Well the detailed (but possibly not fully helpful) answer is in the original article
I = integral r^2 dm
..the sum of all the particles of mass in the arm, times the radius squared
The article also gives ways of measuring it on a finished item
If you want to try to model/predict it then, have a look at
https://en.m.wikipedia.org/wiki/List_of_moments_of_inertia
Looking at the list of defined examples (under the
Moments of inertia section) you could nominally use these to give a prediction.
For example, you could arguably model a tonearm as a combination (sum) of
- main pivot to headshell part of the arm - a rod pivoted at one end
- counterweight stub - a rod pivoted at one end
- headshell - a point mass
- counterweight - a point mass
- or a thick rod pivoted at one end
- or, because the counterweight doesn't actually start at the pivot,
- a thick rod the diameter of counterweight, pivoted at one end, of length to the furthest edge of the counterweight
- minus - the same thick rod, pivoted at one end, of length to the nearest edge of the counterweight
- ...etc.
- building up your model in ever more detail
but all of the above would be approximations
I'd also guess that something like solidworks would calculate the moment of inertia of a design assembly
There is no one single simple formula for a tonearm
You're going to have to calculate it for yourself
You've got to understand the basics of the above maths in order to do it
In summary your options are (other options welcome)
- calculate based on simplified models as above
- use a suitable CAD package (I assume there's one somewhere)
- build it then measure it empirically