Locating the CoG of an object in a 2D system

[Depresion]

Maybe Geometry & Trig isn't the best location for this but it seems like it should be quite simple to solve this problem (it's for a generic home built robot arm)If I know the global mass of a multi body 2d system M_G,the global CoG for that system but onlyin the X axis X_G. In this system there is a rotating object, I knowthe centre of rotation of the object X_R Y_R but not theradius.

Given that, I know the M_G at multiple angles ofrotation of the object can I calculate the Mass and CoG of the object (giventhat it’s CoG is offset from angle of rotation)? If not, if I know the mass of the object can Icalculate the CoG of the object?

I generated a sample dataset in excel by putting 3 objectsinto a 2D grid, then rotating one object about a centre point (with an offsetthat also rotated).

Data Set	1	2	3	4	5
Angle	0	-20	-40	-70	-90
Global CoG X	631	611	583.5	532	598
Object CoG X	2734	2650	2537	2303	2151
Object CoG Y	2779	2651	2560	2514	2549

X_{R = 2329}
Y<sub>R = 1044

MG = 2336.5

Object Mass = 537</sub>


The object rotates with the angle of rotation as it is on apendulum.

(Angles are absolute with 0 being horizontally (+ on the X)and -90 being vertically (- on the Y axis) clearly I won't know the Object CoG values when I'm using the arm but I have them from the quick and dirty excel simulation I knocked up.

I have been banging my head against a wall (figuratively at least so far) with this as it seems like it should be quite simple but I just can't work out where to start looking, or I'm overlooking something,

 

[Depresion]

Ok I was being a bit of an idiot with the if I know the mass of the object thing. If I multiply the system mass by the system CoG then divide by the object mass the results come out within the rounding errors. Still working on the if I don't know the object mass, maybe having the two unknowns is going to make things unsolvable.

 

[Depresion]

Looks like I was optimistic with my thought I had it sussed. The results were pretty close with that data (close enough to think it was rounding errors) set but when I shifted the centre of rotation it all went wrong. Can anyone point me in the right direction as I still have a niggling feeling that this should be simpler than I am making it.