# Coriolis Force

I have been thinking of writing a basic introduction, on this important physical concept — of, what’s a Coriolis Force.

Its one of the interesting ideas of physics, which does not get as much of a mention, unless you just happen to know this, because of your advance footing, in the discourses of Physics; perhaps because its two orders of magnitude smaller, in its strength, compared to relevant forces, in a situation, that involves this small — but, significantly measurable force.

So, what’s a Coriolis Force?

I presume that most of us would be aware of, what’s centrifugal force. Its in this context, that, a Coriolis force is most understandable.

So, what’s a centrifugal force?

A Centrifugal force, comes into picture, every time a centripetal force comes into consideration. A centrifugal force is the corresponding pseudo-force, of a centripetal force. So, for every centripetal force there would be a corresponding centrifugal force. So, we need to shed light on a few things, before we understand, what are Coriolis Force. We need to therefore discuss; what are pseudo forces and what’s a centripetal or a centrifugal force, then a Coriolis force would be clearer, to the root.

So lets begin with Force.

Force has a basic standing in Physics, in connection to, in what frames of reference we need to measure such a physical quantity, as are all other physical quantities to be measured, have a significance, as to; in what frames of reference we are measuring those quantities in.

### Frames of references as velocity vector.

So what are frames of references and why are they important?

Frames of references, are usually a set of 3 perpendicular axes, imagined to be attached, to a body, wrt which an observer makes his measurements, of physical quantities. If a car is moving on a road, one can attach, such a 3-axes system, to the body of the car — or even to an object, on the road, that’s not moving. In other words, the set of axes, will have a velocity or speed of zero, if the axes is attached, to an object, that’s not moving.

Photo Credit; Wikimedia
A Cartesian coordinate system with a specification of velocity is the only thing needed to define the basic Frame of Reference.

Consequently, a set of axes, has a speed v, if its attached to a body — that’s moving at that speed. So the frame of reference, is going to inherit the speed of the object, that it is attached to. As a consequence, a frame of reference, is nothing but an imaginary set of axes, that has an assigned value of speed v. — v stands for velocity, although in a lose sense, because it can supply the value of speed while direction information can be consistent with the larger picture of the description of the problem.

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Frames can be attached to cars, for reference of motion and speed of the car becomes that of the frame of reference.

Photo Credit; Hesston.edu
The frames of reference attached to a car and attached on ground tell us different stories of the same phenomena.

Needless to say, this — frame of reference as a set of imaginary axes, with an assigned value of velocity, is the most general definition of a frame of reference, since this speed or velocity v; can be zero, can be increasing or decreasing, can themselves be rotating, that is, the vector associated with the speed or velocity , will be a function of time, in such a way, that, the tip of the arrow will show different angles, wrt the set of axes.

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Rotation of frames is simply a velocity vector which is doing the same thing.

Since the velocity of the frame of reference, is a generally possible value, in its magnitude and direction, one can say, a frame of reference is simply an arbitrarily possible value of a vector, namely velocity vector.

Photo credit; uwindsor.ca
1. A missile observed from ground, frame has speed 0
2. Missile observed from an airplane, frame has speed of airplane.

Thus you or I or any privileged Physicist, will have the hindsight, to associate this velocity, to any object, in the surrounding that he or she thinks is suitable; for his purposes of measurements. So if NASA or ISRO is sending a space voyage, the most generally possible “velocity of the earth” would serve as, a good definition of a Frame of Reference.

Photo Credit; wp.com
Space Voyage

A frame of reference, would then, no longer be, any different, technically, from the velocity of the earth, howsoever its possible. This can only become more and more general, if we keep on arguing, why earth’s velocity can itself, be measured, from a multitude of other objects in space.

Photo Credit; researchgate.net
An earth centered earth fixed system of frame. This is good for space voyage at least initial stage of it.

This “other objects in space” can be, the centre of the solar system, or the fairly distant star. Depending therefore on the generality of the velocity, of the earth, the frame of reference, will have, more or less precision, associated with it, for a particular measurement, to be carried out. Eg, one may neglect the spin — or self-rotation of the earth, in certain earthbound measurements.

Photo Credit; borderlandresearch.com
The celestial ecliptic plane. This reference frame is fixed with sun, hence its good for measurements that concern beyond our planet and deep into solar system space or even beyond.

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The celestial equatorial system.

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Jupiter’s motion is traced in the background of very distant stars.

Eg For a train jostling between Nebraska and its nearby city of Hullabulla, someone may not deem it relevant, to consider earth’s spin velocity, into the frame of reference that someone chooses, for determining the train’s speed. But in certain measurements, such as the flight touring from NY city to Hamburg, one might consider it a significant effect, to include the speed of self-rotation of earth, which is about half-a-km-per-second.

Photo Credit; eeb.ucla.org Nezlin’s lecture
The movement of water away from point B is influenced by the Coriolis effect and gravity. Earth’s rotation around its own axis is called earth’s spin and occurs at ~0.5 kms/s.

Photo Credit; miamisci.org
The effect of earth’s spin on motion is significant for oceanic and terrestrial transit such as airplane.

So, the concept of a Frame of Reference as nothing but merely a velocity vector, is one of the most generally available definitions of such. Given we see what’s a frame of reference, the next question is what bothered a lot of Physicists.

What are the most basic type of frames of reference?

There are two.

One would be where Newton’s Laws would be valid.
These are called as; Inertial Frames of Reference.
The other would be where the Newton’s Laws would be valid only if we make a small correction to the physical quantities.
These are called as; Non-inertial Frames of Reference.

Newton was certainly lucky, that, Newton’s laws were made to retain its sanctity, by slightly changing the nature or definition of quantities. (put a little more salt, okay now its good)

This becomes more and more difficult, as we pass along the generation, from Newton to Einstein and then from Einstein to Feynman or Bohr. They each make the best in their times, but when we fall back, to the past, to ordain some corrections, to any lingering misunderstanding or inconsistencies, in their theories or understanding, we face bigger and bigger challenges, because the laws themselves have become more and more precise, fulfilling, complicated and self-serving.

So when Bohr’s model was corrected, it was no more a simple trick, but generations of most intelligent scientists harking back and forth between themselves, in the most intellectual rigor possible, the very ideas, which were refined, not just by a dint of technical innovation or heuristic, but also the most uncanny philosophical moorings.

An example in mind of the intellectual rigor and uncanny philosophical moorings, is, the famous debate between Bohr and Einstein about the validity of Quantum Mechanics as a general law of nature.

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Bohr’s atomic radiation.

Photo Credit; phys-astr.gsu.edu
Bohr’s model predictions.

Bohr’s model, while correct, explained only the Hydrogen like configuration, Schrodinger’s and Heisenberg’s and Pauli’s more universal principles, were needed, to see why Bohr’s model was just a necessary consequence, but not completely correct. Bohr’s model met the same fate, as that of Newton’s Laws; at the hands of its successors. We not only stand on the shoulder of giants, but at times, we remodel their shoulders, so that they become more and more suitable, for our purposes. Sometimes we have to make them into staircases and take them again and again, to prove our point.

We will now discuss in detail one by one the following concepts which bear significant conceptual relation among each other.

Inertial Frames of Reference.Newton’s 3 laws.Central and non-central forces.Strong and weak form of Newton’s 3rd law.Principle of angular momentum and momentum conservation.Early unification.

So the type of frame of reference, where Newton’s laws, as they were formulated by Newton are valid, are called as Inertial Frame of References. Since I told you, a frame of reference is nothing but exclusively a particular definition of a suitable velocity; this means, Inertial Frames are frames, that are defined by associating them, with bodies, that move at a uniform velocity. An inertial frame of reference, is an uniform velocity. An uniform velocity is one, that does not change its direction or speed. An unchanging speed or direction is thus, the only characteristic of a Newtonian or inertial frame of reference, — lets abbreviate as IFR.

So, Inertial Frame of Reference is where both, Newton’s 1st and 2

nd

law are valid.

The first law;
objects tend to keep their direction and speed, as long as there are no external agent causing them to change. We say objects have their inertia.
The 2nd law;
the rate of change of momentum of an object, is nothing but that external agent and is called as a Force. — It also has another important and in some sense exclusive definition; the Force is mass times the acceleration.
The 3rd law;
the force acted onto a body — by another body, will always meet an exactly opposite force, by the body that experiences the force, one is called as action and the other is called as reaction. Thus for every action there is an equal and opposite reaction force.

The 1st and 2nd law are valid in the frame of reference in which the velocity of the frame of reference is not changing or uniform. Such a frame is defined to be an inertial frame. Thus, strictly speaking, an inertial frame is one where Newton’s 1st and 2nd law are valid, without further modification, into the quantities or their relationship. A relationship between physical quantities, is often called as a law, if such a relation is an experimentally observed phenomenon in nature.

The 3

rd

law has two forms, in which the law can be stated.

One is of more general form. It is called as a weak form of the law.The other is less general or strict.It is called, for that reason, a strong form of the law.

Lets play some semantics to analogize what we mean by the semantics weak and strong, before we see what they mean in respect of the physical concepts that we are describing.

Weak law;
you are not allowed to move outside.
Strong law;
you are not only not allowed to move outside, but also, you are to cook a meal while you are inside.
In a similar fashion.

Weak form of Newton’s 3rd law;
Forces of interaction, between two exclusive bodies, are equal and opposite.
Strong form of Newton’s 3rd Law;
Forces of interaction, between two exclusive bodies, are not only equal and opposite, but also directed along — that is parallel or anti-parallel, to the line joining the centers of the point bodies that are interacting.

Consequently the forces are also called for the same reason; central forces, if they satisfy the strong form of the 3rd law.

I not only pull you, but also, I pull you toward my center and not in another unattractive fashion. One example of such central forces is therefore the force of gravity. The apple in the absence of wind and other dissipation factor, will fall exactly straight to the ground, because that’s the radial direction, to the center of the earth. If Gravity were not central, the apple upon falling, would trace a path, which is going to be off the straight-down. So in that scenario, we  can happily sit under the apple tree, unmindful of gravity.

If physical analogy to worldly affairs is your forte, I make it a little resilient, by this comparison, Love is not a central force; because of my attraction, my paramour is attracted towards someone else. Love is the reason, why my attraction, towards others, causes them to avoid me, love originates non-centrally for me.

Evidently not all forces are central hence they do not obey the strong form of Newton’s 3rd law always. Also its possible by ordinary forces, such as magnetic forces, to disobey both the weak and strong forms of Newton’s 3rd law. As a consequence, they disobey Principle of Momentum and Angular Momentum conservation.

Forces can be made to be disciplined, by generalizing these force quantities and their existent relations called as laws, in some consistent fashion. For magnetic forces, this is done, by recognizing that, there is an additional form of angular momentum that’s inherent in magnetic forces.

But the central forces and weak and strong form of Newton’s 3rd law, have nothing to do, with the inertial and noninertial frames, just that these are two different examples, of how methods are applied to Newton’s laws, to make them more and more universal.

So sometimes, we have to readjust, either the quantities, and, or — their inherent internal relationships, called as law; to make them more universal. Both of these have been done to Newton’s Laws, to preserve their universality.

Example 1;
Inclusion of pseudo force into the formalism, a quantity is redefined to establish the validity of Newton’s laws.
Example 2;
Redefinition of angular momentum into the formalism, to re-establish the law of conservation of momentum and angular momentum.

This is also a good example of what’s meant by; unification in Physical Laws. The enactment of universality, into physical laws, by making them more consistent, called as unification, is a general attribute of Physical Understanding. Two of them have been performed to Newton’s Laws. The first one, which we will discuss in a bit more detail, is the simplest of its kind.

So we recapitulate; and discuss some more interesting concepts. The Inertial Frame is nothing but a uniform or unchanging velocity — associated with an arbitrary object of choice. Under such a condition, Newton’s 1st and 2nd law do not throw exceptions. — throwing exceptions is not only language of software programming, but early day Physics.

I don’t know when it was made sure that Newton’s laws can be recast by redefining quantities in it, to preserve its sanctity when frames of references are altered. Some amount of historical perspective is  must when we are to have a better grasp of any subject of study.

The noninertial frame or changing velocities, are what make Newton’s 1st and 2nd law behave a little awry. The  1st and 2nd law are not exactly valid, in such frames. It’s a pseudonym to call forces, in noninertial frames as “inertial forces”, its misleading, as these forces are not being considered in an inertial frame, while all we are considering, is, a noninertial frame.

Of-course, non-inertial frames and inertial frames, come hand-in-hand with each other, in many physical situations, but that does not remove the confusion. They should rather be called as “Forces from inertial consideration” because, if anything, the forces exist, irrespective of what frame we are choosing, to carry the measurement.

These forces, might just not retain, the same value, in one frame, that they do in another. So the forces present in the inertial frame, are also active, in the noninertial frame. But there are effects in the noninertial frame, that these forces would not explain, if we are to utilize the Newton’s 1st and 2nd law — or, with added conditions of consistencies and generalities, the 3rd laws as well.

These noninertial effects, would seem to be coming, from nowhere, but these are present, when the frame from which we are measuring the physical phenomena, is having a velocity, that is changing with the instant of time, a reason and definition of calling such frames as noninertial frames.

The velocity can change in two ways, since it has, only two parameters, the magnitude and the direction. When magnitude changes, we say the object is speeding up or down and this change is always so, due to a Force, that’s the essence of Newton’s 1st law and the 2nd law is telling us exactly how.

Speeding up is called as acceleration and speeding down is deceleration, but in general, both are called as acceleration because, acceleration is again a vector and has two parameters. The change in direction of velocity, is also regarded as a change due to acceleration, as acceleration has a direction, which tells us; how the velocity vector will change its direction.

All in all we say, when velocity, a vector, is changing — a change in the inertia of an object that is, its due to presence of an acceleration, and acceleration again has two parameters. So all change in velocity vector, its direction or magnitude, are due to an acceleration. The acceleration is what gives us exactly how much is the force — the agent that causes such a change, as contained in Newton’s 2nd law. Recall that we had stated the 2nd law in two exclusive forms, rate of change of momentum is the force and also mass times acceleration is the force.

Its for the above reason, a noninertial frame is also called as an accelerated frame of reference. Without acceleration, the velocity — or state of inertia, would not change. Hence an inertial frame is an uniform velocity or zero acceleration. And a noninertial frame is a nonuniform velocity or accelerated — that is, non-zero acceleration, frame.

## Pseudo Forces.

The complicacy came into play, in the accelerated or noninertial frame — again a noninertial frame is where the state of inertia — that is velocity, is changing. The complicacy is, there are effects, to be observed in such frames, which are not understandable as such, from the presence of forces in the inertial frame — forces of inertial consideration, in literature these are misleadingly called as inertial forces and what Newton’s 3 laws say these forces of inertial consideration do.

The extra effects are thus called as pseudo effects. They can be understood, in the non-inertial frames, only-if, they are associated with the non-inertial condition — that is, condition; the state of the inertia of the frame of reference is changing. Since this is caused by acceleration of the frame of reference, the clue is to formulate a force, that really does not exist but effectively brings back Newton’s laws into consistency, in explaining these effects.

All we need to do is, consider a new force, as opposed to take a real existing force — real existing forces are what are called as “forces of inertial consideration” or, as per literature; inertial forces — and these are always real, and these new kind of forces, the non-real ones, are equal and opposite to the vector of acceleration — multiplied by the mass of the object, that would experience such an effect.

This force is thus called as Pseudo force. A Pseudo force thus depends on what frame we chose, which is accelerating. A Pseudo force is not an actual force and merely hypothetical, it reinstates the sanctity of Newton’s Laws, just in case it was broken — because the frame of references are accelerated. We must be cautious in choosing the mass which is the mass of the object that is experiencing the effect. That’s not all. The Pseudo Force also depends on the state of inertia of the object in that noninertial frame. Again by state of inertia we mean velocity vector.

Many times authors due to plain-speak mean mass, when they say inertia, that’s only part of the problem, and not the whole problem, the whole problem is velocity (vector) and where we are measuring that velocity from, welcome to Physics, because that where we are measuring velocity from is acceleration (vector) and so on.

So the effect that we would call as pseudo-force effect; has 3 factors, that decide, what this force is going to be.

The acceleration of the frame of reference, a .
This is a vector.
The mass of the object, on which we are considering the effect, m.
This is a scalar.
The state of inertia or actual velocity — and therefore acceleration, of the object, in that accelerated frame
— and not another frame, accelerated or not, notwithstanding, let’s say v and a’.

Now to go back to simplicity, let’s imagine a stone, attached to the end of a string. Earth around Sun is also the near analogue to such a thing, a satellite thrown to Mars, is such a thing and so on, in their near analogy, so the spherical cow or logical umbrella analogy in Physics is not quite worthless.

Photo Credit; shorecrest.org
Mass Attached To a String.

Photo Credit; UCSB.edu,
Earth around sun is an analogy to mass attached to the string

There are two situations.

The situation One;
The situation shows up when, looking at the string and stone being rotated, from a place — which is itself, not moving.

The state of inertia, of that place therefore, entails it to behave like an ideal inertial or uniform velocity frame, whereas the string and stone, do not have an uniform velocity, it’s the frame whose attribute is central to the definition of the inertialness of frame of reference. We see that the stone has a tendency to move in a circle and feels a force which is central to the beginning of the string — perhaps held by a kid. This is called as an uniform or non-uniform circular motion, depending on whether the state of inertia of the stone itself is constant or not. Again by that we mean whether the stone is accelerating in the tangential direction to the string. In any case its accelerating in the central direction.

Photo Credit; Voer.edu.vn
Merry Go Round
(a) noninertial or accelerated due to rotation
(b) inertial, frame attached to outside the rotating body.

So to simplify; the stone will always have a radial or central acceleration for it, to be able to move in a circle. This acceleration is called as centripetal acceleration, as long as we are observing it, from a place which has a zero or unchanging velocity.

But in addition it might have a tangential or perpendicular to string acceleration, this speeds up or down the stone, but the radial acceleration, keeps the stone in circle. These two internal situations of the stone, in our situation one above is called either as uniform or non-uniform — state of inertia of stone, circular motion.

Our situation one is when we are observing the whole circle string stone thing from a place, that is not accelerated, even if the stone itself has either one or two kinds of acceleration — namely radial and tangential, from this place of observation.

The situation two;
The situation shows up, when, the place from where we are observing is in itself accelerating. One example is a dragonfly sitting on that stone.

What would the dragonfly experience?

The dragonfly and in retrospect the stone and the string and anything in contact with these things will simply not experience what Newton’s laws would predict. As I itemized the 3 factors, additional pseudo effects — pseudo in considering Newton’s unmodified laws as sacrosanct, will be experienced by them.

We can take another example, lets say a tiny ball is lying on a circular disc which is rotating. The above 3 factors say the ball will experience a force — pseudo, which is again equal and opposite to the acceleration of the disc, multiplied by the ball’s own mass. That’s what the dragonfly will also experience along with the stone. They experience an outward tendency to fly off, and will do so if the string is cut off. That is, while from a frame which is not moving such as our observing a kid to rotating the string, the stone has a radial inward force acting on it, called as centripetal force, the stone itself has a radially outward — in the instant, force acting on it, as experienced by it. Because we have tied it to the string it is not telling us so.

If off topic analogies do not bother you, this is like people in democracy are tied to constitution and laws, so they can’t speak much. That’s because the stone is accelerating, keeping its own speed constant or not, notwithstanding. This force actually does not exist, but the effect is experienced, by the stone, as its experiencing such a force, from a frame, that’s accelerating. This force is called as a centrifugal force.

Therefore it turns out for reasons of consistency, that the centripetal force, is equal and opposite to the centrifugal force. But they are not action-and-reaction pair of Newton’s 3rd law. Centrifugal force is the radial outward force due to measurements being carried out in the accelerated frame, where the accelerated frame is the one, associated with the centripetal force — in our specific examples, in consequence they are equal and opposite.

The Pseudo force is always equal and opposite to the mass times the acceleration of the frame. Hence Centrifugal and Centripetal forces are equal and opposite, but act on the same object, from different frames of references. Hence these are not two different forces acting at the same instant, but rather an exclusive pair of forces, as per the frame of observation.

But this is the case when the stone, the dragonfly or the ball on the disc, are not moving themselves in the accelerated frame, in which we are considering them. They will experience another pseudo effect, if they were to have a state of inertia that’s changing. That is, if these 3 objects are moving, in their accelerated frame, an additional very small force would be experienced, by them, due to their own velocity and not just due to the velocity — or rather acceleration, of the frame.

So trivia;

there are two kinds of pseudo force

.

Centrifugal force is a pseudo force.
It emanates from the fact that frame of reference has acceleration. Centrifugal force is the exact opposite of the centripetal force, it is the only pseudo force acting on the object if the object is stationary.
The Coriolis force is a pseudo force.
It emanates from the fact that the object in an accelerated frame is not stationary, but has a velocity. It is not the only pseudo force acting on the object, if the object is not stationary, because now there is also a centrifugal force in addition to the Coriolis force.

The suitcase rolling in a turning bus, will not only experience the force that we experience but also another small additional force due to the fact that its not stationary. Gradually it will roll faster.

Because our earth is spinning about itself and also because its revolving about Sun, stationary object and moving objects will experience different pseudo forces. In addition to centrifugal force coming due to acceleration of the earth — spin and rotation around sun, the moving objects will also experience an additional force; called as Coriolis force, two orders of magnitude smaller than the centrifugal force.

For that reason flights and rockets will be slightly off than their non-inertial trajectories, due to the fact that they are already moving in the rotating earth frame. It will be of the % level — that is two orders of magnitude smaller, than the pseudo force already experienced, due to rotation of frame of reference.

One can check for exact formula for Coriolis force; omega*v, where omega gives angular rate, at which the frame is rotating and v is the object’s own velocity. Centrifugal force is omega-squared*r. That is, omega*v is about a percent of omega-squared*r. This is the reason most people do not talk about Coriolis force, but its an important pseudo force concept. We also need to check the veracity of this last statement. But for now, it seems the blog is quite long. Nearing 5 K words. We will review again and again. But your feedback are welcome.