Derive conservation of center of mass position

[Jonsson]

Hello there,

If a mass, m, is placed at one end of an boat of mass M, which is located on a frictionless ocean. If the mass, m, is moved from one end of the boat to the other end, conservation of center of mass position ensures that the boat shifts a small distance in the opposite direction. The conservation of mass is given by: $$\frac{x_1\,m+x_2\,M}{m+M} = \frac{\tilde{x_1}\,m+\tilde{x_2}\,M}{m+M}$$ where each x equal the position of the masses.

How can I derive this formula from more fundamental physics?

Thank you for your time,

Kind regards,
Marius

PS: how can I most easily do inline latex? Is it possible to do something similar to $expr$? Thanks

 

[Matterwave]

This has to do with conservation of momentum. You can show that for the system as a whole to have 0 momentum throughout the motion of the mass m, the center of mass position must not move. This assumes that the external forces are 0 and so by Newton's second law dp/dt=0, and in a frame where p(t=0)=0, dp/dt=0 for all t implies p(t)=0 for all t.

In the above, p stands for the momentum of the entire object (m+M). From this it can be derived that the CoM position must not move because p=(m+M)V_CoM.