So Imagine that a force of 100 N is being applied continuously on face of 5 kg cylindrical rod (1 light year across) causing it to accelerate at 20 m/s^2. If the force was abruptly removed would the rod continue to accelerate a noticeable period of time before coming to a halt? The density of the rod is the same at all points. I believe that logically the rod would continue to accelerate but that begs me to ask how fast this "wave" of "force-information" is travelling? Is it a sound wave (It appears to me) or is it travelling at C? (Here I assume that the speed of sound within the rod is less than C) Given that, here's my follow up question (a bit more interesting I suppose) We define U, as the Net Yank or Jerk * Mass. Where Jerk is the time derivative of acceleration or the third time derivative of position. U = JM or U = [itex]\frac{dA}{dt}[/itex]M So a Yank of 100 Newton Meters per Second is being applied continuously to our same 5 Kg cylindrical rod (1 light year across) (whose rest-density is the same at all points) causing it to increase its rate of acceleration at 20 meters per second cubed. Suppose this Yank was removed? Does that mean this rod would continue increasing its rate of acceleration for a period of time before coming to halt? Thanks! P.S. Sorry for the Grammar mistake in the title
Re: An Moving Body... The back of the rod (where the force was applied) would immediately stop accelerating (or perhaps within a Plank time unit), the front would conceivably would not even have started to accelerate (and depending on the material, if ever) if the applied force stopped before the end would have been reached carrying a wave with a velocity right about the speed of sound.
Good point PassionFlower let me make it clear that the force/yank is being applied continuously until the entire rod is accelerating/increasing in acceleration The reason for the length being a light year is just to make an arbitrary long length that is long enough to notice the "wave" of Force/Yank information being transmitted.
Re: An Moving Body... ...speed of sound in the medium. If the rod were made out of the hardest substance known - diamond, that would still be a glacial 12km/s.
Re: An Moving Body... Yes you are correct. However diamond is no longer the hardest substance known. See for instance http://en.wikipedia.org/wiki/Lonsdaleite
Coming back to the question, does that mean if the entire cylinder is accelerating/jerking then it would take the time of a wave moving at the speed of sound (in the medium) to cover a light year before the entire rod would come to a halt and until then would continue to accelerate/jerk despite not having a force behind it?
You used the term rod coming to a halt twice now, you probably refer to halting the acceleration. But let's make it clear to everybody than when the force stops the rod will continue to move, the back will immediately stop to accelerate and eventually the front as well.
Yes. Except that it does have a force behind it. Every molecule of the rod has the molecules behind it applying an accelerative force. Only when the wave of deceleration reaches it will the force change.
When the acceleration stops the fastest velocity in the object is right there where the force is applied. No other point can achieve this velocity. There will be a transient acceleration in the rest of the object WRT each points actual velocity. At the furthest point the actual velocity is the velocity the end where the force is applied was travelling at well over a year ago. There will be a mildly jumpy deceleration into a lower velocity than when the acceleration force was removed until the entire object is inertial throughout itself. mathal
You seem to have incorrectly assumed the resulting motion. Evidently you are applying a square force pulse (function of time) to the end of the rod. So, decide what the time duration of your force pulse is, then perform a Fourier Transform on that pulse. And forget about hardness--you need a very high Young's modulus and low mass density if you want high values of wave propagation speed and resonance frequencies: v = sqrt(E/rho), i.e., E = modulus, rho = mass density. You'll have to confront the internal damping--set it to zero if you think you can get away with it. You could get a pretty good approximation by discretizing the rod ( use as many descrete points along the rod as you wish--trillions or triillion-trillions of points, etc.). Fourier transform the differential equations of motion--do the whole problem in the frequency domain. Set up the matrix equations, do the eigenvalue-eigenvector problem. The eigenvectors play roles as both mode shapes and columns of the coordinate transformation matrix (takes you from modal coordinates to generalized physical coordinates, and the inverse takes you from physical to modal). Use the matrix transpose to get the modal coordinate representation of the force. In the equations below, [ Y ] is the matrix of eigenvectors (columns), [ M ] is the mass matrix, [ K ] is the stiffness matrix, w is frequency in rad/sec, X is displacement and F is force. [-w^{2}[ Y ]^{T} [ M ] [ Y ] + [ Y ]^{T} [ K ] [ Y ]][ Y ]^{-1}{X(w)} = [ Y ]^{T}{F(w)} This way you can operate on the rigid body translation mode and compression modes independently. Notice that by transforming the point force to a superposition of modal forces, the rigid body modal force is applied simultaneously to the entire rigid body modal mass, so you don't have to worry about traveling stress waves (compression is accounted for separately with the compression vibration modes). The traveling stress wave is just the result of the superposition of the rigid body translation along with all of the modes of compressional vibration. (Was this intended to be a special relativity problem?)
@ bobc2 No it was not intended to be a SR problem but more of a hypothetical question. Nevertheless I will try to use what you've told me to actually solve it now that I see you've taken the time to make such a descriptive response (I'll have to learn a little more linear algebra (though I am familiar with some DiffEQ material and have read about the fourier transform) but that shouldn't be a problem). Thanks for your post!
By the way, the triple matrix products for the mass and stiffness matrices serve to diagonalize those matrices so that now you can work with the equation for each mode separately. You might just ignore the vibration modes and focus on the rigid body translation mode (the zero Hz mode--computationally you might have to use Cholesky springs rather than the perfectly free boundary condition). The displacement and force are in the frequency domain, so after solving for the displacement vs. frequency, you can inverse transform displacement to get the final form of motion vs. time. This will be the modal displacement, from which you can retrieve physical displacements for each point on the rod using the eigenvector coordinate transformation matrix. The modal mass is just the same as the physical mass (within a scale factor). The modal force is equivalent to a complete set of physcial forces being applied simultaneously to all points on the rod (so, you don't have to be concerned about compressions). The forces in the frequency domain are just sinusoidal forces with amplitudes and phase angles for each frequency of the discretized spectrum. But, again, the easy way is to solve for motion in the frequency domain, then inverse transform to get displacement vs. time.
To clarify the role of the longitudinal compressional modes a little further (a little more theory of normal modes--see Goldstein Classical Mechanics)... Notice there will be a set of mode shapes with the front end and far end vibrating in phase (and every mode shape is vibrating at all frequencies of the spectrum, although the response in each mode of course peaks at its resonance frequency), and there will be a second set of mode shapes with front and far ends vibrating 180 degrees out of phase. At the first instant of application of the force, all mode shapes will be in phase at the front end (at all frequencies of the spectrum), and as you move along the rod toward the far end the phases will tend to cancel more and more. At the far end, in the first instant, there will be as many modes in phase (with respect to the front end phase) as out of phase. That's why there is no motion at the far end initially, and the mode shapes superimpose so as to result in a traveling stress wave (all mode shapes are excited over the entire length of the rod simultaneously--regardless of the length of the rod--just like QM Schroedinger waves--except these are really physical). Question for the student: Will all of the vibrating mode shapes eventually cancel each other out, leaving only the rigid body translation motion? Or do you need damping to eventually rid the system of longitudinal compressional vibration?