What's all this I hear about Inertia ?

[Greg Graham]

David Stang wrote:  "But I will reiterate my original point. Mass is the same as inertia. And mass is proportional to weight. So on earth, two things with the same weight have the same inertia. That's the point I had been kind of hung up with."

David, as I will try to demonstrate below, Inertia is NOT the same as mass.  

David also wrote: "Rotational movement is another kettle-o-fish altogether, and has nothing to do with the point I (admittedly ineptly) was trying to make."

David, when you are talking about pianos, rotational movement has everything to do with the point you are trying to make.  Piano parts rotate.  Keys rotate around the balance pin hole.  Hammers rotate around the shank center pin.  

Here are a couple important things to consider:  

Force equals mass multiplied by acceleration  (F = M x A).  A key will require more force when you play loudly because you are trying to move the key faster (greater acceleration).    But... to complicate things, the distance from the key lead to the balance pin has a big influence on the force.  

So, for example...  would it be better to have a lead with mass of 1 gram 10 centimeters from the balance pin, or 2 grams 5 cm from the balance pin?  

>From a static (non-moving) balance point of view, they both do the same thing.  The force of the lead is equal to its mass times the acceleration of gravity (g = 9.8 m/s^2).  This force, commonly known as "weight", acts straight down.  The static torque about the balance point is the weight (force) times the distance between the lead and the balance pin.   2 x 5 x g = 10g, or 1 x 10 x g = 10g.  You can cancel the g.  The acceleration of gravity is the same no matter where you put the lead.  I think this is what you were suggesting. 

Here is the fine point you may have missed:   The acceleration due to key movement is NOT the same everywhere on the keystick.

When you start to move the key, you are adding new acceleration to the key.  The velocity of the lead 5 cm from the balance is half what it is at 10 cm, because it is only moving half the distance in the same amount of time.  Velocity = distance divided by time (V=d/t).  Acceleration is the change in velocity over time (A=V/t) which is the same as distance divided by time squared  (A=d/t^2).   

At the front of the key, you could have a dip of 10 mm.  A very loud blow would be fast and cover this distance in a short time.  Let's guess a tenth of a second, or 0.1 sec.  The acceleration of a lead halfway between the keyfront and balance pin would be 5 mm / 0.1 sec / 0.1 sec, which is 500 mm per second squared, or 0.5 meters per second squared (0.5m/s^2).  A lead three quarters of the way from the keyfront to the balance pin would accelerate 2.5mm/0.1sec/0.1sec = 0.25m/s^2.  

The 1 gram lead at 10 cm requires a force from the key to cause it to accelerate 0.5m/s^2 which we can calculate using F=ma, or 1 gram x 0.5m/s^2 = 0.5 gram-meters per second squared.  The 2 gram lead at 5 cm  likewise is calculated, 2 gram x 0.25m/s^2 = 0.5.   Same force, right?  Yes, but acting at different lever lengths.  The torque equals force x distance.  The lead at 10 cm produces torque of 0.5 x 10 = 5, but the lead at 5 cm produces only 0.5 x 5 = 2.5.   

This is the importance of inertia.  The key with heavier lead closer to the balance pin feels lighter at high volumes than the key with lighter lead farther from the balance pin.  

Static (weight) balance involves mass times distance.  

Dynamic (Inertial) balance involves mass times distance SQUARED.

The faster you try to move, the less important gravity becomes, and the more important inertia becomes.  

But this is all theory.  For practical application, please see
http://www.stanwoodpiano.com/articles.htm 
and read how all this can be used in real piano work.  

But please don't say mass is the same as inertia.  

One last thing:  You said "There would be less wasteful bending and
more dynamic range if there were less weight in front. Correct me if
I'm wrong here,...."

Well, think of this:   If all the lead was at the front of the key, your finger would be acting directly on the lead, so the key would not have to bend at all to get it moving.  Bending is not the issue.  Acceleration is the issue.  The lead at the front of the key would have to move the entire distance of the keystroke, rather than a fraction of that if placed closer to the balance pin.  

Regarding dynamic range, there is only one way to increase it:  Make it possible to play louder.  The bottom end is zero, and never changes.  Silent = zero.  Maximum volume comes with maximum hammer momentum, which is mass times velocity.  Heavier hammers moving faster.  High inertia in the keys may limit the player's ability to accelerate a heavy hammer to maximum velocity, but heavy hammers require heavy counterweights in the keys to keep static touch weight reasonable.   Maybe higher static touchweight would be OK if the trade off was lower dynamic touchweight (lower inertia)?  Maybe key bending and hammer shank bending can be made to work with the performer to whip the hammer to higher velocity than a rigid lever would?  

This is a deep topic, and truly IS worthy of study, Journal articles, test equipment, and all of that.

As in most of life, there is a simple answer for everything, and it is usually wrong.

Greg Graham, RPT (and Mechanical Engineer 20 years ago)