This is a follow-up to my previous question, in which I am now trying to calculate the acceleration of the cart (as before, the block surfaces are frictionless).

To do this, I first need to find the tension on the string.

I came up with the system of equations:

$$T=m_{1}\left( a-a_{M} \right)$$ $$T-m_{2}\text{g}=-m_{2}a$$

Since $ m_1 $ moves right while $M$ moves left, and where $a$ is the acceleration of $m_2$.

However, since the tension created is responsible for the acceleration of the $M + m_2$ system, isn't this equation also valid?: $$T = a_M\left(M + m_2 \right)$$

Clearly the solutions to both are not the same, so one (or possible both) of the above are incorrectly accounting for the forces acting.

Furthermore, once this tension is found how do you account for the normal force between $M$ and $m_2$ that also affects the acceleration?

Alternatively, is it possible to solve this using conservation of momentum or by using center of mass?