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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?

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$$ T-mg=ma\\ 2mg-T=2ma\\ a=\frac{g}{3} $$ $$ T=\frac{4mg}{3}--(1) $$ Now coming to center of mass, $$ 3mg-2T=3ma_c\\ 3mg-2\left(\frac{4mg}{3}\right)=3ma_c\\ a_c=\frac{g}{9} $$
You have made mistake in considering net external force,
You have to consider tension.

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  • $\begingroup$ thank you for replying but we are considering only the blocks and string connected to blocks,and not the pulley,the force applied by pulley on the system must be zero(it is a massless string so net external force must be zero).so how does the force applied by the string connected to ceiling act on the system?(two blocks and string connected to blocks) $\endgroup$ – Ajay Sabarish Nov 11 '15 at 8:48
  • $\begingroup$ 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). $\endgroup$ – BowlOfRed Nov 11 '15 at 9:18
  • $\begingroup$ sir,thank you for replying but we are taking the string and blocks as one system,so tension becomes an internal force and hence cancels out,and please explain how is the string connected between the ceiling and pulley exerting a force on our system? $\endgroup$ – Ajay Sabarish Nov 11 '15 at 9:27
  • $\begingroup$ @AjaySabarish Tension on the pulley is an external force which needs to be considered because it acts at the center of mass of the pulley system against gravity. Note that the string from the ceiling is not the same as that which connects the blocks. $\endgroup$ – Sathyaram Nov 11 '15 at 11:07
  • $\begingroup$ sir,thank you for replying.tension acts on the pulley but pulley is not a part of our system and the pulley does not exert any force on string connecting blocks.i very well know iam wrong i just want to know why?please tell me why is the external tension force acting on our system and explain that part alone in detail,sir. $\endgroup$ – Ajay Sabarish Nov 11 '15 at 11:36

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