement Formula
Angular displacement (Δθ) = $$\frac{\Delta s}{r}$$

where, As is the linear displacement and r is the radius.

Angular Displacement Units

(ii) Angular Velocity Definition:
The time rate of change of angular displacement (Aθ) is called angular velocity.

Angular Velocity Formula
Angular velocity (ω) = $$\frac{\Delta \theta}{\Delta t}$$
Angular velocity is a vector quantity

Angular Velocity Units

Relation between linear velocity (v) and angular velocity (ω) is given by
v = rω

(iii) Angular Acceleration Definition:
The rate of change of angular velocity is called angular acceleration.

Angular Acceleration Formula:
Angular acceleration (α) = $$\lim _{\Delta t \rightarrow 0} \frac{\Delta \omega}{\Delta t}=\frac{d \omega}{d t}=\frac{d^{2} \theta}{d t^{2}}$$

Angular Acceleration Units

Angular Acceleration Dimensional Formula
dimensional formula is [T-2].

Relation between linear acceleration (α) and angular acceleration (α)
a =r α
Relation between angular acceleration and linear velocity
α = $$\frac{v^{2}}{r}$$

Non-uniform Horizontal Circular Motion
In non-uniform horizontal circular motion, the magnitude of the velocity of the body changes with time.
In this condition, centripetal (radial) acceleration (aR) acts towards centre and a tangential acceleration (aT) acts towards tangent.
Both acceleration acts perpendicular to each other.
Resultant acceleration, $$a=\sqrt{a_{R}^{2}+a_{T}^{2}}=\sqrt{\left(\frac{v^{2}}{r}\right)^{2}+(r \alpha)^{2}}$$
and

$$\tan \phi=\frac{a_{T}}{a_{R}}=\frac{r^{2} \alpha}{v^{2}}$$
where, α is the angular acceleration, r is the radius and v is the velocity.

### Kinematic Equations in Circular Motion

Relations between different variables for an object executing circular motion are called kinematic equations in circular motion.
(i) ω = ω0 + αt
(ii) θ = ω0t + $$\frac{1}{2}$$αt²
(iii) ω² = ω0² + 2αθ
(iv) θt = ω0 + $$\frac{1}{2}$$α (2t -1)
(v) θ = $$\left(\frac{\omega+\omega_{0}}{2}\right)$$t
where, ω0 = initial angular velocity,
ω = final angular velocity,
α = angular acceleration,
θ = angular displacement,
θt = angular displacement at t seconds and t = time.

Motion in a Plane (Projectile and Circular Motion):
In this chapter or under this topic, we are going to come across the motion of the object when it is thrown from one end to another end. This practice is said to be projection. Also, when an object is moved in a circular motion, then the equation of the motion is derived here. We will learn here about centripetal force and centripetal acceleration in detail with formulas. Also learn the force applied in everyday life motion of the particle in a vertical circle.