Topic: System of Masses
Concepts: Atwood machine, Moments of inertia, Pulleys with rotational inertia

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Solution:

We first find the acceleration in an Atwood machine where the pulley has moment of inertia $I$. Applying Newton’s 2nd law to each mass and the pulley,

Assuming there is no slipping $\alpha=a/R$, we have

Adding this equation and the first two equations yields

so the acceleration is

The ratio between the accelerations in systems $A$ and $B$ is

Since pulley $A$ has mass $m+M$ and is a uniform disk,

The mass of a disk is proportional to its area $m \propto r^2$ so a disk with radius $r/2$ has mass $m/4$. Thus, the moment of inertia of the removed hole is

so for pulley $B$,

Substituting this into the ratio of accelerations,

so the answer is A.