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.

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.

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.

Relation
v = ωr

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 (at2undefined+ 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

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.

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 [usundefined+ tanθ / 1 - ustanθ])

Vertical Circular Motion
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
Angular displacement in nth second
 * 1) ω = ωo + αt
 * 2) θ = ωot + (1/2)αt2
 * 3) ω2 = ωo2 + αθ

θn = ωo+ (α/2)(2n-1)