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Old Aug 21st 2017, 10:11 PM   #1
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D'Alembert and accelerating inclined plane

Just when I think I have figured this out the answers in the solutions book prove me wrong yet again. I am trying to understand the steps here.

Here is what I got:

$\displaystyle F_{applied} = M \ddot{x} + mg$

$\displaystyle F_{inertial} = \frac{m \ddot{s}}{\sin \alpha}$

I reasoned for the second "inertial" forces that the block could only move along one surface so only added one term. My intuition told me that the block could slide backwards but I thought this was accounted for in the applied force.

This is all surprisingly very difficult to think in terms of generalised co-ordinates when you don't know what the mathematical D'Alembert framework is supposed to be doing for you and what you still have to do for yourself!! Clearly, I still don't get that.

Ok, I better state my questions:
  • If we are using generalised co-ordinates why do we have to correct angles with trig?
  • Why do we not also correct with trig for applied forces?
  • Is there a straightforward (mechanical) way of determining the signs of the accelerating terms?
  • Is each set of connected particles, moving along one curve, one and only one generalised co-ordinate, even if one of the connected particles are hanging vertically of a "cliff"?
  • If we keep having to add in trig terns then in what sense does generalised co-ordinates free us from the euclidean co-ordinates?
  • Can the holonomic constraints be piecewise defined? (This seems implied by using a wedge with one particle hanging over the edge in earlier problem.)
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D'Alembert and accelerating inclined plane-d-alembert-2.09-problem.png   D'Alembert and accelerating inclined plane-d-alembert-2.09-solution.png  

Last edited by kiwiheretic; Aug 21st 2017 at 11:21 PM.
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Old Aug 22nd 2017, 04:57 AM   #2
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This is all surprisingly very difficult to think in terms of generalised co-ordinates when you don't know what the mathematical D'Alembert framework is supposed to be doing for you and what you still have to do for yourself!! Clearly, I still don't get that.
Perhaps that is because dear old D'Alembert did not introduce his principle in the way your textbook probably does.
That is simply because generalised coordinates and Hamilton's Principle were not introduced for another century after D'Alembert.

A good way to study D'Alembert is to do the same problem in the Newtonian and D'Alembert way side by side.

In fact, you can start with the Newtonian equations of motion, motivate then deduce the energy principles, the principle of virual work and D'Alembert's principle in short order. Then you can go on to derive Lagrange and Hamilton from this base and further to Noether, holonomy, symmetry invariants from this base.

But the original D'Alemebert principle was a mathematical transformation to reduce a more difficult dynamic problem to an equivalent easier static one, where the equations of equilibrium can be employed.
This was done by introducing imaginary or fictitious forces, sometimes called the inertial reaction(s) or the kinetic reaction(s).

Translation from old French

"When any particle is acted upon by an external force, the resultant of this external force and the kinetic reaction of the particle is zero"

Applying this should help with signs and trigonometry/geometry.

If you want to know more we should discuss it in a separate thread, rather than mess up this one about a specific problem.
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Last edited by studiot; Aug 22nd 2017 at 05:01 AM.
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Old Aug 22nd 2017, 07:02 PM   #3
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Originally Posted by studiot View Post
Perhaps that is because dear old D'Alembert did not introduce his principle in the way your textbook probably does.
That is simply because generalised coordinates and Hamilton's Principle were not introduced for another century after D'Alembert.
Yeah, I may switch to the Goldstein text as Calkin seems to be throwing me too many curve balls. (Everyone seems to be recommending Goldstein and seeing this if for self edification rather than actual course work I am at liberty to switch texts). Perhaps switching to generalised co-ordinates prematurely has resulted in having to sacrifice intuition over methodology which clearly hasn't been working for me.

Ok, I will start a new thread on "What precisely are inertial forces?"

Last edited by kiwiheretic; Aug 22nd 2017 at 07:32 PM.
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Old Aug 22nd 2017, 07:46 PM   #4
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Originally Posted by kiwiheretic View Post
Yeah, I may switch to the Goldstein text as Calkin seems to be throwing me too many curve balls. (Everyone seems to be recommending Goldstein and seeing this if for self edification rather than actual course work I am at liberty to switch texts). Perhaps switching to generalised co-ordinates prematurely has resulted in having to sacrifice intuition over methodology which clearly hasn't been working for me.
Do you really want to waste all that time with this principle? Its probably one of the least useful things I've seen in physics. Not useless, just less used.


Originally Posted by kiwiheretic View Post
Ok, I will start a new thread on "What precisely are inertial forces?"
There seem to be two uses of that term. One is in what you've been studying. The other is described here

Inertial Force

An inertial force is a force which is present due to the change on ones frame of reference. I.e. suppose that S is an inertial frame of reference and let there be a body which is moving freely in that frame, i.e. constant velocity. Now consider the frame S' which is accelerating relative to S. Observers in S' will observe the body to be accelerating and thus appear to be moving under a force. That kind of force is called an "inertial force". Notice how that kind of force results in an acceleration which is independent of the mass of the body, just like gravitational forces are. That's what led Einstein to GR!

Inertial forces were important to Einstein's work on GR. In fact in GR the gravitational force is an inertial force. Read the Einstein quotes in that page I just gave above
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Old Aug 22nd 2017, 08:17 PM   #5
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Originally Posted by Pmb View Post
Do you really want to waste all that time with this principle? Its probably one of the least useful things I've seen in physics. Not useless, just less used.
If you can assure me I don't need it as a prerequisite to working with Lagrange's equation.

An inertial force is a force which is present due to the change on ones frame of reference. I.e. suppose that S is an inertial frame of reference and let there be a body which is moving freely in that frame, i.e. constant velocity. Now consider the frame S' which is accelerating relative to S. Observers in S' will observe the body to be accelerating and thus appear to be moving under a force. That kind of force is called an "inertial force". Notice how that kind of force results in an acceleration which is independent of the mass of the body, just like gravitational forces are. That's what led Einstein to GR!
Sure, but none of the problems I have been working with could conceivably working with fictitious forces arising from accelerated frames of reference so I am quite bewildered regarding the pedagogy.
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Old Aug 22nd 2017, 08:33 PM   #6
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Originally Posted by kiwiheretic View Post
If you can assure me I don't need it as a prerequisite to working with Lagrange's equation.
I can not only assure you of it, I promise you that you don't need it.


Originally Posted by kiwiheretic View Post
Sure, but none of the problems I have been working with could conceivably working with fictitious forces arising from accelerated frames of reference so I am quite bewildered regarding the pedagogy.
You are wrong. Whenever you work with the gravitational force you are working with an inertial force because the gravitational force is an inerital force. There's a great book out by Landau and Lifshitz on mechanics in which they formulate the Lagrangian for a non-inertial frame. That text is well-known and highly regarderd in the physics community.

A text which you might like which is also well-known and highly regarded is Classical Mechanics by John R. Taylor.

I should have mentioned this one to you rather than Goldstein since Goldstein is a graduate level text while Taylor is an undergraduate text. In fact Goldstein is used at the University of Lowell in Lowell, MA for a text in their graduate course. I would have loved Taylor in my undergrad course in mechanics if it had been out when I was an undergraduate.
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