Oops... Better late than never. I had edited this email and set it aside in my drafts folder, rather than sending it. Then I thought I had sent it... Oh well.. FAIW, you asked for clarifications on what I had written in my last email... ----------------------------------- Hi Ric, Ric: Depends on what you are talking about. Inertia is per definition the tendency of an object to stay at whatever velocity it is at, and that in the same direction.. And an object will do just that unless some force(s) acts upon it to make it do otherwise. >> Sarah: That's a more elaborate definition of inertia than is really necessary. *Mass* is the quantitative expression of inertia, at least in the sense of translation -- i.e. from point A to point B. Units are in kilograms, for instance. The terms "inertia" and "mass" are approximately interchangeable. Then there is rotational inertia, which is expressed as sort of a "rotation mass." Rotational inertia is the sum of masses, times the square of distances of those masses from the point of reference. A gyroscope, for instance, has a lot of rotational inertia. The units of rotational inertia would be kg-m^2, for instance. Ric: So what I see is this... we use x amount of force to accellerate a key to a max of say 1 m/sec and in that millisecond or so we are talking about here... whatever mass in the key is brought to up to that velocity. There are two things that are going to counter the keys inertia.. the first is gravity, the second is the load on the capstan. The flip side of this is that the inertia of the key (and lead) is going to try and force the other two to change their velocities. Obviously its going to have a hard time affecting the earths mass, but the load on the capstan is another thing entirely. >> Sarah: I think where we're miscommunicating is that you are using the term "inertia" in the same way that I would use the term "kinetic energy." They are actually different. Kinetic energy has a velocity component to it, while inertia does not. We're also talking about collisions and mechanical coupling in different ways. If I might, I think this is how you're thinking about the keystick/wippen energy coupling: The finger delivers energy into the keystick. The keystick moves. Because of its mass, it has a certain amount of kinetic energy. Then (roughly instantaneously) the capstan delivers energy to the wippen/hammer. The inertia of these elements provides resistance against the capstain's motion, slowing the keystick. This is how I picture the situation: The capstan and wippen are already in contact. The wippen and hammer shank may as well be coupled, being in such close proximity. When you depress the key, the entire mechanism moves as one unit. Movement of the elements is driven by transfer of force (not inertia) through the system: The keystick applies force to the wippen. The wippen applies force to the hammer shank. All elements are accelerated together. To oversimplify somewhat, the force the keystick applies to the wippen is equal to and opposite of the force applied by the wippen to the keystick. The total accelerational force on the keystick is the force of the finger, minus the force of the wippen (which acts in an opposite direction). (I'm oversimplifying by not accounting for leverage or rotation, but simplicity is better for illustration. Bear with me...) The sum of these forces is the force that accelerates the keystick. Assuming constant force on the keystick, the total energy delivered into the keystick is force times distance (10 mm), which is the kinetic energy that gets dissipated into the front rail. If the keystick were completely massless, there would be no differential between the finger's force and the wippen's force, and thus there would be no wasted energy. If the mass of the system were dominated by the keystick, then the force differential would be quite high, hence a large degree of energy wastage. Now, the same principles apply to the transfer of force and energy through the wippen to the hammer shank. In the end, the system is characterized by a total inertia, comprised of the rotational inertia of the keystick, the rotational inertia of the wippen (sans jack), translational and rotational inertia of the jack, and rotational inertia of the hammer/shank/knuckle. Effectively, these "feel" and "act" like translational inertia at the front end of the keystick. All these elements are set into motion at the same time, and each element that moves acquires kinetic energy, attributable to its rotational inertia and its rotational velocity. When the keystick reaches the end of its travel, its kinetic energy (differential force, times distance) is delivered into the front rail. When the wippen reaches its end of travel, its kinetic energy (again, differential force, times distance) is transferred primarily into the letoff button (??), and when the hammer reaches its end of travel, its kinetic energy (this time, force times distance, since there is no oppositional force, simlistically speaking) is transferred into the strings, with considerable wastage in the friction of the felt. ------------------- Sarah: You're sitting on the hood of a car. The car runs into a massive concrete barrier at 5 MPH. You fly off of the hood, over the barrier, and head-first into another barrier. Now repeat the experiment with a freight train. Same speed -- 5 MPH. We'll assume we can build a barrier to stop even this massive machine dead in its tracks. You fly off the top of the freight train into a barrier. Q: Which is worse? A: Neither. They're both exactly the same to you. You still collide with the second barrier at 5 MPH, with the same kinetic energy -- your body mass, times your velocity squared. The inertia of the car vs. the freight train has nothing to do with the severity of your impact. Likewise, the kinetic energy of the car vs. the freight train has nothing to do with your impact. Ric: Really Sarah.. I dont see the analogy here... Sarah: Freight train/car = keystick. First concrete barrier = front rail. You = wippen/hammer. The second wall = the strings. The "ouchy" effects of colliding with a second barrier are an indication of your kinetic energy. Both you (the wippen/hammer) and the freight train/car (keystick) are set into motion at the same time. My point is that no matter the scenario, the same amount of energy will be transferred to the strings. In the case of the freight train, there's much more wasted energy in the form of unneccessary kinetic energy in the vehicle beneath the seat of your pants. --------------- Sarah: The moral of this little thought experiment is that it is the velocity of the key, not the inertia, that affects how fast the wippen moves. Ric: Ok... drop a 20 pound lead on the key... now tell me ... is it the leads velocity... or is it the leads inertia that will get things moving. Sarah: LOL! Now you're talking my language! ;-) This is a different sort of situation, though. You're talking about a collision -- which doesn't actually happen. Fair enough, but let's talk apples and oranges: Let's compare a 10 kg bowling ball and a 10 g marble. Drop both from the height of a meter. Kinetic energy in both will be the product of gravitational force times distance (1). However, the kinetic energy of the bowling ball will be much greater, since the gravitational force on the bowling ball will be directly proportional to its mass. Considered another way, the bowling ball and marble will fall at the same rate and will therefore have the same velocity squared. Since kinetic energy is equal to 1/2 mass times velocity squared, the bowling ball's kinetic energy will be a thousand times that of the marble. Fair enough. There will be a lot of key repair to do in one of those examples. Now let's consider what happens with the key: First assume the key has a good bit of inertia. The bowling ball will squash it, and the piano will go "BANG." The marble will strike it, and the piano will go "bing." Yes, there's a difference. However, look at the velocity of the keystick during this process. The keystick will move at roughly the precollision velocity of the bowling ball but at only a fraction of the precollision velocity of the marble. This is a velocity difference, and that accounts for the "BANG" vs. the "bing." Now, if you want to equate the bowling ball / marble to the keystick and look downstream from this point, fair enough. Remember, however, that the keystick and the wippen do not "collide," as with this example. BUT IF THEY DID... (assuming the finger accelerates the key, then leaves the finger, whereupon the key's capstan contacts the wippen heel)... OK, using the above example, the "bowling ball" keystick has a lot of kinetic energy and "whams" the wippen heel. The "marble" keystick has much less kinetic energy and "pings" the wippen heel. Velocity is the same, true... until you look at wippen velocity... but that's another story. However, let's equalize these two situations and take them back to an energy in/ energy out sort of analysis. Let's take our entire setup on a ride on the space shuttle, at some enormous expense to the American taxpayer. Now, in our microgravity lab, we accelerate the bowling ball over a distance of, say, 1 cm, with a force of 1 Newton. We similarly accelerate the marble over a distance of 1 cm with a force of 1 Newton. Both now have the same kinetic energy. The bowling ball moves very slowly towards the wippen, which is a super light 10 g, for simplicity sake. The marble moves very quickly towards an identical 10 g wippen. When these balls hit their respective wippens, the wippens bounce off of them. We're interested in finding out the wippen velocity. When the marble strikes the wippen, the kinetic energy is transfered totally to the wippen (assuming elasticity). The marble is then halted, and the wippen moves forward at the same velocity of the marble. (It's like those executive playtoys -- the racks of five hanging steel balls that click back and forth.) When the bowling ball hits its wippen, not much will happen to the velocity of the ball. The wippen will bounce off it and move forward at approximately twice that velocity. This is all well and good, and it might appear that the bowling ball is more effective at moving its wippen. However, consider that the bowling ball wasn't moving very fast in the first place. Kinetic energy being equal, the marble would move Sqrt(1000), or approx 32, times as fast as the bowling ball. In the end, the ratio of wippen velocities would be 1:16, with the marble's wippen moving much faster. But energy is conserved. Where does the energy go? Well, our astronaut friend doesn't need to catch the marble, as it halts where it strikes the wippen. He/she does, however, have to catch the bowling ball before it strikes the delicate navigation console. To bring the ball to a halt, he/she would have to apply the equivalent of 1 Newton of force (or a bit less, actually, considering that around 6% of the ball's kinetic energy was lost to the wippen) over 1 cm of distance. Ric: Lets take one of your overstated examples... if velocity is the only determinant here.. then ok...say you have this feather moving along at a velocity of 50 metes per second... and it runs into your freight train... just how fast do you think the freight train is going to move in response. Ric: What this really illustrates is that its best to stick to examples that closely resemble our piano action to begin with. Sarah: No, no! This is a good example. It illustrates the difference between a system of collisions and a system of mechanically connected elements! A feather moving at 50 m/s wouldn't move a freight train very much, granted. However, glue the feather to the caboose and THEN accelerate it to 50 m/s. NOW you've got the train moving! ;-) Ric: Grin... Im not convinced. In fact... so far I'm more inclined to think the other way around. It will be a delight to test your claims against the experiment I mentioned earlier. Because if I am wrong... then the spring counter balance will do more work, and if things turn out the other way around... well... you've got some explaining to do :) Sarah: Dunno... I think this is all a process of a thousand tweaks. We've probably gotten to tweak # 503, with 497 to go! I'm certain inertia is an important part of the equation, but how much inertia and where? Well, that's a matter of tweaking too. ;-) Peace, Sarah
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