Answer to Question #150826 in Classical Mechanics for Subhajit Das

Question

Masses on table
Two masses, A and B, lie on a frictionless table, as shown.
They are attached to either end of a light rope of length l which
passes around a pulley of negligible mass. The pulley is attached
to a rope connected to a hanging mass, C. Find the acceleration of
each mass. (You can check whether or not your answer is reason-
able by considering special cases—for instance, the cases MA = 0,
or MA = MB = Mc.)

Answer 1

We can consider blocks on the table as a single mass. Newton's second law gives

$$(M_A+M_B)a=T.$$

For the hanging mass:

$$T-M_Cg=M_Ca,\\ (M_A+M_B)a-M_Cg=M_Ca,\\\space\\ a=g\frac{M_C}{M_A+M_B-M_C}.$$

Check: if mass C is 0, the acceleration of A and B is zero (no hanging mass).