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So when I was reading about Newton's laws, and my textbook (Sears and Zemansky's University Physics) gave the classic examples of when we might be tempted to create an additional "centrifugal force" and later went on to say that since an object going around in a circle is actually accelerating, we do not, rather can not include it in our free body diagrams (FBDs).

They say that the term "centrifugal force" will no longer be used anywhere in the text and strongly advised the reader against using it too. Now I was swayed by the argument "since it is a non inertial frame, we cannot avail of the privilege of using Newton's laws and hence cannot draw FBD like our "common sense" tells us to.

Everything was smooth sailing until I saw people all over the Internet use the concept to explain basic phenomena that I could actually explain without using it. This led to me to a rabbit hole, and now I am stuck with this so called "pseudo force". It's obviously not a force, why depict it as such. Even the wikipedia article on the topic says that the concept was used to explain events in general relativity, but it is no longer needed now. I also think it's "anti-physics" to analyse forces when in reality they don't even exist! I would really appreciate if someone could explain whether we actually need them? If yes, can we not use other methods to analyse motion in a non-inertial frame?

P.S.: As may probably be clear, I am not a graduate student yet, so I could really do with an explanation that does not go into the nitty gritties of general relativity. Also sorry for such a long question.

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    $\begingroup$ Does this answer your question? Does centrifugal force exist? $\endgroup$
    – Puk
    Mar 25, 2023 at 8:03
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    $\begingroup$ E.g. because it is sometimes simpler. $\endgroup$
    – Qmechanic
    Mar 25, 2023 at 8:07
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    $\begingroup$ In physics, force is what a force gauge measures. Textbook writers sometimes confuse the abstractions we use to capture physics with actual physics. But a force gauge may measure a centrifugal force in its frame. It's perfectly physical. $\endgroup$
    – John Doty
    Mar 25, 2023 at 20:23
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    $\begingroup$ For the same reason that we still use Newtonian mechanics, even though we know that technically it's wrong: it's simpler, more convenient and (arguably) more intuitive. $\endgroup$ Mar 25, 2023 at 22:18
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In practice, in newtonian mechanics, pseudoforces are used, because they're simple and convenient.

At the end of the day, they're only inertial terms that you're "disguising" as forces. You can choose to keep those inertial terms raw, or treat them as forces. It makes no difference mathematically, so it's a matter of opinion.

There is nothing "anti-physics" about them: they provide a mathematical model that correctly predicts the behavior of systems. Also, if you're standing in a bus that slows down, you will feel a force pushing you forward. Even though this force has a special status in mechanics, it's natural to treat it as a force instead of introducing new tools.

If your book promotes an alternative way to handle non inertial frames, keep in mind that:

  • Other people from other backgrounds might not use this alternative way, so you'll have to adapt anyway.
  • That sort of alternative way rarely makes things simpler. So ask yourself if you really find it pleasant to use.

It's only in general relativity that discussing the nature of pseudo-forces really bring something interesting to the table.

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    $\begingroup$ Yep, and in GR the force of gravity is just as real (or unreal) as centrifugal force. $\endgroup$
    – John Doty
    Mar 25, 2023 at 20:07
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    $\begingroup$ To quote the classic adage. "All models are wrong, but some are useful." $\endgroup$
    – Izzy
    Mar 26, 2023 at 14:06
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In an inertial frame $(t, x)$ Newton's law takes the familiar form \begin{equation} F=m\frac{d^2 x}{dt^2}\;, \tag{1}\label{1}\end{equation} where$^{1,2}$ $x(t)$ is the coordinate of a particle subject to force $F$.

Now write this in terms of a rotating coordinate frame $(t',x') = (t, R(t)x)$ (with $R$ a rotation matrix) and, voila, you will discover new terms - schematically these are \begin{equation} F = m \bigg(\frac{d^2x'}{dt'^2}+ f(x',\frac{dx'}{dt'})\bigg) \equiv m \frac{d^2x'}{dt'^2} - \sum_i F_i\;, \tag{2}\label{2} \end{equation} for some function $f$, and the right hand side serves as a definition of $F_i$ — these terms you can call ‘pseudo forces’ if you want. Note that these new terms really belong to the kinematical part (right hand side) of equation \eqref{1}, rather than the left hand side, so are not true forces in the usual sense. But clearly these terms exist, regardless of what you call them. That we can call them pseudoforces is because when they are moved to the left hand side of \eqref{2}, we can pretend (hence the prefix pseudo) that they really are forces and that Newton's law in its original form \eqref{1} holds. Then we can draw force diagrams and so on and everything is easy to understand.

Clearly in certain cases, it is much easier to think directly in terms of a rotating frame, rather than to go out to an inertial frame and then return to the rotating frame, which would be rather silly if you already have written Newton's second law in a rotating frame.

There should be no rabbit hole — the concept is very simple. There is certainly no need to mention general relativity, although if you read your Weinberg you may see some sort of connection. Any mechanics textbook which avoids the use of centrifugal forces is a little dubious in my opinion - after doing a few practical examples you will quickly convince yourself that such notions make certain 'rotating' problems much more direct to solve and intuitive.

The prefix ‘pseudo’ is used elsewhere in physics, as in ‘pseudovector’, usually when the prospective object can be treated as if it were something else. For example, the magnetic field (or angular momentum) are pseudovectors. They aren't really vectors (in the physics sense) — a more precise treatment regards them as components of a 2-form. However, if you are solving a basic dynamical problem, at least by hand, it would be silly not to treat them as if they are vectors.

Footnotes

$1.$ Note the two uses of $x$, one as the coordinate and the other as an embedding into this coordinate system - this is a common notational abuse but useful in practical problem solving.

$2.$ More generally the force $F$ could be time-dependent.

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  • $\begingroup$ Well, I agree with most parts of your answer. Ultimately, I think the problem is more due to our nomenclature than due to physical reasons. It's just like pseudo forces should be called something else, preferably along the lines of "non-inertial frame adjustment". Just like how EMF should not be called electromotive force, except textbooks don't warn you with the term "fictitious force" like how they do with EMF. $\endgroup$ Mar 25, 2023 at 8:57
  • $\begingroup$ We have: in the equation of motion for a rotating coordinate system the centrifugal term and the coriolis term are added. The magnitude of the centrifugal/coriolis term depends on the angular velocity of the rotating coordinate system with respect to the inertial coordinate system. That is: the act of using a rotating coordinate system is dependent on an prior established reference: the inertial coordinate system. If one doesn't use the inertial coordinate system as underlying reference one doesn't have a theory of motion. It's not about simpler; it's about: what is the underlying reference? $\endgroup$
    – Cleonis
    Mar 25, 2023 at 9:11
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    $\begingroup$ @Science_notfound I have edited. Note that they are usually called 'pseudoforces', which semantically is not the same as 'force'. As hyperlinked in my answer, the word pseudo refers to the idea that these terms can `pretend' to be forces, so is quite apt in my opinion. And of course, in practical calculations in rotating frames, one treats them exactly like forces. If you want to call them something else, it better be short, snappy and memorable. Also, physics is littered with anachronisms and badly named objects, at some point you just have to get over it. $\endgroup$ Mar 25, 2023 at 13:02
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    $\begingroup$ It's like the confusion that "imaginary" versus "real" numbers causes. Both are products of human imagination. $\endgroup$
    – John Doty
    Mar 25, 2023 at 20:09
  • $\begingroup$ You've managed to insert a philosophical remark which is not held by everyone. First, you don't need to bring in complex or real numbers to this discussion at all. Second, the point is nomenclature, not metaphysics about whether or not real numbers or pseudoforces belong in your imagination or not. Besides, it's probably not imagination when you get thrown to one side during a car skid. But let us leave metaphysics aside. $\endgroup$ Sep 17, 2023 at 17:43
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In practice, you can't do without using pseudo forces a.k.a. "fictitious forces". The analysis of even quite simple classical mechanics problems can become extremely complicated if you don't afford yourself the freedom to work in non-inertial frames.

It is true that the formalism involving pseudo forces brings with it its own complexity due to having to derive the expressions of pseudo forces. But in the Langrangian formalism of classical mechanics it's easy to work in arbitrary frames.

In general, it's wrong to not want to use a certain mathematical formalism because some of the mathematical quantities are unphysical. You can just as well say that you don't like to use negative numbers, and you can indeed work with only positive numbers, but that doesn't make life much easier. Similarly, you can avoid using complex numbers too.

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The whole distinction between real and pseudo forces is a bit misunderstood. The forces that we call "pseudo" are effectively the forces that are explained by the current theory.

Gravity is explained in general relativity. What we observe as gravity is "actually" just objects following straight lines (geodesics) along spacetime. And indeed, it is called a pseudo force (or fictitious force) in GR.

Would it be useful to entirely stop using gravity just because we can explain it in other terms?

To see that "real" and "pseudo" is just a matter of model, let's define a new model of mechanics. Define the force of inertia to be $F_I = -m/a$. Now we have physics in which all forces on an object always sum to zero. In this model the Newton's second law is a pseudo law — we just explained that it's "actually" just inertia and the objects are just following the zero-force trajectory at all times. Is Newton's second law suddenly fake?

I saw people all over the Internet use the concept to explain basic phenomena that I could actually explain without using it

I also think it's "anti-physics" to analyse forces when in reality they don't even exist!

Forces are a way to describe what's happening to an object. You can do all of mechanics without ever defining the concept of a force, it would just be more clumsy. Just like when you're refusing to use the inertial forces. Do forces exist or are they all fake?

Also remember that you can think of plenty of senses in which pressure, temperature and loudness does not exist. Yet we can use them in useful calculations and even feel them.

In general sense I see no use in arguing the existance of these terms. They all exist as tools. None of them exist in the sense of being defined by a supreme authority.

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    $\begingroup$ This is probably the best answer on this site to the classical question about the existence or not of inertial forces. +1 $\endgroup$ Mar 26, 2023 at 11:11
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Pseudo forces are a tool, which can be useful in certain situations. Sometimes it is simpler to work in a non-inertial frame and so introduce pseudo forces; sometimes not.

The key thing to remember is that they are not real forces, so they should not be used in any explanation of motion. For example, if we say

A satellite that is travelling sufficiently quickly stays in orbit because centrifugal force counter-balances the pull of gravity.

then this is definitely wrong, because it suggests that there are two forces acting on the satellite. The correct explanation (at least, from a Newtonian point of view) is

A satellite is in free fall and there is only one force acting on it, which is gravity. If the satellite is travelling sufficiently quickly then its free-fall path misses the Earth and it stays in orbit.

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    $\begingroup$ Nonsense. Of course they are useful in the description of motion. Coriolis force is the easiest path to understanding a Foucault pendulum. $\endgroup$
    – John Doty
    Mar 25, 2023 at 20:13
  • $\begingroup$ @JohnDoty Useful in calculations, but if you say "The plane of a Foucault pendulum rotates because of the Coriolis force" then that is wrong because it elevates the Coriolis force to the status of a real force. The plane of a Foucault pendulum rotates because its suspension point is moving along a non-inertial (accelerated) path. $\endgroup$
    – gandalf61
    Mar 26, 2023 at 8:47
  • $\begingroup$ You are using "wrong" and "correct" as if there was a supreme definition of force, but in GR gravity is just as pseudo as the centrifugal force and the satellite is moving along a "straight" line as no "real" forces act on it... $\endgroup$
    – Džuris
    Mar 26, 2023 at 10:12
  • $\begingroup$ @Džuris Yes, I realise gravity is a pseudo force in GR, which is why I qualified the explanation of the satellite's motion with "from a Newtonian point of view". Using gravity in an explanation may be wrong from a GR point of view, but it is less wrong than using both gravity and centrifugal force. $\endgroup$
    – gandalf61
    Mar 26, 2023 at 10:44
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    $\begingroup$ @gandalf61 In your whiteboard formulation, Coriolis isn't a real force. In the physical world, force is what force does, and Coriolis force is as real as any other force. The frame in which you do your measurements matters. Your whiteboard formulation is a mathematical model of physics: it is not physics. $\endgroup$
    – John Doty
    Mar 26, 2023 at 12:47
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There is a relation with the Copernican revolution.

I will use a historical perspective to present a view of what is at hand.


In ancient times astronomy was at a very high level in various parts of the world, I will focus on ancient Greece.

In our everyday experience we see that if you give things a push they always come to a standstill again. In terms of ancient theory of motion that is the expectation for any motion.

That is, the expectation was that for motion to continue a force must be supplied continuously.

Therefore to the scholars of ancient Greece it was not possible to consider the possibility that the Earth is rotating. They had a good idea of the size of the Earth. If the Earth would be rotating then something whould have to push the air to move along with the Earth. But nothing is available to push the air like that. If the Earth would be rotating then there would be a permanent east-to-west wind, and that was not the case. So that ruled out the possibility that the Earth was itself rotating.

It wasn't so much selfishness that made the scholars of ancient Greece believe that the Earth is stationary. Without the modern notion of how inertia works the conclusion that the Earth is stationary is unavoidable.


What happens if there is zero friction?

It was Galilei who argued that if there is no friction then motion will continue indefinitely.

That is a fundamental shift of thinking about motion. The new insight was: it is the case of uniform motion in a straight line that does not require a force.

In the years after Galiei scholars came to recognize the following aspects of inertia:
-uniform motion in a straigh line will continue indefinetely
-to cause change of velocity a force is required


Huygens used this principle to great effect.
Imagine two marbles of equal mass colliding. First consider the case where you are using a coordinate system such that the Common Center of Mass (COM) of the two marbles is stationary. Then the collision will be symmetric: both marbles will have their velocity reversed.

Now use a coordinate system that is co-moving with one of the marbles. Huygens reasoned: after the collision the COM of the two marbles must still be moving at the same velocity. Therefore relative to this coordinate system the first marble will come to a standstill, and all velocity is transferred to the other marble.

A consequence of the principle of inertia is that what counts is not the velocity vector itself, but the magnitude of change of velocity.

Huygens then went on two examine cases of marbles with unequal mass.


The mechanics of the solar system: Copernican revolution

If the principle of inertia is granted then it follows that the celestial bodies of the solar system must collectively be orbiting their Common Center of Mass.

Newton used the inverse square law of gravity to infer the mass ratio of the Sun and Jupiter (Comparing orbital periods of Jupiter's moons to the orbital periods of the planets.) While the Sun has far more mass than Jupiter, the COM of the Sun-Jupiter system is actually a bit outside the Sun. In that sense the Sun is orbiting too.

(In Kepler's theory of celestial motion the Sun was an unmoving object.)

When you use an inertial coordinate system to represent the motions of the celestial bodies of the solar system then the motion of the celestial bodies can be accounted for in terms of the inverse square law of gravity.


In retrospect we can see the Copernican revolution as a shift to making inertia the prime organizing principle for understanding motion.

Galilei got the Copernican revolution going, Newton completed it, by showing that all of the celestial motion can be accounted for in terms of:

-the principle of inertia
-inverse square law of gravity


Further reading:
discussion of the principle of inertia by Kevin Brown, on the website Mathpages:

Inertia and Relativity

I very much recommend reading the entire series that is available on that website: 'Reflections on Relativity'

More general recommendation: don't demand of yourself that you understand all of the information there. Even if you don't understand all of it, it will still stimulate your thinking.

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I don't I think if it's anti physics to use pseudo forces as they are quite incredibly significant in non inertial frames to convert the motion into inertial frame. And using pseudo forces gives an easier approach to many problems.

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    $\begingroup$ @chasly-supportsMonica You can't have a ruler that's exactly 1m long, either. "Real" numbers are as much a product of human imagination as imaginary numbers are. $\endgroup$
    – John Doty
    Mar 25, 2023 at 21:36
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    $\begingroup$ @chasly But I can have an imaginary voltage. And, of course, we use complex numbers to describe quantum wave functions. Still, both "real" and "imaginary" numbers are a bit treacherous as models of approximate quantities. Mathematics isn't physics, and the objects of mathematics, as useful as they are as models, are not physical. They don't exist in reality. You can have a ruler, but "one" is an abstraction. $\endgroup$
    – John Doty
    Mar 25, 2023 at 23:02
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    $\begingroup$ @chasty We can calculate √-1 without difficulty. It's really no different than 1/3 or π, just a mathematical object with well-defined properties. $\endgroup$
    – John Doty
    Mar 25, 2023 at 23:35
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    $\begingroup$ @Solomon Slow - You say, "if I enter -1 and press the square root key, it displays 0.0000 i1.0000" That isn't a calculation, it's just a different way of writing the same thing. A calculation would look like this: √1009 ≈ 31.7647603 $\endgroup$ Mar 26, 2023 at 0:53
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    $\begingroup$ @Solomon Slow - "I don't understand what you are trying to tell me." All I am trying to tell you is that you can't calculate the value of √-1. You are trying to contradict me. I say that you can calculate, i.e. perform an algorithm, that will change the symbolic notation √2 for example, into a series of numerals to any degree of precision you require. There is no such algorithm that will change the symbolic notation √-1 into a series of ... $\endgroup$ Mar 26, 2023 at 11:24
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Q: Why do we still use pseudo forces?

Two ways that other answers have already touched on can be elaborated on using orbital mechanics.

First is intuition: when a capsule is docking with a space station in Earth orbit, their speeds and directions are almost exactly but not quite matched. If you forget this their relative motion makes no sense; thrust forward and you slow down and move to a higher orbit. "Come along side" out of the plane of the station's orbit and 23 minutes you'll smack right into it!

So while a modern guidance computer will probably numerically integrate all real forces and cut no corners, someone trying to maneuver manually will need to pay homage to the early space pilots and intuit what will happen based on pseudo forces.

Second is (as other answers suggest), shuffling terms around to make solutions easier to understand, remember, parse, characterize, or put into historical context, which is a good thing!

Who's going to tell Lagrange otherwise? :-) For example the restricted three body problem and those five Lagrange pointss and their associated Lissajous and halo orbits are a lot easier to wrap one's mind around and solve for and approximate and perturb and draw stable/unstable manifolds for (for getting to, from and between orbits) if one works in a rotating frame with a pseudopotential.

If we didn't do that, we'd never have such intuitive (humor) diagrams like this!

This answer to Space Exploration SE's What sort of orbital elements are used to describe halo orbits? links to Robert W. Farquhar's hundred page tome The Utilization of Halo Orbits in Advanced Lunar Operations, NASA Tech. Note D-6365.

halo orbits from Robert W. Farquhar's hundred page tome "The Utilization of Halo Orbits in Advanced Lunar Operations" http://www.ieec.cat/hosted/web-libpoint/papers/dichmann-doedel-paffenroth.pdf NASA Tech. Note D-6365

and E. J. Doedel et al, (2007) Elemental periodic orbits associated with the libration points in the circular restricted 3-body problem International Journal of Bifurcation and Chaos 17, 2625 (2007) (also in Researchgate):

cropped and annotated from E. J. Doedel et al, (2007) "Elemental periodic orbits associated with the libration points in the circular restricted 3-body problem" https://doi.org/10.1142/S0218127407018671 International Journal of Bifurcation and Chaos 17, 2625 (2007)

cropped and annotated from E. J. Doedel et al, (2007) "Elemental periodic orbits associated with the libration points in the circular restricted 3-body problem" https://doi.org/10.1142/S0218127407018671 International Journal of Bifurcation and Chaos 17, 2625 (2007)

cropped and annotated from E. J. Doedel et al, (2007) "Elemental periodic orbits associated with the libration points in the circular restricted 3-body problem" https://doi.org/10.1142/S0218127407018671 International Journal of Bifurcation and Chaos 17, 2625 (2007)

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Netwon's 1st and 2nd Laws outright fail if you omit pseudo forces

As an example, imagine a car at a traffic light that just turned green. Let's examine the forces and accelerations on the driver, and on her purse on the passenger seat, in two reference frames: the inertial reference frame of the street, and the accelerating reference frame inside the car.

For simplicity, we will assume the seats are frictionless. It's possible to elaborate the problem and include friction, but the outcomes are the same. We can ignore the forces in the vertical direction and will only consider the horizontal forces.

Before we continue, the most common mistake I've found that learners make is to go back-and-forth between the two reference frames. They often take forces or use arguments that exist in one of the reference frames, and incorrectly apply them in the other reference frame, leading to wrong results. Pick one reference frame and stick with it until you have finished your analysis of that reference frame, then continue to the other reference frame.

Also, I've found that it's better to explain pseudo forces first with an example in horizontal linear motion. Then I elaborate the idea with a vertical motion elevator example. Once my students understand the concept, then they are ready to combine it with circular motion. Too many instructors try to do it all at once, which results in cognitive overload for many students. They get wrapped up in the centripetal/centrifugal distinction instead of focusing on the important ideas.


First, consider the forces and accelerations as seen by an observer in the inertial reference frame of the street:

  • There is zero net force on the purse. Indeed, there are no forces on the purse -- we said there is no friction, and there are no pseudo forces in an inertial frame.
  • The acceleration of the purse is zero. To the person on the street, the purse simply appears at rest and remains at rest. The car however is accelerating underneath the purse.

These two previous statements (zero net force, zero acceleration) follow the 1st and 2nd Laws.

  • There is a forward net force on the driver. The back of her seat (part of the car) pushes her forward. There are no other horizontal forces, so the force by the seatback equals the net force.
  • The driver accelerates forward. Indeed, she appears to accelerate with the car.

If you do the math, you find out that the net force and acceleration are exactly what you would expect from the 2nd Law. The 1st Law does not apply because neither the net force nor acceleration are zero.

Sir Isaac is happy.


Now switch your reference frame to inside the car. It is an accelerating reference frame, so every object now sees an additional pseudo force in the opposite direction (backwards) of the reference frame's acceleration (forwards).

  • The net force on the purse is now nonzero. There is now one horizontal force on the purse: the pseudo force, which pushes backwards.
  • The purse accelerates backwards. We see the purse move back along the seat.

If you do the math, you find out that the net force and acceleration are exactly what you would expect from the 2nd Law. The 1st Law does not apply because neither the net force nor acceleration are zero.

  • The net force on the driver is now zero. She has two horizontal forces on her: the seat back pushing her forward, and a psuedo force pushing her backward. If you do the math, you find out that these forces are equal and cancel each other out.
  • The driver's acceleration is zero. She appears to remain in place in her seat.

These two previous statements (zero net force, zero acceleration) follow the 1st and 2nd Laws.

Sir Isaac is still happy.


"But they're not real forces." Okay, let's see what you get when you omit pseudo forces from the net force. It doesn't matter in the inertial frame, since those pseudo forces are always zero. So reconsider the accelerating frame without pseudo forces:

  • The net force on the purse is zero, as there are no horizontal forces.
  • We see the purse slide backwards on the seat, so it has nonzero acceleration.

A nonzero acceleration with a zero net force is not possible under the 1st and 2nd Laws.

  • The net force on the driver is nonzero and forwards, caused by the seatback pushing on her.
  • We see the driver remain in place, so her acceleration must be zero.

Under the 2nd Law, a nonzero net force cannot cause zero acceleration.

Sir Isaac is unhappy. You must include the pseudo forces in the net force.


But pseudo forces do not obey the 3rd Law

What does the 3rd Law actually mean?

If object A pushes on object B, then object B pushes back on object A with an equal and opposite force.

So for example, the seat back pushes forward on the driver. The driver then pushes backward on the seat back, with an equal amount of force.

But what about the pseudo force on the driver? We know it's a backwards force. What object is actually causing this force?

Students usually reply "the seat back". But we already accounted for that force, and furthermore it is a forward force, not a backward force.

I also get "the air" as an answer. Nope, you can repeat this experiment without air and it makes no difference. Not the air.

Sometimes I get "the gas pedal" or "the floor of the car". You can repeat this experiment without such contact (e.g. self-driving car) and you will see that it makes no difference.

The correct answer is there is no object creating this force. It is a result of being in an accelerating reference frame. That's why it is a "fictitious" or "psuedo" force!

So if there is no object creating the force, the 3rd Law fails. There is nothing to push back on with an equal and opposite reaction.

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