Centripetal and Centrifugal Forces

DEFINITIONS :-

1) Centripetal Force :- a center-seeking force that causes an object to move in a circular path.

For example, suppose a ball is tied to a string and swung around in a circle at a constant velocity. The ball moves in a circular path because the string applies a centripetal force to the ball. According to Sir Isaac Newton's first law of motion, a moving object will travel in a straight path unless acted on by a force. So, if the string were suddenly cut, the ball would no longer be subject to the centripetal force and would travel in a straight line in a direction tangent to the circular path of the ball (if not for the force of gravity).

As another example, suppose a person is riding on a merry-go-round. As the merry-go-round rotates, the person must hang onto the ride to keep from falling off. Where the person grasps the ride, a centripetal force is applied to the individual that keeps the person moving in a circular path. If the person were to let go, he or she would travel in a straight line (if gravity were absent).

In general, the centripetal force that needs to be applied to an object of mass m that is traveling in a circular path of radius r at a constant velocity v is mv²/r.

When a ball is whirled in a circle, it is accelerating inward. This inward acceleration is caused by a centripetal, or center-seeking, force supplied by the tension in the string. The required force is equal to mv²/r, where m is the mass of the ball, v is its velocity (speed and direction), and r is its distance from the center of revolution. 

2) Centrifugal Force:- An apparent force that seems to pull a rotating or spinning object away from the centre. Its existence depends upon the frame of reference. As it is a pseudo force, obviously it comes into play only when the viewer is in a non - inertial frame.

THE MISCONCEPT :-

Often, centripetal force is confused with centrifugal force. While centripetal force is a real force,—that is, the force is due to the influence of some object or field—centrifugal force is a fictitious force.

A fictitious force is present only when a system is examined from an accelerating frame of reference. If the same system is examined from a non-accelerating frame of reference, all the fictitious forces disappear.

For example, a person ( viewer himself ) on a rotating merry-go-round would experience a centrifugal force that pulls away from the center of the ride. The person experiences this force only because he or she is on the rotating merry-go-round, which is an accelerating frame of reference. If the same system is analyzed from the sidewalk next to the merry-go-round, which is a non-accelerating frame of reference, there is no centrifugal force. The individual on the sidewalk would only note the centripetal force that keeps the individual moving in a circular path. In general, real forces are present regardless of whether the reference frame used is accelerating or not accelerating; fictitious forces are present only in an accelerating frame of reference.

THE WRONG ASSUMPTION :-

Now what some people do is that, they say the body remains in its place while going in circular motion because c.p. force and c.f. force cancel each other. 

Thats not right !!! C.p. force is not a new type of force acting on the body undergoing circular motion. When a body undergoes circular motion, a force ( one among the four forces of nature, e.g. gravitational, electrostatic, magnetic etc, etc ) always acts on the body perpendicular to its motion.

That force has a magnitude of its own (according to the law which defines that force) and that magnitude is just equal to mv²/r. Thats it.

But when anyone observes the circular motion from any non-inertial frame,

the accl^n of the body w.r.t that frame = {vct(F) - vct(f)} / m.

where vct(f) = pseudo force = centrifugal force = m x acclⁿ of. the non inertial frame of reference.

and vct(F) = force acting on the body as seen from inertial frame = m x acclⁿ w.r.t inertial frame = force which causes circular motion ( Force perp. to velocity) which acts as c.p. force ( = mv²/r) in case of inertial frame.

Now if acclⁿ of the body w.r.t the non - inertial frame = 0

then, vct(F) = vct(f)

=> magnitude of Force causing circular motion  = magnitude of c.f. force.

this is the case when the body appears to be at rest..

NOTE:- here we are not writing the force causing circular motion as c.p. force, as we will not be observing the body to go in a circle (which is possible only if we are observing the motion from an inertial frame).

So c.p. force and c.f. force can never act together on a body.