# Does the rope stretch when the pulley of Atwood's machine is accelerated?

Let's say I've an Atwood's machine in which I have two point masses which are different. Now, when I accelerate the pulley, the two masses accelerate at a different rate which can be shown by calculation. The proof is given in Kleppner's book of mechanics, where he showed the acceleration to be $$\begin{equation}\frac{(2A+g)M2-M1g}{M1+M2}\end{equation}$$ for mass 1 and likewise for M2.

Note that it's different from the general Atwood's machine because the pulley itself is being accelerated upwards. In the ideal case, we assume that the rope cannot be stretched. However, in this case, as the two masses are accelerating at different rates, the rope must stretch to support the two masses, right? So, are we giving up our assumption of the rope being inexpandible here or is something wrong with my concept?

There is no flaw in the question. It's all about frames of reference.

If you watch the motions of both the objects from the pulley's frame of reference then the 2 objects will have the same acceleration magnitudes (in opposite directions). But if you observe the motion from the ground's frame of reference, the acceleration:

1) Of bigger mass: A-a

2) Of smaller mass: A+a

If A has been directed upwards.' a' is the acceleration of the masses from pulley's frame of reference. You may calculate that using a pseudo force.

This is called relative motion. I would encourage you to watch a few videos on relative motion and then try to solve this question. It is a very easy one.

• So, from an inertial frame, will I observe that the rope is being stretched? – Tahsin Choudhury Jan 27 '19 at 11:41
• Nope. It's just the illusion due to frame of reference. – Suven Jagtiani Jan 27 '19 at 12:11