Inertia, was "Grand Touch"

[William R. Monroe]

Comments below:

> William,
> 
> Vladan is correct.  The mass vs  force concepts are, unfortunately, not
> intuitive.  It's not made any easier by the fact that we use the same units
> for both mass and force.  (real physicists use different units)  The
> readings you refer to, taken at the front of the key, are force (or weight,
> if you will).  A given value of key front weight could be the result of an
> infinite number of combinations of mass and distance from the fulcrum. 
> Might be a large mass close to the fulcrum, might be a smaller mass further
> from the fulcrum.  So measuring front weight does not tell us the mass.  To
> determine the rotational moment of inertia, it is necessary to know where
> the mass is, and how big it is.  

Right, I agree.  My question is more directed at the concept of whether more mass closer to the fulcrum yields a different MOI than less mass farther from the fulcrum.  Is the fact that some of the weight of the key/lead combination is supported at the fulcrum at all relevant in our real world example where the pianist plays at the front of the key?  That is, if we change the point at which we apply the force on the key, will that alter our calculations of inertia?


Also, to Vladan (anyone) again, in your example of the two differing leads at different places, my brief experimenting seems to support this concept.  That is, I just weighed off a keystick and tared the scale to zero with no weights added.  Adding a 10g weight at 5cm from the fulcrum gave me a FW of 3.5g.  To get the same 3.5g FW with a 5g weight, that weight needed to be not 10cm, but 11.9cm from the fulcrum.

So, following your example calculations, I have:
10g*5cm*5cm = 250gcm^2
vs.
5g*11.9cm*11.9cm = 708.1gcm^2

Egads, it's worse than I thought.  A much greater inertia with less weight further from the fulcrum, as you predicted, with same FW.  Now, I only question if this is the appropriate formula for calculating this relationship.  I'm not suggesting you are wrong, I simply don't have as clear an understanding of all that we should consider for this problem.  As I said, I'll go read Stephen's paper now, and see what I can glean from that.

Thanks,
This is fun,
William R. Monroe


> > > I=m*r^2
> > >
> > > I=moment of inertia
> > > m=mass of the lead
> > > r=distance from the balance rail hole to the lead
> > >
> > > Note that the formula takes the square of "r".
> > >
> > > Now, let's look at our options for lead placement.
> > > Suppose that we can use a 12 gram weight placed 10 cm
> > > away from the balance rail hole to get the desired
> > > balance weight.  As an alternative, we could also use
> > > a 24 gram weight placed 5 cm away from the balance
> > > rail hole and still have the same balance weight.  Is
> > > there a difference between the two?
> > >
> > > Yes, there is.  Looking at the formula above, the
> > > inertia increases with "r" squared.  In the first
> > > case, the inertia will be 12*10*10=1200 gcm.  In the
> > > second case, the inertia will be 24*5*5=600 gcm.
> > > Placing the 24g weight 5cm away from the balance rail
> > > hole reduced the moment of inertia by 50%.  That's a
> > > lot.  You can minimize the inertia by placing the
> > > weights closer to the balance rail hole and increasing
> > > the amount of lead accordingly so that the balance
> > > weight meets the target.
> > >
> > > There is a fundamental trade-off between balance
> > > weight and key stick inertia.  If we add a lot of lead
> > > to make the balance weight low, the action will feel
> > > very light, but only when you move the key slowly.  If
> > > you try to play a loud note, all that lead will have
> > > to be accelerated and the key will feel heavy.
> > >
> > > If you put very little (or no) lead into the key and
> > > accept a high balance weight, the action will be heavy
> > > when you measure it with your weights, but it will
> > > feel light when you play fast/hard/loud.
> > >
> > > The question is: What is better?  My own preference is
> > > for a heavier balance weight with less inertia.
> > >
> > > Vladan
> >
> 
> 
> 
> 
-------------- next part --------------
An HTML attachment was scrubbed...
URL: https://www.moypiano.com/ptg/pianotech.php/attachments/20060712/8ca96b0a/attachment-0001.html