Circular Motion

Circular Motion In case of Circular motion speed of the body remains constant but the direction of motion always changes or we can say that its velocity doesn’t remains constant. The direction of motion or velocity at any instant of time is given by the tangent drawn at any point on the circle. The component of acceleration along the direction of velocity remains zero. In case of circular motion force always acts on the body towards the center of the circular path, this force is known as Centripetal Force. Therefore the direction of acceleration also acts towards the center of the circular path, this acceleration is known as Centripetal Acceleration, Its direction changes but the magnitude remains constant.

Acceleration = Angular velocity × Linear velocity a=ωv Linear velocity = Angular velocity × Radius of the circular path v=ωr a = ω2r Thus Force acting in case of circular motion = Centripetal Force = F F = Ma = M ω2r F = (Mv2)/r

Motion of a Satellite: Satellite: Satellite is a body which revolves around a Planet in a definite orbit with definite speed. The velocity of the Satellite in an orbit is called Orbital velocity. When a Satellite revolves around the circular path, the necessary centripetal force is provided by the Gravitational force. Mass of the Satellite = m Distance of the Satellite from the Earth = h Orbital velocity of the Satellite = v Mass of the Earth = M Then From the Newton’s law of Gravitation (Mv2) / R = (GMm)/r2 v2 = (GM)/r v = √ [(GM)/r] If the radius of the Earth = R r = (R + h) v =√ [(GM)/(R+h)] Acceleration due to gravity = g = (GM)/R2 gR2 = GM Hence v =√ [(gR2)/(R+h)] We see that the orbital velocity v doesn’t depend upon the mass of the Satellite.

Motion of a car on a level circular Road: Suppose a car of mass m is moving on a circular track of radius r. If the coefficient of friction between the tires and the road is μ, the maximum velocity with which the car can turn of radius r is given by v.

N = mg Frictional force = f = μmg = (mv 2)/r Velocity of the car = v = √ (μgr)

Motion of a car on a Banked circular Road: Suppose a car of mass m is moving on a banked circular track of radius r. If the track is banked through an angle θ then

Or,

Nsinθ = (mv2)/r Ncosθ = mg tanθ = v2/ (rg) θ = tan-1[v2/ (rg)] ______________________________________