Cartridge Loading and Compliance Laws

One more twist to the plot is that the cartridge specs may not be accurate to real world application. eg. the compliance of the cartridge may be given at a high frequency point and could be much different at low frequency. My SME tonearm conveniently lists the mass and compliance in the manual. So I can use the website calculators to determine my total system resonance frequency. I had a Denon cartridge at one time. It made my system Fn equal about 16Hz because my SME arm has low effective mass. It sounded good, but even the slightest warp in the record would cause the needle to skip. The stylus and tonearm were very animated when observing it closely while playing a record. I solved the issue by adding a 4 gram mass from England made for the Denon/SME combo. (It had threaded holes to fit the SME headshell and was designed to be glued to the cartridge. It also raised the tonearm making it easier to get the right VTA.) The 4 gram mass made Fn about 10Hz as I recall. The tracking was much more stable. A low system Fn such as 5-6Hz can make your turntable more susceptible to footfalls or other low frequency vibrations, including warps in the record. In the end, the calculation is only as good as the data supplied by the tonearm and cartridge manufacturers, but test records are available that can be used to determine actual system Fn.

Correct, if the tables are turned, then you are out of luck. The only choice is a new tonearm or cartridge. Some might say dampening the tonearm would help, and it might. Only thing is, dampening is masking the underlying problem. There still might be consequences or undesireable attributes with dampening. That is different from isolation. Isolating the turntable and tonearm from airborne and mechanical vibrations is always beneficial.

Good point. When talking about effective mass, we are really talking about inertia and the moment of inertia is proportional to square of the distance from the axis of rotation. So, yes, more counterweight mass closer to the pivot will reduce effective mass.

Lighter screws are definitely better because that mass is at the farthest point from the pivot. So that affects the inertia of the tonearm. Aluminum screws are preferred for both less weight and the non-magnetic properties. Don't use plastic screws. Plastic creeps, ie. stretches, and you won't be able to maintain the torque with plastic screws.

When we talk about tonearm effective mass, we are really talking about inertia. And when we talk about cartridge compliance, we are really talking about spring rate. Not to sound confusing, but high compliance is low spring rate and low compliance is high spring rate. So the resonance frequency of the tonearm/cartridge system is simply the square root of the cartridge’s spring rate over the sum of the masses. It is the classic spring/mass equation. The system will become very excited at its resonance or natural frequency meaning it will mistrack or even skip.
The moment of inertia of the tonearm about its pivot is I=mr2. So moving the counterweight further from the pivot point increases the inertia (effective mass) by the square of the distance whereas keeping the counterweight at the same distance and increasing the mass is a one to one change in inertia. That’s why using a heavier counterweight closer to the pivot reduces the tonearm effective mass.
The tonearm is on a pivot bearing with as little friction as possible, so it is important to have a tuned system that does not get excited by vibrations from the record/platter or vibrations external to the turntable. The stylus is connected to a suspension which gives it its spring rate. As the stylus moves up/down and left/right on the record it is pushing against the tonearm. As long as those motions or vibrations are not near the tonearm’s resonance point, the tonearm remains stable.
When you pluck a spring, it resonates at its natural frequency. A string nearby tuned to the same frequency will vibrate on its own because it is excited by the other string’s vibrations. That is called sympathetic vibration. The same holds true for the tonearm/cartridge system. The 10Hz +/-2Hz natural frequency is a rule because it works. Music ranges from 20Hz- 20kHz. So if the tonearm/cartridge system is tuned too closely to 20Hz, then the arm could become excited when playing music on the record that is close to 20 Hz. 10Hz being half of 20 keeps the tonearm/cartridge system safely away from the range of the music. Now at the other extreme, footfalls and other low frequency sources in the 5-6 Hz range will excite the tonearm if it is tuned too low. So again, 10Hz being double of 5 helps reduce the tonearm’s sensitivity to footfalls and such.
I hope this all makes sense.

Question 1: The eff. mass is a product of the counterweight mass (plus the overall arm mass)and distance squared from the pivot point. So increasing the counterweight mass increases eff mass, yes. Increasing the counterweight mass and then moving it closer to the pivot in order to maintain the same VTF will reduce the eff mass.
Question 2: The tracking is affected by a number of factors, but just considering the natural frequency of the cartridge/tonearm system the location of the counterweight is only a part of the final result. The ideal range remains around 10Hz.

Picture the stylus like a wheel on a car. The stylus has micromotion as it tracks the groove on a record and that motion is absorbed by the suspension. Just like the wheel on a car moves over small bumps in the road but the car remains fixed. In both cases in regards to the micromotion and inertia of the stylus and the wheel, they are very small compared to the mass of the tonearm or mass of the vehicle. So that micromotion causes little or no motion in their relatively massive counterparts. But when the stylus moves over a warp in the record, for example, now the entire tonearm must move in response to that warp. Consider that a macro-motion. In this case, if the tonearm has too much interia, the tonearm raises up to ride over the warp but takes too long to come back down resulting in a skip. Similar sitation in a car- a dip in the road or bump in the road causes the car to move up/down, but if the car has too much inertia, then it leaves the road surface. In both cases, a car or a tonearm, the spring rate and the effective mass affect how they respond to those macro-motions.