Using Newton's Second Law to analyze the Atwood's machine

Consider the modified Atwood's machine:

Two masses (cart m_A, hanging block m_B) are connected by an ideal string passing over an ideal pulley, as shown. Note that an "ideal string" has the same tension force T everywhere along its length, but that force pulls

to the right, on the sliding cart m_A
up, on the hanging mass m_B

Let us assume for now that the friction on the horizontal surface is negligible. Can you figure out all the forces acting on the cart? Fill in the table.


        force         x           y
       -------------------------------


                                      
    





       -------------------------------
        total

Now, please write out expressions for the forces acting on the weight.


        force         x           y
       -------------------------------


                                      
    





       -------------------------------
        total

Two equations, two unknowns

You should end up with a set of two equations:

one relating the total force on the cart to its acceleration along the track,
one relating the total force on the weight to its acceleration down towards the floor.

You should also end up with two unknown quantities

T the tension in the string connecting cart and weight
a the magnitude of the acceleration of the cart -- which is the SAME as the magnitude of the acceleration of the weight

If you are careful with your signs, you should be able to solve these two equations for the two unknowns. In other words, given the masses of the cart and weight, you should be able to predict their acceleration.