# Rope and rotational mechanics: why does my solution method work?

i recently come across a physics problem involving rotation and rope with mass attached to it the (1) first way to solve this kind of problem that i come aross is to find the rope tension by using $$mg -T = ma$$ then torque would be equal to moment of inertia x angular acceleration however the second solution (2) of this kind of question i found very interesting is to think of the system as a mass that attached to the rotaional stuff and then think of torque as (moment of inertia of the rotational stuff +moment of inertia of the object using $mr^2$ when $r$ is radius ,not the length to bob) x angular acceleration .

now my question is why does the second solution work?

In method 1 block A moves in a straight line. In method 2 block A is fixed to and rotates with the disk. In both cases you have applied $\tau=I\alpha$.
If the disk and block A have moments of inertia $I_D$ and $I_A$ about their own centres of mass, and block A is attached rigidly to the disk with its centre of mass at a distance $b$ from the axis of rotation, then using the Parallel Axes Theorem the total moment of inertia of the composite object is $I=I_D+I_A+mb^2$ where $m$ is the mass of block A. Your method assumes that $b=r$, the radius of the disk, and $I_A=0.$ That is, you are assuming that block A is a point particle.