----- Original Message ----- From: "Richard Brekne" <Richard.Brekne@grieg.uib.no> To: "Pianotech" <pianotech@ptg.org> Sent: Saturday, December 27, 2003 6:59 AM Subject: Re: Inertia and Physics.. Paul Chick > In a fairly well known book about pianos there is a claim made for > justifing the use of very (extremly) light hammers. The basic argument > used is that since F = ma, and since lowering mass while applying the > same force (assume from the finger) the acceleration of the hammer will > increase proportionally. Seems to me this argument is full of holes. But > assume for a second that the given two hammers, one 3 grams lighter then > the other were accelerating at different rates such that at the exact > moment of impact F would be the same for the two. It is true that an object with less mass will accelerate more with the same force applied to it. But again we need to use angular equations, not F=ma. Since the hammer revolves about a pivot we need to use T = I[alpha] (torque equals moment of inertia times angular acceleration). This equation can be rearranged to alpha = T/I So as the moment of inertia, I, gets smaller, we get more angular acceleration, which makes sense. Assuming that you accelerate through the same angle, the angular velocity when the hammer is released by the mechanism is greater with the lighter hammer But speed isn't necessarily everything; we also have less moment of inertia now. The kinetic energy of this hammer (which is the energy we have to work with when we strike the string) would be KE = 1/2 Iw^2 (one-half the moment of inertia times angular velocity squared). So let's say we cut the moment of inertia of the hammer to 1/3 what it was. Then we would get three times the acceleration. We calculate the angular velocity at release with this equation: w = sqrt (2 [alpha][theta]) where theta is the angle of travel (which didn't change). If we triple alpha, the angular velocity, w, will only increase by the *square root* of three (1.73). But then when we plug it into the kinetic energy equation it squares again and we get our 3 back. And since we only have 1/3 the moment of inertia now, the 3 and the 1/3 cancel. The upshot is that as long as you apply the same torque through the same angle you get the same kinetic energy no matter what the m.o.i. is. > Is the WORK done by the hammer the same ? > How would you use WORK, MOMENTUM, and/or KINETIC ENERGY to describe the > impact of the hammer and string ? Work is done *to* the hammer by the mechanism. Work is W = T [theta] (work equals torque times angle) And is in energy units (joules or foot-pounds). If you knew the torque that was being applied to the hammer while it accelerated you could multiply this by the angle (in radians) and get the work done on the hammer. In this case you would find that it is equal to the kinetic energy at release. So we didn't do all that work for nothing! > Another thing I've heard from time to time is the claim that the only > thing the hammer brings to the string (aside from its elasticity) is its > MOMEMTUM. Would you agree ? No. The hammer has momentum before and after it strikes the string, but this pretty much useless trivia since the string doesn't really go anywhere. The energy imparted to the string manifests itself as vibration, not linear motion, so momentum doesn't help us much. Energy does. If we neglect the effects of gravity on the hammer we can tell how much energy it lost on impact by calculating the difference in its kinetic energy. Since the m.o.i. doesn't change we just need to know the angular velocity right before impact and right after impact. Plug them into the KE formula and subtract one from the other and you have the energy lost in the impact. Where this energy goes is another story. Some of it goes into vibrating the string, but some also goes into heat, vibration in the mechanism, etc. As I explained in an earlier post, this is where the center of percussion becomes important. Don A. Gilmore Mechanical Engineer Kansas City > > Cheers > RicB > _______________________________________________ > pianotech list info: https://www.moypiano.com/resources/#archives
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