Why is the price of new tonearms so high

Assume a standard "taco warp" so the warp frequency is 7 rad.s^-1.

Hi Mark - please, what's a "taco warp"? I'm assuming it's not related to female tonearm connectors :) . . . but seriously, while I can't really conceive of a standard shape to the minor warps on my records (for those that actually get played, not the obviously 'defective' sort) . . . just going by the tempo of the excursions, I'm guessing that 1-3 Hz is about the frequency range of most of them. So your 7 rad/sec (a little over 1 Hz) is a good figure, but probably a little far away from resonance for calculating good maximum/minimum changes in VTF.

If the inertial mass were 25 g (say 15g arm plus 10g cartridge) and the warp were 5mm high, the maximal variation would be around 6.2 mN. A similar calculation allows a maximal warp tolerance to be derived for any arm / cart combination as a function of VTF.

But the record doesn't act directly on the effective mass of the arm/cartridge - the compliance of the cartridge is in series. So for the actual change in VTF, you'd need to add a scaling factor based on the cartridge/arm resonance, no? Also, the point of maximum/minimum force in the warp cycle will change with warp frequency, as will the phase relationship between the force exerted by the record on the stylus and the force exerted by the cartridge on the tonearm will change as the warp frequency approaches the primary resonant frequency.

Which makes me think that in most cases, the minimum/maximum cases of effective stylus VTF are unlikely to occur very near the tops/bottoms of the warp excursion, meaning that we're also going to see these extremes as occuring at a slightly altered VTA, the extent of which depends on the slope of the warp, hence its amplitude and frequency. Now my head's starting to spin . . .

The important point is that it has nothing to do with the balance of the arm but is stictly related to the moment of inertia (and the mass of the cart).

Yes, this is indeed the important point . . . but I would clarify that it's the moment of inertia from the effective mass of the arm/cartridge combination, coupled with the resonant behavior of the cartridge compliance interacting with this effective mass.

But the big point with static vs. dynamic-balance is . . . what exactly is issue that the dynamic-balance system is trying to solve? Is it always used as an attempt to improve the constancy of VTA with tonearm position? In the tonearms I've set up, it seems that many are position-sensitive (remove the mat from under the force gauge and get a different reading, etc.), and many are not . . . and this doesn't necessarily correlate with whether or not the particular tonearm has a dynamic-balance system. So it seems to me that it's more in the overall execution than anything else.

thus the inertia in a say 15 grams effective mass 12" tonearm with a given cartridge is always larger then in a 15 grams 9" tonearm with the very same cartridge.

No. This is the classic "which is heavier, a pound of lead or a pound of feathers?" axiom. It's just that 12" tonearms tend to have higher effective masses than their 9" counterparts of the same make and "model", because they're bigger.

Inertia is increasing with the distance to the center of movement.

This would be true if we were assuming a constant angular acceleration about the vertical tonearm pivot, but we're not. We're assuming a constant linear (okay, circumferential) acceleration at the end of the tonearm.

Again . . . if the moment of inertia, applied (circumferentially about the tonearms' pivots) to the end of two different tonearms is different . . . then their effective mass is NOT the same. QED.

Furthermore I was referring to the inertia and you are referring to the effective mass.

Yes, we are indeed talking about the same thing. The "effective mass" of a tonearm is the inertial mass of the end where the cartridge bolts on. The interchangability of inertial mass and gravitational mass is fundamental to classical physics . . . as P=mv and F=ma . . . mass = inertia.

I believe (think, know, have had it checked at the technical university Munich in 1995 with precise laser graphics - choose one), - and this is backed by technical papers of the record industry too - that there is a (although tiny in distance) constant vertical movement while playing a record.

Well, yeah . . . any simple analysis of the tonearm/cartridge resonance envelope shows that in the audioband, if there is vertical modulation, there must be vertical movement of the headshell.

But that's not what we're talking about here . . . . we're talking about the reflexion of VTF as it varies with vertical headshell position, which is why Mark is analysing this in terms of record warps. And here the question is exactly about record imperfections, NOT audio-related vertical modulation.

So I think we do see a vertical angular movement - not constant, but even worse alternating in direction - even if not always apparent to the eye.

Yes, angular movement . . . but the force we're talking about is being applied to the end of the tonearm, which is where the effective mass is measured. The angular force vector around the vertical bearing will of course change with all different manner of tonearm-design factors (including effective length) but is irrevelant to the cartridge between two tonearms that have the same effective mass.

Quod erat demonstrandum in realitas mobilis versus modelus in spiritus ?

Oder . . . herrum sitzen und daumen drücken?

I don't get your last comment (unless Mark's comment that effective mass = moment of inertia divided by square of effective length is wrong). If Mark's equation is right, the two could be different and still result in an identical third (effective mass) value.

No, the equation isn't wrong; I believe that Mark was using it to make a point about how little variation there is in the deflection of a dynamic VTF spring, and thus a correspondingly extremely small change in the VTF at the end of the tonearm as a result of this change in spring deflection. For this calculation, the tonearm length is of course relevant.

The misconception that I understood from Dertonarm's comments is that given two tonearms of different effective length, but identical effective mass . . . that they will somehow present different forces against the cartridge when tracking vertical undulations and warps:

Only if the moving mass is homogenous distributed in the whole moving corpus - which is not the case in a tonearm with mounted cartridge.
Brings up again the picture of the Micro Seiki and other turntables which increased their moment of inertia by moving most of the mass towards the outer rim.

You are correct in that the effective mass at one end of a tonearm is the result of its static mass, length, and distribution of mass along the length. But the 'effective mass' specification takes all that into account, and if you change two of these factors (i.e. length and static mass) to acheive the same effective mass at the cartridge . . . the cartridge doesn't care one iota.

To use the platter analogy . . . you can either increase the rotational mass by adding a BUNCH of weight close to the center, or a whole lot less weight at the edge. But which ever way you do it (practical considerations aside), the platter can have the EXACT same rotational mass either way. As long as the "effective mass" at the circumference of the platter is the same, they will exhibit the same inertial characteristics.

The point to note is that all this is dependent on the moment of inertia of the arm and the compliance of the cartridge. It has nothing to do with the method by which VTF is applied.

Absolutely agreed - and we're on the same page as far as the proper electrical model for the cartridge resonance equations.

If the moment of inertia of the arm and cartridge combination is 1.26 x 10^-3 kg.m^2 the maximal torque transmitted to the arm is 1.38 x mNm which is equivalent to a force of 6.2 mN acting at a distance of 225mm.

Sure, absolutely - and calculating the rotational torque of the tonearm is necessary for evaluating the particulars of the downforce spring itself . . . but that's about all.

I guess some of the assertions got jumbled around in various different posts . . . but my point is mainly that to calculate deviation of VTF across a warp . . . if you have an accurate figure for effective mass from the cartridge's point of view, then neither the rotational torque nor the length of the tonearm matters - the tonearm can be viewed simply as a pure mass that moves only on two axes.