I am very glad to read your question. I had a similar one in a thread that I started. Unfortunately I did not get a satisfactory answer. So I put my thinking cap on...
Here is my basic theory.
When your tone arm is horizontally balanced the effective mass of the counter weight end is equal to the effective mass of the cartridge end. The problem is you do not know the effective mass of either.
Small note before proceeding, my definition of effective mass is the mass that would be measured at the tip of the tone arm if the arm itself had a mass of zero and a weight were hung a the tip of it.
What do we know? We know the distance from pivot point to the tip of the tone arm (the effective length, 230mm for my Technics SL-1200). We also know or can easily determine the distance from the pivot point to the center of mass of the balance weight. And finally we know the mass of the balance weight.
Definition of variables
Mass of Balance Weight end of the tone are = Mbal
Distance from pivot point to center of balance weight =Dbal
Mass of tone arm = Mtarm (unknown)
Effective Lenght of tone-arm = Dtarm
In the horizontal balanced position the equation is:
(Mbarm x Dbarm) - (Mtm x Dtm)= 0
We now have one equation two unknowns
For the second equation I propose to move the balance weight back by a small easily measurble increment. I have a technics SL-1200, which has a very adjustable and user friendly tone arm. The pitch of the balance weight is 10mm and there are 40 graduations on weight, thus by turning the weight by 2 graduations it moves back 0.5mm.
I not familiar with SME thus you might have to use a vernier caliper to accurately measure your change in balance weight position.
Once the weight is moved back the cartridge tip of the tone arm will rise by a few millimeters (10 to 20) before finding it new point of equilibrium (or not at all, if hits its end stop or the point of equilibrium has shifted to the vertical position, in which case you must move the balance weight back by a smaller increment).
Now the difficult part, you must measure the change in height of the tip of the tone arm. With the change height and some high school trigonometry you can determine the angle the tone arm has shifted (actually not really required) and the new vertical distance between the pivot point and the various other points.
Now you have:
Mbal = no change
Dbal = initial position
Dbal1= new position
Mtarm = no change
Dtarm = initial position
Dtarm1 = new position
Mbal X dbal - Mtarm x dtarm = 0
Mbal x dbal1 - Mtarm x dtarm1 = 0
Mbal x dbal - Mtarm x dtarm = Mbal x dbal1 - Mtarm x Dtarm1
Mtarm = Mbal x[(dbal1 - dbal)/(dtarm - dtarm1)]
Now for some caveats!
1- In theory this great but in practice it requires the precise and accurate measurement of small distance (0.5mm) of parts that are not fixed but balanced. So measuring technic is everything.
2- futher to the point above small measuring errors can have a large impact on the end result.
3- I have ingnored the mass of the tone arm on balance weight end, my feeling is that its mass is small compared to the balance weight and distance is small from the pivot thus the change in distance should (this has not been verified mathimatically) have negligible effect on the final result.
I am very curious to hear feed-back from the forum. I must go to sleep now but I will share the results of my measurements tomorrow.
Here is my basic theory.
When your tone arm is horizontally balanced the effective mass of the counter weight end is equal to the effective mass of the cartridge end. The problem is you do not know the effective mass of either.
Small note before proceeding, my definition of effective mass is the mass that would be measured at the tip of the tone arm if the arm itself had a mass of zero and a weight were hung a the tip of it.
What do we know? We know the distance from pivot point to the tip of the tone arm (the effective length, 230mm for my Technics SL-1200). We also know or can easily determine the distance from the pivot point to the center of mass of the balance weight. And finally we know the mass of the balance weight.
Definition of variables
Mass of Balance Weight end of the tone are = Mbal
Distance from pivot point to center of balance weight =Dbal
Mass of tone arm = Mtarm (unknown)
Effective Lenght of tone-arm = Dtarm
In the horizontal balanced position the equation is:
(Mbarm x Dbarm) - (Mtm x Dtm)= 0
We now have one equation two unknowns
For the second equation I propose to move the balance weight back by a small easily measurble increment. I have a technics SL-1200, which has a very adjustable and user friendly tone arm. The pitch of the balance weight is 10mm and there are 40 graduations on weight, thus by turning the weight by 2 graduations it moves back 0.5mm.
I not familiar with SME thus you might have to use a vernier caliper to accurately measure your change in balance weight position.
Once the weight is moved back the cartridge tip of the tone arm will rise by a few millimeters (10 to 20) before finding it new point of equilibrium (or not at all, if hits its end stop or the point of equilibrium has shifted to the vertical position, in which case you must move the balance weight back by a smaller increment).
Now the difficult part, you must measure the change in height of the tip of the tone arm. With the change height and some high school trigonometry you can determine the angle the tone arm has shifted (actually not really required) and the new vertical distance between the pivot point and the various other points.
Now you have:
Mbal = no change
Dbal = initial position
Dbal1= new position
Mtarm = no change
Dtarm = initial position
Dtarm1 = new position
Mbal X dbal - Mtarm x dtarm = 0
Mbal x dbal1 - Mtarm x dtarm1 = 0
Mbal x dbal - Mtarm x dtarm = Mbal x dbal1 - Mtarm x Dtarm1
Mtarm = Mbal x[(dbal1 - dbal)/(dtarm - dtarm1)]
Now for some caveats!
1- In theory this great but in practice it requires the precise and accurate measurement of small distance (0.5mm) of parts that are not fixed but balanced. So measuring technic is everything.
2- futher to the point above small measuring errors can have a large impact on the end result.
3- I have ingnored the mass of the tone arm on balance weight end, my feeling is that its mass is small compared to the balance weight and distance is small from the pivot thus the change in distance should (this has not been verified mathimatically) have negligible effect on the final result.
I am very curious to hear feed-back from the forum. I must go to sleep now but I will share the results of my measurements tomorrow.