Key Inertia

[Phillip Ford]

>Phillip,
>
>Your example is perfect, but neglects one important thing.

Then I guess it wasn't perfect.

>
>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.

Yes, I see your point.  But since you mention it, when we're talking 
about inertia we're concerned about acceleration, not distance or 
velocity.  In order for the hammer to move 10 times the angle of the 
key in the same time, its velocity would have to be 10 times that of 
the key, not its acceleration.  What would be the ratio of the 
accelerations? (I think once upon a time I would have been able to 
figure this out).

>
>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.

I think it has to be adjusted by the acceleration ratios.  If 
everything were moving at constant velocity it seems that there would 
be no inertial effects.

Phil Ford

>
>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
>
>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
>