Key Inertia

[Mark Davidson]

Phillip,

Since acceleration is just change in velocity/time, and velocity is change in distance/time, I guess I would expect the
acceleration, velocity and distance ratios to all be the same.

As to where the square thing comes from, well I don't claim to completely understand how it's derived, and I certainly don't claim
its obvious, but the term for this type of "perceived" inertia is "reflected inertia".  If you search on that you will find all
kinds of stuff, mostly about electric motors and gear boxes.  They consistently say to divide the inertia by the square of the gear
ratio (they're usually gearing down, not up).

Here's a link full of not very comprehensible explanation:

http://www.tech.plym.ac.uk/sme/desnotes/gears/gearaccel.htm

I don't follow everything in there, but the result makes sense to me for the following reason: if the resistance at the key results
in energy in multiple moving parts, then the resistance is due to the work being done on each of those parts.  So the resistance
should break down the same as the energies.

-Mark

> >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
> >