Take the case of Atwood machine where masses of blocks are $m$ and $2m$.

The acceleration of individual masses are $\frac g3$. now the acceleration of center of mass of masses and string system is $$\left|\frac{\left(\frac{mg}{3}-\frac{2mg}{3}\right)}{3m}\right| = \frac g9.$$

There is only one force acting on the masses and string system-gravity.The net gravitational force is $3mg$ and therefore the acceleration of center of mass must be $\frac{3mg}{3m}=g$.

Acceleration of center of masses derived are not matching why?

$$T-mg=ma\\ 2mg-T=2ma\\ a=\frac{g}{3}$$ $$T=\frac{4mg}{3}--(1)$$ Now coming to center of mass, $$3mg-2T=3ma_c\\ 3mg-2\left(\frac{4mg}{3}\right)=3ma_c\\ a_c=\frac{g}{9}$$
• Being massless only means that we can ignore both its weight and it's inertia. It doesn't mean that it can't transmit forces through tension. We can consider each leg of the string to cause a force of $T$ to be felt at each end (one end a block, and one end the pulley). Commented Nov 11, 2015 at 9:18