Acceleration of system related to rolling motion and pulley

[Steve4Physics]

Please forgive me for going off-topic but the image in @Lnewqban ’s Post #29
https://www.physicsforums.com/attachments/manlift-jpg.305490/
immediately reminded me of an (IMO funny) Irish folk song.

For anyone so inclined - and having three minutes to spare - here it is (with lyrics).
Edit: The only song I know about Atwood machines!

Otherwise, please ignore.

 

[Lnewqban]

Thank you, @kuruman.
I would pick items 1 and 3, because those are the only ones having linear accelerations (here assuming that that is the type of acceleration requested to be calculated by the problem).

Item 5 must be considered as well because it determines the fraction of the weight of item 1 that is resisting the weight of item 3 (driving force in the system), or tension in item 4.

The rotational inertia of items 1 and 2 also resist the effect of the driving force; therefore, both need to be considered.

 

[haruspex]

Thank you, @kuruman.
I would pick items 1 and 3, because those are the only ones having linear accelerations (here assuming that that is the type of acceleration requested to be calculated by the problem).

Item 5 must be considered as well because it determines the fraction of the weight of item 1 that is resisting the weight of item 3 (driving force in the system), or tension in item 4.

The rotational inertia of items 1 and 2 also resist the effect of the driving force; therefore, both need to be considered.

You originally proposed taking the vector sum of the two accelerations, which is bizarre.

In post #23, you mention COM, so you might be proposing the acceleration of the mass centre of the system consisting of the two linearly moving masses. If so, @kuruman already offered that in post #17, and I agreed that was defensible.
But I could also interpret your post as meaning the weighted average of the magnitudes of the two linear accelerations, which also has some validity. That interpretation seems to be supported by posts #29 and #32.

But the whole discussion is fruitless. The question does not define "the system" (why exclude the massive pulley?), it does not mention magnitudes and does not mention COM. If we're playing guess the question setter's intent, it is clear s/he overlooked the ambiguity, so may well have just been thinking of the descending mass.

@songoku has solved the problem as far as is possible.

 

[Lnewqban]

You originally proposed taking the vector sum of the two accelerations, which is bizarre.

Would you mind explaining why do you believe so?

 

[haruspex]

Would you mind explaining why do you believe so?

Because it has no physical meaning, as illustrated by my example in post #16.

 

[songoku]

Thank you very much for all the help and explanation Lnewqban, erobz, kuruman, haruspex, Steve4Physics, jbriggs444, malawi_glenn