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1. Do All Torque-Free Systems Necessarily Exhibit Polhode Motion Or Is It a Special Case for Specific Geometries (E.G

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1. Do All Torque-Free Systems Necessarily Exhibit Polhode Motion Or Is It a Special Case for Specific Geometries (E.G Ph610 Fall 2012 Students’ questions #6 1. 1. Do all torque-free systems necessarily exhibit Polhode motion or is it a special case for specific geometries (e.g. Poinsot's Ellipsoid)? 2. How good of an approximation is the classical precession without accounting for a variable precession and variable obliquity, the Earth being non-rigid and non-relativistic precession effects? 3. Is there a straight-forward way of identifying which convention is being used for Euler's angles? The convention seems to be switched between quantum and classical books. Do either fields use the x-convention that mathematicians use? 2. Questions 9/30 The notation for the tutorial of problem 6.1 confuses me. Your arrows above the I indicate a tensor but this is a scalar. It is the "moment of inertia about the axis of rotation" Goldstein p. 192. The moment of inertia tensor is the between the dot product. So in part a) when you ask for the rotational inertia tensor, it's confusing as to which I you want. I've tried to compare the terminology with Goldstein, Marion, and Taylor mechanics, as well as various websites. Rotational inertia tensor is not in common use, but I think you mean the moment of inertia tensor. For part c), or in general, I don't understand how include time. My tensor in part a) is supposed to be I(t) at t = 0, but seems identical to what I'd get for the I independent of time. I don't understand how to make it time dependent. Do I just multiply dt onto each matrix element? 3. 1. How difficult is it to extend our analysis of rotating, rigid objects to non- rigid objects? How is it done? 2. How would you explain to a child the reason that a top precesses? I have been asked this question before and found that my physical intuition for the problem was lacking... 3. I know guns/bullets are designed so that the bullet is rotating as it travels to its target, presumably for purposes of accuracy. Is the idea behind this simply that giving the bullet an angular momentum helps to ensure that its direction does not change, or the reason lies in a more complicated analysis of air flows around the bullet? 4. 1. Regarding the football examples from class: For the imperfect spiral, I understand why the angular velocity vector precesses around the axis of the football, but I don't understand why the overall angular momentum vector (which is constant when there is no torque) wasn't directed along the axis. It especially seems this should happen in the case that the precession of the angular velocity is centered around the axis. Could you help clear this up? 2. An undergraduate E&M professor told me that in some MHD problems, it's useful to have the coordinate system run along the field lines. Do you know of any other examples in which non-orthogonal coordinate systems are useful? 5. 1. The question is about notations: in the HW 6 what does it mean \vec L = \hat x_i I_ij (t) \hat x_j \vec omega. Why can't we apply that \hat x_i and \hat x_j are orthogonal? Should the equation just be L_i = I_ij omega_j ? 2. How should we treat the free top problem if atoms are no longer at fixed position but oscillating around their equilibrium positions and the oscillating frequency depends on angular velocity of the top. 6. 1. angular velocity is dependent on the origin? 2. the strategy in question 1-(d) is valid only when t is small? for general t function of w, should we have to use the euler-lagrangian equations? 3. the equation 4-82, what if there is a translation motion of body relative to the fixed frame? then should we additional term for that? 7. 1) I am under the impression that the Moment of inercia tensor is always symmetric. Is it true? and if it is not can you show me an example. 2) In Problem 1 part c "Find the rotational inertia tensor at a later time dt" my answer does not depend on dt, only up to second orden. How can I interpreter this result (if it is correct)? 8. 1) When we consider problems of rotations about fixed axes are we assuming all forces are central? If we didn't assume this and the potential energy was dependent on the direction of r then wouldn't the inertial forces contribute to the total angular momentum? 2) In regards to the problem of the precession of the spinning top, we assumed the angular velocity was constant but it isn't completely obvious to me why this can be assumed. Can you perhaps explain why the angular velocity can't be anything other than constant? 9. 1. I know that rigid body is an idealized system that do not work in quantum mechanics. From that, I am assuming that there is a correspondence principle between classical rigid body and quantum mechanics that justifies the study of rigid body system just like the Ehrenfest's theorem. If there is such general theorem, can you explain it? 2. I am curious about semi-rigid body system where the rigid body slightly expands around the axis of rotation(possibly shrinks along the axis). If such system exists, can we still apply what we learn in the class to study the system? 10. 1) Can we present evolution of material point as a transformation of coordinate system? 2) Which restrictions should we have in order to present it as a combination of rotations? 3) Which condition we should have for presenting two rotation on two different axes as one another rotation? .
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