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Old Aug 22nd 2017, 08:08 PM   #1
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What precisely are "inertial" forces?

This concept seems to be hanging me. It's used in a lot of D'Alembert problems and even on various texts on the web.

The wiki page on "D'Alembert's principle" points to a page called "fictitious forces". https://en.wikipedia.org/wiki/Fictitious_force

Except the wiki page on this starts talking about accelerated frame's of references which aren't the sort of problems I have been working on.

So some questions...
  • Are D'Alembert forces real or fictitious forces ?
  • What makes these forces special and why do we need them?
  • What advantage is D'Alembert's method over just applying Newton's second law?
  • Does it make sense to use D'Alembert's method without employing generalised co-ordinates?
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Old Aug 22nd 2017, 08:24 PM   #2
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Originally Posted by kiwiheretic View Post
This concept seems to be hanging me. It's used in a lot of D'Alembert problems and even on various texts on the web.

The wiki page on "D'Alembert's principle" points to a page called "fictitious forces". https://en.wikipedia.org/wiki/Fictitious_force
WARNING! Whomever wrote that page made a mistake. The inertial forces from D'Alembert's principle are not the same thing as inertial forces in the Wikipedia page it links to.

Originally Posted by kiwiheretic View Post
Except the wiki page on this starts talking about accelerated frame's of references which aren't the sort of problems I have been working on.
Exactly. Didn't I mention this to you before? The term inertial force is an overloaded term which means it has two different meanings. Context determines which one you want.

Originally Posted by kiwiheretic View Post
Are D'Alembert forces real or fictitious forces ?
Real

Originally Posted by kiwiheretic View Post
What makes these forces special and why do we need them?
One inertial force which is referred to as the Coriolis force is because its what causes certain weather phenomena. See: https://www.nationalgeographic.org/e...riolis-effect/

Another inertial force of renown is the centrifugal force. Its what gives the spaceship inhabitants in the movie "Martian" the effect of gravitational field so they can walk around that part of the ship. It's for reasons like these that they were of great importance to Einstein when he was working on GR

Originally Posted by kiwiheretic View Post
What advantage is D'Alembert's method over just applying Newton's second law?
In certain instances it can make the formulation of the mathematical problem easier to formulate.

Originally Posted by kiwiheretic View Post
Does it make sense to use D'Alembert's method without employing generalised co-ordinates?
Not really. You can certainly do it. You just make it less generalizable and thus harder to apply.
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Old Aug 22nd 2017, 08:36 PM   #3
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So then does D'Alembert's principle have anything useful to say about problems involving inclined planes, rolling balls, frictionless sliding blocks and pulley's where no accelerated frames are involved?
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Old Aug 22nd 2017, 08:43 PM   #4
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Originally Posted by kiwiheretic View Post
So then does D'Alembert's principle have anything useful to say about problems involving inclined planes, rolling balls, frictionless sliding blocks and pulley's where no accelerated frames are involved?
I don't know what you consider useful. As I said, sometimes it just makes solving a problem easier. Is that useful?
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Old Aug 22nd 2017, 09:49 PM   #5
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By making a dynamics problem a static one by putting the problem in an accelerated frame of reference?
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Old Aug 22nd 2017, 10:08 PM   #6
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Do you recall the book I mentioned by Lanczos on the variational principles of mechanics? Its a great book on D'Alembert's principle and addresses what you wish to know.

I should state that while I have read and studied D'Alembert's principle of virtual work in graduate school it was a very long time ago and I don't recall much of it. I did reread that part of Lanczos and recommend you find it online and read that section. I cannot stress that enough. Let me quote part of it here so you get how great that text is. From page 93
D'Alembert's principle is fundamental in still another respect; it makes possible the use of ,moving reference systems, and is thus a forerunner of Einstein's revolutionary ideas concerning the relativity of motion, explaining - within the scope of Newtonian physics - the nature of those "apparent forces" which are present in moving systems.
The author also points out that Hamilton's principle can be derived from D'Alembert's principle by a mathematical transformation. Hamilton's principle is restricted to holonomic systems whereas D'Alembert's principle applies to both holonomic and non-holonomic systems. My back is killing me now so I'll let you look those terms up yourself.

By the way. When I suggest taking a look at a particular text/book its always for a very good reason. Some books explain certain things exceptionally well. I'd quote them but it takes time and its painful for me to post lately since my HDTV bit the dust. I used it as a monitor which made things easy to read and thus post. Now I'm restricted to smaller monitor and have to sit on the floor making it very painful for me to post right now. The idea here is that reading that text is better all around rather than having me quote it or try to recall things from that dusty part of my memory.

On another note - I have three cats. Two are brothers. Right now one is licking the butt of the other. Now that is what I call brotherly love. Lol!

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Old Aug 22nd 2017, 10:33 PM   #7
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Thanks, I'll try the Taylor's book first and work through some of the problems in Chaper 6 Calculus of Variations, Chapter 7 Lagrange Equations and possibly Chapter 8 Two Body Central Force problems and then decide where to from there.

I am hoping this pursuing the "holy grail" of "not relying on co-ordinate systems" is really worth it and the advantages of generalised co-ordinates hasn't been oversold.
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Old Aug 22nd 2017, 11:16 PM   #8
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Originally Posted by kiwiheretic View Post
Thanks, I'll try the Taylor's book first and work through some of the problems in Chaper 6 Calculus of Variations, Chapter 7 Lagrange Equations and possibly Chapter 8 Two Body Central Force problems and then decide where to from there.

I am hoping this pursuing the "holy grail" of "not relying on co-ordinate systems" is really worth it and the advantages of generalised co-ordinates hasn't been oversold.
Generalized coordinates is extremely important.
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Old Aug 22nd 2017, 11:25 PM   #9
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Originally Posted by kiwiheretic View Post
Are D'Alembert forces real or fictitious forces ?
It's not a good idea to refer to inertial forces as "fictitious". There's significant caution in the physics literature against doing that, although this is a point of disagreement among physicists.

The point on thinking of inertial forces as not being "real" is addressed in that text I mentioned, i.e. The Variational Principles of Mechanics - 4th Ed., Cornelius Lanczos, Dover Pub., page 98.
Whenever the motion of the reference system generates a force which has to be added to the relative force of inertia I’, measured in that system, we call that force an “apparent force.” The name is well chosen, inasmuch as that force does not exist in the absolute system. The name is misleading, however, if it is interpreted as a force which is not as “real” as any given physical force. In the moving reference system the apparent force is a perfectly real force, which is not distinguishable in its nature from any other impressed force. Let us suppose that the observer is not aware of the fact that his reference system is in accelerated motion. Then purely mechanical observations cannot reveal to him that fact.
From Introducing Einstein's Relativity, by Ray D'Inverno, Oxord/Clarendon Press, (1992) page 122
Notice that all inertial forces have the mass as a constant of proportionality in them. The status of inertial forces is again a controversial one. One school of thought describes them as apparent or fictitious which arise in non-inertial frames of reference (and which can be eliminated mathematically by putting the terms back on the right hand side). We shall adopt the attitude that if you judge them by their effects then they are very real forces. [Author gives examples]
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Last edited by Pmb; Aug 22nd 2017 at 11:27 PM.
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Old Aug 23rd 2017, 09:57 AM   #10
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OK so let us make a start moving on from Newton with a worked example.
Hopefully we can answer your questions along the way.


Here is the question.

A particle of mass m is suspended from the vertex of a right circular cone, of vertical angle 2a, by a light inextensible string of length L, so that is lies on the slant surface of the cone, at radius r from cone axis.
The cone is rotating about the vertical axis through the vertex taking the particle along with it, at constant angular velocity w.

Assuming a no slip condition between the cone and the particle compare the dynamics by Newton and D'Alembert, establishing the string tension and the normal reaction between the cone and the aprticle.

Sketches of the geometry of the setup appears in Fig1.

With both methods ,as always, we draw free body diagrams (for the particle in this case). These appear in Fig2.
Since the methods are different it is not suprising to find that corresponding FBDs are different.

We see that the particle is held up against gravity by the string tension (T) and the normal reaction (N) and rotates in a circle in a horizontal plane.
The friction between the cone and the particle is not considered because it is perpendicular to both the vertical and the radial directions of this circle. (This is true in both analyses)

For Newton's analysis the particle is in motion and therefore obeys Newton's motion laws. In particular we want N2, Force = mass times acceleration.
Since this is a vector equation we can resolve it vertically and horizontally in the horizontal direction.

This is done in the left hand column in Fig 2.

For D'Alembert's analysis the famous centrifugal force has been added to reduce the particle to horizontal equilibrium. It is already in vertical equilibrium.

The D'Alembert analysis is shown in the right hand column.

Comparisons to note are.

Vertically the particle is neither rising nor falling.

Newton says The net Force adds to zero so the acceleration is zero
D'Alembert says because the aprticle is in vertical equilibrium the net force is zero

Both methods see the same vertical forces and produce the same equations.

Horizontally in a radial direction The radius neither increasing nor deacreasing.

Newton says There is a net Force which equals the mass times the radial acceleration.
D'Alembert says because the particle is in horizontal equilibrium the net force is zero, but there is an additional horizontal force acting, called the centrifugal force.

So the methods offer different horizontal net forces and correspondingly different equations.
But these equations turn out the same, when rearranged.

I will set aside the issue of is the centrifugal force real or imaginary till next time, but observe that this is one force, much used by engineers to simplify various areas of mechanics for example.

Fluid mechanics in pipes and turbines.
The variation of apparent gravity at the equator and the poles.
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What precisely are "inertial" forces?-dalembert2.jpg  
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