On October of 2014, prompted by the
travel ban on ivory,
Eric Swanson posted an article about the feasibility of swapping and ivory bow tip for a silver one.
He argued that the extra weight of the silver tip would not significantly alter the bow's balance point.
If one did care about the slight shift, and wanted to rebalance the bow, he pointed out that extra mass could be easily added to the winding.
(If you've never given much thought to the names of bow parts, see Fig. (1) below.)
Fig. 1(a) Bow tip with ivory plate protecting the wood.
Fig. 1(b) Wire winding (left) and thumb grip (center) protecting the wood from the players hand.
The question then became, can the balance point shift be easily calculated?
Can the extra mass needed in the winding be easily calculated?
Is the precision used by the average bow maker good enough to make the calculations, or even the rebalancing, useful?
This post explains the physics behind the balance point and rebalancing, and answers these questions.
If you like spoilers... the answer to the first two question is "yes", and to the last "probably not".
To read the article, scroll down or use the table of contents.
To discuss the balance point of a bow, it is useful to understand it from a physics standpoint.
Physicists refer to the balance point as the center of mass.1
It is the point where, for any object of any shape, if you provide support directly under that point, the object will balance. Mathematically, instead of worrying about how gravity is pulling down on different areas of an entire object, we can imagine the force of gravity pulling down on the object at its center of mass. This equivalence allows for substantial simplification of many physics problems, including the one we have here.
There are many pages on the internet that can give more details about why this is legitimate; e.g. see wikepedia's entry.
While I will do my best to use words when possible, using variables and equations often allows for more clarity.
I will define variables as I go, but for convenience I list them all here.
The distances are shown in Fig. (2), below.
Fig. 2 Violin bow with positions labeled.
1. Technically, we want to use the center of gravity. The difference between center of gravity and center of mass is subtle and makes absolutely no difference in the situation here. If you are doing bow repairs or playing Mendelssohn near a black hole though... Well, this would be the least of your problems.↩
Fig. 3(a) A stick of wood showing the center of mass.
Fig. 3(b) The same stick with two objects hanging from it.
What if we take all measurements from the right end of the stick?
Fig. 4 Same set up as in Fig. (3) but with all measurements taken from the far right end.
What if the stick is not of uniform shape and/or mass?
We can do all these calculations with a stick of unusual, asymmetric shape and mass distribution as long as we know where its center of mass is. This is very helpful when thinking about alterations of bows. The original, unaltered bow has a center of mass we can easily determine just by balancing it. You can balance it on your finger, or for better precision, a pencil or rulers edge.
One way of correcting the bow's balance point (returning its center of mass to the original point) is simply trial and error.
One needs add mass to the end opposite of the tip.
A simple place to do this is in the winding. A bow maker can replace the original winding with a slightly thicker wire adding more mass.
They could also extend the winding, further adding to the mass.
By marking the balance point of the bow before replacing the tip, they can then simply add coils to the new winding one at a time until the balance point is restored to its original point.
This trial-and-error method is very practical and easy to implement.
Its limitation is that it is only as good as the bow maker's ability to measure the balance point in the first place.
As discussed, this could typically have an error of a millimeter or two, comparable to the shift induced by the change in tip material
