Vector problem

[Michael Wathen 556-9565]

This operation of reducing the size of the knuckle is
guaranteed to increase the touchweight and it should do so rather
dramatically.  If I understand the procedure correctly it will
reduce the "action arm" of the hammershank.  The "action arm" is
a term is physics that defines the distance from a center to a
point where a force is applied along with a consideration of the
angle where that force is applied.  Mathematically

      Torque = (D * Sin (A)) * F

In terms of our action D is the distance from the center to
where the force F is applied.  A is the angle that that force
makes with the line along which D is measured. F is the force.
In the piano D is from the center pin to where the jack contacts
the knuckle. "A" is generally 90 degrees.  The "action arm" is
(D * Sin (A)) which in our case is(D * Sin (90 degrees)) .
Since Sin ( 90) = 1 then our "action arm" is just D. So if you
reduce the size of the knuckle you will reduce the "action arm"
length which in turn will require the Force that the Jack
applies to the knuckle to be greater which can only occur with
more Force at the key.

Here is another thought I would like to add to Ken Sloane's
comments about this problem.  He has it exactly right.  His test
will work.  When the top of the capstan does not line up with
this line then it is forced to slide somewhat.  This is "kinetic
friction" which almost always is lost energy from the system
being given away in heat.  When the capstan is in the correct
place there is no kinetic friction but there should be some
friction otherwise the parts will not move.  This friction is
generally referred to as "rolling friction".  If you think of a
car wheel you want some friction especially on an icy day.
Friction can be our friend or our enemy.  In terms of our action
the place where the capstan meets the whippen can be looked at
as like  cog wheels transferring their motion.  Although there
are no "cogs" you can imagine by such thinking  the interaction
of the two surfaces having friction as producing the same
result.  It is for this reason that I'm willing to bet that
polishing capstans is a waste of time.  The only time that it
would produce any effect at all is when there is "sliding
friction"; that is, the capstan meets the whippen off of our
ideal line.

Michael Wathen