iMechanica - linear momentum
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enUse Only the Angular Quantities in Analysis? Three Sample Problems to Consider...
https://imechanica.org/node/8313
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/277">energy</a></div><div class="field-item odd"><a href="/taxonomy/term/584">mechanics</a></div><div class="field-item even"><a href="/taxonomy/term/989">mathematics</a></div><div class="field-item odd"><a href="/taxonomy/term/5216">linear momentum</a></div><div class="field-item even"><a href="/taxonomy/term/5217">momentum</a></div><div class="field-item odd"><a href="/taxonomy/term/5218">angular momentum</a></div><div class="field-item even"><a href="/taxonomy/term/5219">conservation</a></div><div class="field-item odd"><a href="/taxonomy/term/5220">independence</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>
A recent discussion at iMechanica following my last post here [<a href="node/8288" target="_blank">^</a>] leads to this post. The context of that discussion is assumed here.
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I present here three sample problems, thought of almost at random, just to see how the suggestions made by Jaydeep in the above post work out.
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<strong>Problem 1.</strong> For simplicity of analysis, assume a gravity-free, frictionless, airless, field-free, etc. kind of a physical universe, and set up a suitable Cartesian reference frame in it such that the xy-plane forms the ground and the z-axis is vertical. (Assume a right-handed coordinate system.)</p>
<p>Consider a straight rigid rod of zero thickness and length 2a, initially positioned parallel to the x-axis and at a height of h, translating with a uniform linear velocity, v, in the positive y-direction.</p>
<p>Suppose that the translating rod runs into a rigid vertical pole of infinite strength, zero thickness and a height > h so that a collision between the two is certain. </p>
<p>If the collision were to occur at the midpoint of the rod (i.e. at its CM), it would simply begin translating back with a -v velocity. </p>
<p>However, assume that the collision occurs at a distance d (0 < d < a) from the CM. This will impart an angular motion to the rod after the collision.</p>
<p>Derive the equations of motion for the rod before and after the impact, using (a) the usual method of analysis (having both the linear and angular quantities in it), and (b) the method / lines of analysis suggested by Jaydeep.</p>
<p>Comment on the linear and angular quantities before and after the impact in both the cases.</p>
<p>Make any additional assumptions as necessary.
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<br /><strong>Problem 2.</strong> Repeat the Problem 1., using both the usual analysis and Jaydeep's method, now assuming that the rod is linear elastic with infinite strength. All other assumptions and data remain the same; in particular, the pole continues to remain rigid.
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<br /><strong>Problem 3.</strong> Provide a detailed derivation for an FE model of the beam element, replacing the three rotations and three translations by six rotations so as to conform to Jaydeep's method.
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<strong>Asides:</strong> BTW, I plan to solve none. But I will make sure to have a look at any solution(s) that are offered, but without promising in advance that I will also comment on them.
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With that said, solutions and comments are most welcome!
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--Ajit
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PS: Also posted at my blog here [<a href="http://ajitjadhav.wordpress.com/2010/05/27/use-only-the-angular-quantities-in-analysis-three-sample-problems-to-consider/" target="_blank">^</a>]
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<em>Update on May 29:</em> Corrected a typo. Now the Problem 1 reads: "...a uniform linear velocity, v, in the positive y-direction. ..." This is the way it was intended. Earlier, it incorrectly read: "...a uniform linear velocity, u, in the positive x-direction. ..."
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[E&OE]
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</div></div></div>Thu, 27 May 2010 16:06:49 +0000Ajit R. Jadhav8313 at https://imechanica.orghttps://imechanica.org/node/8313#commentshttps://imechanica.org/crss/node/8313Are Linear and Angular Momenta Interconvertible?
https://imechanica.org/node/8288
<div class="field field-name-taxonomy-vocabulary-8 field-type-taxonomy-term-reference field-label-hidden"><div class="field-items"><div class="field-item even"><a href="/taxonomy/term/277">energy</a></div><div class="field-item odd"><a href="/taxonomy/term/584">mechanics</a></div><div class="field-item even"><a href="/taxonomy/term/989">mathematics</a></div><div class="field-item odd"><a href="/taxonomy/term/5216">linear momentum</a></div><div class="field-item even"><a href="/taxonomy/term/5217">momentum</a></div><div class="field-item odd"><a href="/taxonomy/term/5218">angular momentum</a></div><div class="field-item even"><a href="/taxonomy/term/5219">conservation</a></div><div class="field-item odd"><a href="/taxonomy/term/5220">independence</a></div></div></div><div class="field field-name-body field-type-text-with-summary field-label-hidden"><div class="field-items"><div class="field-item even"><p>To the best of my knowledge, the two momentum conservation principles, namely, the conservation of linear- and angular-momentum, operate completely independent of each other. For an isolated object, there is no possibility of conversion of one form of momentum to the other.</p>
<p>Today, when I Googled on this topic, I found that most pages agree with my position above. Yet, to my great surprise, I did run into a page written by an engineer here [<a href="http://cnx.org/content/m14342/latest/" target="_blank">^</a>] claiming that the linear momentum is only a special case of the angular momentum... Here is the relevant excerpt (bold emphasis mine):</p>
<p> Conservation of angular momentum is generally believed to be the counterpart of conservation of linear momentum as studied in the case of translation. This perception is essentially flawed. As a matter of fact, this is a generalized law of conservation applicable to all types of motions. We must realize that conservation law of linear momentum is a <strong>subset </strong>of more general conservation law of angular momentum. <strong>Angular quantities are all inclusive of linear and rotational quantities.</strong> As such, conservation of angular momentum is also all inclusive. However, this law is regarded to suit situations, which involve rotation. This is the reason that we tend to identify this conservation law with rotational motion.</p>
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Of course, I think the above argument is flawed, but that's a different story... (I do plan to alert the author right away.)</p>
<p>Coming back to the main issue I want to discuss here: As far as I know, we have these two forms of momenta, and their conservation is independent of each other.</p>
<p>Now, as far as conservation principles go, we also have the principle of conservation of (to simplify the matters, say, the mechanical) energy.</p>
<p>Most, if not all, college-level physics text-books begin illustrating the principle of conservation of energy in reference to only the translational motion. Then, typically, these books go on to directly generalize the energy principle to all types of motion. A few text-books might spend some (though inadequate amount of) time to point out the relation between the two conservation principles: energy and momentum. But most authors (and teachers) simply skip over this entire issue, and directly or indirectly deploy the Argument from the Authority in impressing upon the student that the energy principle is at least desirable if not superior to the momenta principles. (Even Resnick and Halliday, e.g., think of energy conservation as one of the unifying themes of physics, and base the organization of their text around this theme, and so.)</p>
<p>Consider the motion of an object of non-symmetrical shape and non-uniform matter density, e.g., an inter-galactic probe, and suppose that its batteries (or other power-sources) have run out (or have become defunct). Let us also neglect any external forces such as gravity and friction for the time being. For the motion of an isolated object like this, in the general case, there would be both rotations and translations.</p>
<p>Now, such an object can be assumed to have a constant mechanical energy. But that is only one part of the story. The fact is, since each linear and angular momentum of this object is conserved, the respective components of its linear and angular energy also would be conserved independent of each other.</p>
<p>The question I had in mind is this: Not just in the above example, but considering the totality of your knowledge, do you know of any single physical situation or context in which the linear form of momentum transforms into the angular form, or vice versa?</p>
<p>One notable example I can think of is that frustrating problem of pushing an egg with the sharp tip of a pencil. The egg refuses to get pushed; it simply rotates in place. ... Even a point-sharp pencil could have imparted a translational motion if there were no friction between the egg and the supporting surface. It's this friction which helps convert all your applied point force into a torque. ... But even otherwise, whatever be the effectively applied torques and forces, once they set an object in motion in a frictionless world, it would be impossible to convert rotations/spins into translations, and vice versa. ...</p>
<p>...If you wish to have a present-day media-friendly sound-byte about it: the spin will keep spinning without getting anywhere. (Apart from being media-friendly, it also describes the media rather well, doesn't it?)</p>
<p>Or, if you prefer to think about the whole situation in terms of energy, let me ask: for an isolated (unconstrained and unforced) physical object, what is it that prevents a transformation of the translational form of the energy into the rotational form or vice-versa?</p>
<p>So, it's one matter to say that the total energy is conserved; it's another matter to say that the specific fractions of the linear and angular parts of the total energy also stay unaltered.</p>
<p>I think that physical scientists and engineers should have already acknowledged the second part as a principle in its own right: the principle of impossibility of interconversion of the linear and angular parts of motion of an object i.e. its momenta/energies.</p>
<p>Here is a request: If you find any explicit recognition of this principle in the prior literature (easily possible right since Isaac Newton, Jr.,'s Cambridge days), then please do point it out to me. Thanks in advance.</p>
<p>Incidentally, let me add one more observation. Why the above cited author might have thought that the angular momentum is more general.</p>
<p>Consider a solid object having its center of mass (CM) at a point P. If you apply a "force" F at a finite perpendicular distance s from CM, then what the "force" actually acts as is a torque, of magnitude T = Fs (and appropriate vector direction). Consider a smaller distance, and the torque gets reduced even if the applied "force" is the same. In differential terms, we may say: dT = Fds. When ds tends to zero, the torque dT tends to zero, but the F remains finite. At ds = 0, the object is imparted a force, not a torque. Considering this mathematical part, there is likely to be a kinetic argument that the concept of torque encompasses that of force, or, in kinematical terms, angular motion includes linear motion.</p>
<p>The argument is false. There is a certain reason why I have been putting quote marks around "force." When you consider application of a "force" with a finite s, in theory, you are considering the effect that the "force" applying agent, if acting singly on another hypothetical object, would have on that hypothetical object---the effect of linear acceleration. But this entire situation still is hypothetical---not actual. In the given actual situation, we assume that there is a constraint or at least a friction-providing surface on the other side of CM, and this couple of forces then goes on to produce a torque, which in turn produces angular acceleration. To treat just one of the forces out of context, is an error.</p>
<p>Further, as I keep pointing out in my writings, there also is a subtle point about the essential difference between the infinitesimal, the finite, and the zero. Most people tend to confuse between the case that the ds tends to zero and the case that the ds is zero. Not just beginning students and common engineers but also mechanicians with PhDs. By way of example, consider my explicit position in the discussion on point vs particle at iMechanica here [<a href="node/5214" target="_blank">^</a>], and also the lack of any explicit support to this position in that discussion. If you consider a material particle to have zero volume (or mass), you are essentially committing the same error as that involved in directly setting ds equal to (and not tending to) zero, in the above example.</p>
<p>Both errors arise from misunderstanding the difference of an infinitesimal from a zero. Further, I also suspect, an over-emphasis in looking merely at the mathematics also is a reason.</p>
<p>In general: you cannot substitute mathematics in place of physics. I am "inherently" very cautious of any characterization of physics as being inherently mathematical---as famously done by Dick (Richard) Feynman, and also as stated in the Table of Contents of a forthcoming book by David Harriman [<a href="http://www.amazon.com/Logical-Leap-Induction-Physics/dp/0451230051" target="_blank">^</a>]. I do look forward to reading it.
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In the meanwhile, comments and corrections regarding my position(s) in this blog post are certainly welcome.
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Note added on May 24, 2010: Also posted at my other blog [<a href="http://ajitjadhav.wordpress.com/2010/05/23/are-linear-and-angular-momenta-interconvertible/" target="_blank">^</a>]
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[E&OE]
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</div></div></div>Sun, 23 May 2010 09:48:43 +0000Ajit R. Jadhav8288 at https://imechanica.orghttps://imechanica.org/node/8288#commentshttps://imechanica.org/crss/node/8288