Seeking intuition on movement of COM

[fahraynk]

Basically a weight moves from starting point on a ramp down to the bottom of the ramp. The ramp is on wheels so the ramp will also move. The formula is based on the center of mass of the combined object. Center of mass does not move.
XA0 = position of small mass before it slides down ramp
XB0 = position of COM of ramp before box slides
XA,XB= new position after box slides

$$X_{A0}M1+X_{B0}M2 = X_{COM}(M1+M2)\\
X_{A}M1+X_{B}M2 = X_{COM}(M1+M2)\\
X_{A0}M1+X_{B0}M2 = X_{A}M1+X_{B}M2$$

My question is where does this come from? Assuming I did not know this through repetition that the COM would not move... what physical laws would you apply to come to this conclusion?

If the box is sitting on the ramp, there should be a force normal to the ramp as M1gSin(theta). Isn't this the force on the ramp whether the mass is moving or not? If the ramp is not moving before the box moves that means friction between the ramp and floor is holding it in place. So... that will still be canceled out whether the box is moving or not. Where does this center of mass formula come from??

 

[Baluncore]

Conservation of energy.
Conservation of momentum.

 

[Khashishi]

You can do a free body diagram. It's not a closed system because you have gravity pulling down, and the ramp is being supported by the ground which provides an upward normal force. But you can see that gravity and the normal force are completely vertical. There is no horizontal component to the external force on the system, so there is no change in the horizontal component of the center of mass.