# Find how the center of mass changing

I have a system that contains a rod and a disk on it, balanced on a point. On the other side, there is a small mass on the rod:

Assuming every object (including the rod) has uniform mass density. Given the mass of the disk, rod, small mass, distances of each from the center of mass, and the rod length.

I want to calculate the center of mass position as a function of the distance of the small mass from the center of mass.

I tried to solve it simply by multiplying mass and distances. However, to calculate the rod mass and the distance from the center mass, I made these assumptions:

Rod mass on any side was calculated by the partial length on that side, i.e. if 30% of the length is on that side, the mass is 0.3 times the rod mass.

I'm not sure how to calculate the distance of the road from the center mass. I assumed the part of the rod is a point of mass, and the distance of that point is the middle of the length.

This is what I came to:

$$d_s * M_s + (\frac{L_{rod}-cm}{L_{rod}}*M_{rod}) (\frac{L_{rod}-cm}{2}) = M_{disk}*d_{disk}+ (\frac {cm}{L_{rod}}*M_{rod}) \frac {cm}{2}$$

$$d_s$$ and $$M_s$$ are the distance from the center mass and the mass of the small mass on the right. I'm not sure if that is true. I did that as a part of a lab report. We neglected the change of center mass when moving the small mass (because it was light compared to the system), but this could be one of the errors of the experiment.

you want to obtain the center of mass positions as a function of the geometry $$~L,a~,b~$$ and the masses $$~m_i~$$ . the equations are
$$\sum_{\rm torque (A)}=m_3\,d_3-m_1\,d_1-m_2\,d_2=0\\ \frac L2-(d_1+d_3 )=a\\ a+b+d_2+d_3=L$$
you obtain 3 equations for the 3 unknowns $$~d_1~,d_2~,d_3~$$ the solutions is a function of $$~m_i~,L~,a~,b~$$ which are knowns