[Jeff D]

Hi Rivkin

I would like to address some of your questions and hope it can add to Ian's excellent answers.

Originally Posted by Rivkin

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

Inertia is the tendency for a body at rest to stay at rest or stay in uniform motion. It requires an external force to alter this state. With our sword example I think its relevance is more in the handling of the sword than in its ability to cut as per Jen's original question. So I would agree with you that for all intense purposes it would be OK to consider it the center of mass.

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 ?).

I am not sure what model the engineers used, however I think it would be safe to use a bar with a clamped (fixed) end (similar to a hilt) as a model. Using this model the formula is;
F(1) = 0.162 {a/(L squared)} {the square root of(Y/d)} where a=thickness, L is the length of the bar, Y is Young's module (which is a variable of the elasticity of the material), and d is the density.

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

Using my model I would not consider the tang in the Length formula as the hilt and tang would be the fixed portion of the bar.

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.

I think again we are confusing some of the issues. The vibrations only affect Jen's question in reguards to the energy transfer. The less vibration, the more energy transfered from the kinetic energy of the swing into cutting energy. If there is a lot of vibration then a lot of energy is wasted. The vibration will obviously affect the handling as well. I think a complex bar with varing thickness, fullers etc. would set up a series of standing waves that would cancel each other out so that there will be minimal palpable vibrations.

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.

I am not sure of what your question here is but I think you are correct in that the center of percussion is where the blade strikes the sword, and should there for be where the resistance of the object balances.

Hope this adds a bit.
Jeff