Key Inertia

[Phillip Ford]

>Phillip,
>
>Your example is perfect, but neglects one important thing. 
>
>The piano key moves approximately 2 degrees while the piano hammer moves
>approximately 20 degrees.  That's a factor of 10 difference in 
>angular acceleration.
>
>In a coupled system such as this where the parts move at different 
>angular speeds,
>the inertia (i.e. the inertia of the hammer as felt through the key)
>must be adjusted by the square of the speed ratio.  That's a factor of 100.

Oh, and an additional comment.  I don't understand where this square 
factor comes into play for inertia.  I see that it comes into play 
for energy.  I thought the governing equation for angular 
acceleration was T = I a, where T = Torque, I is moment of inertia, 
and a is angular acceleration.  What I would call inertia is T.  If 
you want to accelerate the action, T must be applied by your finger. 
T is proportional to a, not to the square of a.  So, if the hammer is 
indeed being accelerated 10 times as much as the key, it seems that 
the inertia of the hammer needs to be adjusted by a factor of 10, not 
100.  This is also assuming rigid coupling among all the components. 
In fact there is compliance from felts, knuckle, shank bending, etc. 
which may cause the hammer acceleration to be less than a factor of 
10.

>
>I agree that the inertia of the hammer hasn't changed, but it feels larger due
>to the action leverage, and that gearing is part of what causes the energy to
>go into the hammer rather than the key.
>
>So now your inertias become
>
>Inertia of hammer about its center  = 10 x 13 x 13 x 100 = 169000 g cm^2

That would make this 16900, still much lower than the key inertia in 
this example.

Phil Ford

>
>Inertia of key lead about its center = 50 x 23 x 23 = 26450 g cm^2
>
>Think about a flywheel turned by a gear with a crank attached.  If you change
>the gear ratio, the resistance to the crank (which is all due to 
>inertia) changes
>even though the inertia of the flywheel has not changed.  That's 
>whats going on here.
>
>Now I'm not really trying to say keyleads are insignificant (well I 
>guess I did),
>just trying to make the point that they're not AS significant as 
>people assume.
>
>-Mark