Sliding weights

[Ian]

Hi Kirrill:

You raise some interesting points and I will try to deal with them as best I can. My college physics is but a distant memory!

Both of my local contacts went on vacation on Friday, and will be out of the office for the next four weeks. Academics do very well with vacation time. I will do my best.

Ian.

Originally Posted by Rivkin

1. With given definitions the center of gravity and the center of inertia will be the same.

As described above, the respective moments vary with distance of the various components of mass from the handle, raised to the power of 0 (the zero moment, which is simply the total mass); with distance raised to the power 1 (the first moment); and with distance raised to the power of 2 (the second moment). The center of gravity relates to the first moment, the center of inertia relates to the second moment. If mass is distributed uniformly along the length of the rod, then I believe that the center of gravity and center of inertia will be the same. When mass is distributed unequally, then the two will be different. The difference will be demonstrated by the two tests I listed.

Originally Posted by Rivkin

2. If its possible, I would really like to see the formula they use for sword's frequency as a function of length (do they consider it a string ? a thin and long prism ?).

Need the experts for this one. I believe they modeled this as a solid rod.

The frequency we are talking about, then, is the resonant frequency of a solid rod, which (if I recall correctly) for a given diameter varies with the density of the material and its length. When we talk about a string, there is also a factor for the rigidity of the material or tension applied (a taught string resonates at a higher frequency than a slacker string). The resonant frequency is fixed for a rod of given dimensions and homogeneous construction. The amplitude of the vibration varies with the distance the rod is struck away from the resonant node.

An interesting example is the aluminum (aluminium) baseball bat, which has an outer aluminum shell and an inner core that is air-filled. Striking a ball with such a bat produces a brief, high-pitched "ching," and a lower-pitched "thunk." The higher pitched sound reflects the resonant sound of the metal shell, and the lower-pitched sound comes from resonance in the air-filled chamber.

These sounds are hard to distinguish with the human ear but apparently have been measured with sophisticated recording equipment. The low frequency sound is just a few hundred cycles per second, approaching the limits of detection for the human ear.

Originally Posted by Rivkin

3. Concerning tang not being a b.c., or even a separate body, I would prefer to hold a vastly different opinion.

For full tang construction, there should be no boundary condition because the tang is essentially an extension of the blade. This is the same situation as a baseball bat, and the handle presents no boundary condition in that example.

As I mentioned above, there may be dampening of the vibrations by materials around the tang. For partial tang construction, I am unsure how much of a boundary condition there may be. It probably varies with the width and length of the tang, and again the wrapping materials will be important in how much dampening of the vibrations might occur for the user.

Originally Posted by Rivkin

4. Concerning waves propagating in swords and nodes - propagating waves usually do not have nodes. When people talk about nodes, they usually speak about standing waves, i.e. steady state solutions etc.
I suspect that the logic was that if sword can be considered a string, than a full wavelength standing wave will have a node in the middle, but it will basically be true only for even halfwavelengths mode... Plus I'm really too lazy to calculate the modes of a string with a variable mass, so I don't know how big percentage of the waves will have nodes at the center of mass.

This one requires the experts. The test described above speaks to a property of standing waves, I think, but the center so defined also identifies the "sweet spot" which relates to properties of propagated waves also -- at least that was how it was explained to me.

With respect to analogous models, I believe that a string as we usually think of it is probably not the correct one. A string can have variable tension. If we exert enormous tension on a string, and essentially make it highly inflexible or "rigid," then we may approach a more representative model. A metal rod has a high degree of rigidity, which is essentially constant for the purposes of this discussion.

Originally Posted by Rivkin

5. Concerning the center of percussion - as far as I remember (and I remember it very poorly), the center of percussian is when you hit it, all the momentum is transfered into the rotation movement of the sword, without any daggling down or up.

Could be, but we need the experts for this one too.