Addendum to Coriolis Force; Definition of Centripetal Force.

Centripetal Force

… Hi **. First off I wanted to answer this earlier. So I apologize for giving a bit delayed answer. But I did think of giving you a suitable answer even while I was walking. Secondly your ideas are highly juxtaposed with each other I am afraid, so I will try my best in making them simpler if I could.

We need to understand first that Force can be categorized into two types. One is called tangential or collinear force. This component of the force is always along the direction of motion and changes speed of an object. It can change direction once the velocity of the object has become zero. Its not centripetal force. It can never make an object go in a plane or 3 D trajectory, as the motion is limited to only one dimension. The object can only go back and forth.

Now look at the other component. Its called a radial force. Its always perpendicular to the direction of motion. This force is called centripetal force, always. Note that its different from what we call central forces.

In consequence, both tangential and radial forces can be central.

This radial force also called centripetal force does not change the speed of the object. The speed is changed by the tangential force. But the radial or centripetal force changes the direction constantly. As a consequence the object is seen to be moving in a circle if observed from a standpoint called fre of reference which in itself is not accelerated.

Because the object moves in a circle, always, under the influence of a radial force the radial force is also called as centripetal force.

Mathematically once the object has a constant speed v but goes around in a circle of radius r, the acceleration is easy to see; from geometry, its calculated to be: a = v²/r. Hence from Newton’s law the radial or centripetal force is: F = mv²/r. Centripetal force is therefore not as arbitrary as tangential force.

The tangential force changes speed and when speed becomes zero direction but only in 1-D. But the radial or centripetal force changes only direction maintaining the speed as long as the force is constant. It therefore makes circular motion possible.

If both tangential and centripetal forces are present the effect is a curvilinear motion, the object moves in a plane. If the radius of the circle is infinite the centripetal force is zero and the object moves in 1-D acted upon by the tangential force and appears to be moving straight.

Hence the apple falling does not have a centripetal or radial component to it’s motion. What it does have is force of gravity acting tangentially to its motion which constantly speeds up the apple. From zero velocity it attains a larger value.

Hence if we consider the apple from a distant star reference to make pur earth an ideal zero-acceleration or inertial frame the centripetal component will be zero as Apple’s trajectory will be a straight portion of an infinite circle.

Once we understand this much it’s to be noted that the circular motion and resulting centripetal motion as given by: a = v²/r is satisfied in a inertial frame.

But if a circular motion is to be described from a pedestal which is let’s say fixed to the object (here the apple) itself then we can no longer say our frame of reference is inertial. Evidently the object in circular motion is accelerated (in case of apple falling down the apple is tangentially accelerated in a straight down motion). In this case the Newtons Law framework is to be reformulated by introducing the concept of the pseudo force.

The pseudo force here is -mg. Note that the pseudo force here is opposite of the force of gravity. Since there is only a tangential force which is mg.

But in a case where we consider circular motion (and not the apple example) the radial force that helped the object to keep a circular path was called a centripetal force given as mv²/r. Hence the corresponding pseudo force is given by -mv²/r. This is called a centrifugal force. This centrifugal force is exactly opposite of the centripetal force. This is necessary because we no more observe the motion as a circular path, but formulate what will be the force experienced by the object in circular motion. Eg the object itself experiences a force opposite to the radial force called centripetal force.

While the apple straight-down experienced -mg the stone in a circle would experience -mv²/r. Both are pseudo forces but the first one is a tangential pseudo force. The 2nd one is a radial pseudo force and called centrifugal force. Hence the centrifugal force is equal and opposite to the centripetal force.

Centripetal force is a real force. Centrifugal force is a pseudo force exactly equal and opposite in direction to the centripetal force.

Similarly tangential force is a real force. But tangential pseudo force is a pseudo force exactly equal and opposite to the tangential force.

Now Coriolis Force is another example of pseudo force. That is it’s not same as any of the 4 forces we discussed above. It’s not centripetal force. Its not centrifugal force. Its not tangential force and it’s not tangential pseudo force.

Its a pseudo force that comes from the fact that there is a centripetal acceleration given by

a = ω²r. The Coriolis force is then a pseudo force given by: ωv where v is the speed of the object.

If the object is stationary v = 0 and Coriolis force is evidently zero. But all the other forces may not be zero.