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What do you mean by rigid body?Can anyone tell me why rigid body does not exist?

  1. Oct 15, 2012 #1
    Hello,
    Does anyone knows why rigid body does not exist.So in order to know its history which textbook can i refer to ???? For example to know classical mechanics properly you have to go to Newtons book known as Principia. In order to know rigid bodies who should we refer.
     
  2. jcsd
  3. Oct 15, 2012 #2
    The rigid body is an idealisation, although it's normally a very accurate model. A real solid is made of atoms and molecules bonded together by mutual electrostatic forces, and these forces are not transmitted instantly (but at the speed of light). For example, if you have a long rod and impart a torque on it at one end, the far end will not begin to move until the forces have been transmitted down the rod. Until this point, the near end will be moving whilst the far end will remain stationary, and the rod is therefore bending.

    For a ruler or stick, this is nearly instantaneous so we don't usually worry about problems with the rigid body assumption, but there are examples where it does become problematic. For example we sometimes use this assumption to model the rotation energies of molecules, but as a molecule rotates more it tends to become more stretched by centrifugal forces. This stretching increases the moment of inertia of the molecule, and leads to inaccuracies in the calculations.
     
  4. Oct 15, 2012 #3
    Good morning mremadahmed, welcome to Physics Forums.

    I am not sure what question you are asking.

    Do you understand what a rigid body is or do you want to know?

    A body, rigid or otherwise may undergo the actions of external forces or transformations (of coordinate systems).

    If every length of line between every two points, A and B on and within the body remains the same (ie A and B remain the same distance apart) we say that the body has undergone a rigid body motion or transformation.

    The every is important because it is possible to have a motion or transformation where some lines remain unchanged and some do not. This is not rigid body motion.

    It is not necessary for the orientation (direction) of the lines to remain the same.

    If they do remain the same the body has undergone a rigid body translation.
    If they do not, the body has undergone a rigid body rotation, although one line or point called the axis or centre must remain the same.

    Or are you looking at the history of the subject and already understand what a rigid is?

    Try

    A History of Strength of Materials by Timoshenko
     
  5. Oct 15, 2012 #4

    Andrew Mason

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    Just to follow up on what MikeyW has said, the maximum speed at which the force applied at one end of a body can be transmitted to the other end of the body is finite: c, the speed of light. A force is propagated through the body by slight changes in inter-molecular distances within the body. Since no change in force can propagate through a body faster than the speed of light, molecules in one end of the body will move before the molecules in the other end can respond.

    AM
     
  6. Oct 15, 2012 #5
    Maximum transmission speed through a solid is for p (pressure wave) component of sound. Typically in range 3-5*103m/s: http://www.potto.org/gasDynamics/node73.html Compare to speed of light c = 3*108m/s.
     
    Last edited: Oct 15, 2012
  7. Oct 15, 2012 #6

    AlephZero

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    There is no logical reason why a rigid body should not exist, within the framework of classical mechanics. But as Andrew Mason implied, we know by experiment that classical mechanics is only an approximation.

    Whether the speed of propagation of an effect through a body happens at the speed of light or orders of magnitude slower isn't relevant to the argument. The important point is that the speed is finite.
     
  8. Oct 15, 2012 #7

    russ_watters

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    By definition, force propagation speed is the speed of sound.
     
  9. Oct 15, 2012 #8

    AlephZero

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    Of course, but within the assumptions of classical (Newtonian) continuum mechanics there is no theoretical reason why the speed of sound could not be faster than the speed of light.
     
  10. Oct 15, 2012 #9

    russ_watters

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    I think it is counterproductive to discuss such what-ifs when the OP was asking about reality. Several people have given either a directly wrong or implied wrong answer to the OP's question by mentioning the speed of light.
     
  11. Oct 16, 2012 #10

    Andrew Mason

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    I disagree. There is no principle that limits the speed at which atoms can interact except the limit on the speed of light. The forces within a rigid body are electrical and nuclear (gravity being negligible). To be perfectly rigid a force applied to one part of the body must be transmitted instantaneously. But we know that the force applied to an outer atom is transmitted to the adjacent atom by electrical means. The force applied by an electron in that atom to the protons and neutrons in the nucleus is transmitted electrically and by the strong nuclear force.

    AM
     
  12. Oct 16, 2012 #11
    As a side-note, this got me wondering why the speed of sound in a solid isn't the speed of light, since all the force carriers for internal interactions do travel at that speed. By what mechanism is the perturbation being slowed from the speed of light to the speed of sound?

    I can understand this with air since air molecules aren't in continuous contact, so it may take some time for high pressure "front" to move through the gas. But I don't see why this should happen in a solid lattice.
     
  13. Oct 16, 2012 #12
    I think even in classical mechanics, rigid body is a paradoxical concept. I remember doing a mechanical question about considering a rigid thin rod with one end attached to the ground, inclining at angle θ, in the gravitational field undergoing free fall. The results from simple force analysis on any arbitrary point of the rod is contradictory to the results obtained from conservation of angular momentum because a rigid rod fails to include the distortion.
     
  14. Oct 16, 2012 #13
    For a longitudinal sound wave moving along a slender solid rod, propagation speed is given by v = √(E/ρ), where E = Young's modulus and ρ = volume mass density. Slightly higher v applies when propagation is through an essentially unbounded solid (e.g. seismic waves). For explanations and derivations, see e.g. "www.scitechpub.com/r/samples/9781891121920.pdf" [Broken] Recall that an EM wave is propagation of the EM field itself - and is c in vacuum only. This is far removed from inter-atomic residual electrical forces - strongly modified by QM principles, acting as 'springs' between atoms/molecules acting as 'masses' having inertia. The sound wave is a moving disturbance of crystal lattice (3d array of 'masses' connected via 'springs') from equilibrium.
    And afaik about the only situation in the universe where sound approaches but never equals light speed is in the case of neutron stars - hardly your average engineering situation.

    @Andrew Mason in #10:
    Recall the OP asked why a rigid body was impossible. Don't you think it best to refer to what is known about real materials in answering that?
     
    Last edited by a moderator: May 6, 2017
  15. Oct 16, 2012 #14
    Thanks very much. Do you know a link to the previous chapter? The velocity equation derivation was apparently given at an earlier stage.
     
  16. Oct 16, 2012 #15
    Sorry about that - I picked an example that was just sample from a 'please add to cart' textbook. Here is the best I can find:
    http://en.wikipedia.org/wiki/Speed_of_sound - quite accessible maths.
    http://en.wikipedia.org/wiki/Structural_acoustics - covers different propagation modes
    http://www.google.com.au/url?sa=t&rct=j&q=&esrc=s&source=web&cd=18&cad=rja&ved=0CFAQFjAHOAo&url=http%3A%2F%2Fcourses.physics.illinois.edu%2Fphys406%2FLecture_Notes%2FP406POM_Lecture_Notes%2FP406POM_Lect3.pdf&ei=NXh9UI7YGs20iQfn8oGADQ&usg=AFQjCNHTtYemuyrZQyfomsAYcKjBt7LfEg - some visual stuff included. Mostly on sound through air.
    "www.pma.caltech.edu/Courses/ph136/yr2004/0411.1.K.pdf" [Broken] - Fairly mathematical treatment that included examples and some figures. Good coverage.
    http://www.google.com.au/url?sa=t&rct=j&q=&esrc=s&source=web&cd=6&ved=0CEoQFjAF&url=http%3A%2F%2Fwww.smf.phy.cam.ac.uk%2FPublications%2FStrength%2520papers%2F607StrWalleyEncycMateronline.pdf&ei=uHJ9UN-8BqueiAfCooDQAw&usg=AFQjCNETF2yVYvy6BBnrf400kFcjVYp2iQ&cad=rja - broad sweep of subject.
    Hope this helps.
     
    Last edited by a moderator: May 6, 2017
  17. Oct 16, 2012 #16
    The mechanism is related to the inertia of the material comprising the rod, basically, ma.

    If you divide the length of a rod into increments, typically between x and x + dx, and treat each increment as a free body, then the tensile force acting at x is Eε(x)A, and the tensile force acting at x + dx is Eε(x+dx)A, where E is the Young's modulus, A is the cross sectional area, and ε (x) is the tensile strain at x. The strain at x is related to the displacement u at x by ε = du/dx.
    The amount of mass between x and x + dx is ρAdx, and the acceleration is utt. If you include this all in Newton's second law, you get

    utt=√(E/ρ) uxx

    This is the wave equation for a wave with velocity √(E/ρ). Including the mass times acceleration of each increment of mass along the rod is what slows the speed down from c to the speed of sound.
     
  18. Oct 16, 2012 #17

    Andrew Mason

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    There are at least a couple of factors that affect the speed of propagation of a force through a collection of atoms.

    The first is that in order to transmit the force, the separation between the nuclei in adjacent atoms must change. But a nucleus has significant mass and there is a "phase" difference between peak electrical force on a nucleus and peak change in separation between "adjacent" nuclei. The speed of transmission of a force through a body should depend on how strong the force is compared to how massive the constituent particles are. In a neutron star, for example, the force between neutrons is the strong nuclear force so the speed of "sound" in a neutron star approaches the speed of light.

    The second thing is that when coupled atoms or coupled parts of an atom move slightly there are restoring forces created by that motion which cause the atoms/parts to oscillate. But it is not a simple oscillation because these oscillations are constrained by quantum mechanical laws (eg. they are quantum harmonic oscillators).

    The third thing is that the force transmission is diffuse - it is not just one little atom but large groups of atoms that move very slightly together. Macroscopically the force is transmitted by oscillations of groups of molecules (phonons), and an analysis of their behaviour involves somewhat different aspects of quantum mechanics.

    AM
     
    Last edited: Oct 16, 2012
  19. Oct 16, 2012 #18

    Andrew Mason

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    Why? Using everyday materials one can show that it does not occur but that does not prove that it is theoretically impossible ie. that there is no upper limit to how rigid a body can be. There is a theoretical upper limit on how rigid a body can be and it appears to me that it is limited only by the speed of light .

    AM
     
  20. Oct 16, 2012 #19

    russ_watters

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    MikeyW and Q-reeus's discussion is informative of the problem. To summarize:

    Bringing up electromagnetism implies that the signal transmitted by hitting a rod with a hammer is electromagnetic in nature and therefore should propagate at a high fraction of C, just as electricity does. But the fact that the force between the atoms arises due to electromagnetism does not imply that the signal of smacking one into the next (and the next...) is also electromagnetic. It isn't: it is purely mechanical, modeled highly accurately as a collectin of little masses connected by springs.

    Now, as said, the speed of propagation is determined by density and elasticity. So getting the speed of sound to approach the speed of light would require either an extremely low density or extremely high modulus of elasticity. And this is where the issue becomes one of chemistry and materials science. These are the sciences that discuss the strength of molecular bonds. I'm just not sure it is all that useful to ask 'why aren't chemcial bonds orders of magnitude stronger?' or 'why aren't atoms orders of magnitude lighter'? Reality is they aren't and that's why the speed of sound is so low.

    And a point of order: the OP didn't actually ask about the speed of sound, only about rigid bodies. So only half of the issue is on the table, slightly differently put: 'Why aren't chemical bonds infinitely strong?'

    Where to look for an answer depends on the depth desired. Just saying that material strength and rigidity is determined by the strength of the chemical bonds holding it together might be enough for the OP. To go beyond the mechanics of materials/materials science explanation would probably involve chemistry and even QM. But ultimately, we'll get to a place where we'd have to say that molecular bonds or the electromagnetic force that causes them just are what they are.
     
  21. Oct 16, 2012 #20

    russ_watters

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    The limit comes from the strength of the bonds holding the material together.
     
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