Calculate Acceleration of a Single Sided Atwood machine With Pulley Mass | Newtons Laws & Rotation

Set up a system of equations using force as well as torque to solve for the linear acceleration of a block hanging from a pulley which has both mass and radius.
Apply Newton's 2nd Law to the hanging block to relate the weight (mg) of the block to the tension in the string as well as the linear acceleration of the block.
Then apply Newton's 2nd law in its rotational form ( Torque equals Inertia x angular acceleration ) to the rotating pulley to relate the string tension to the angular acceleration of the pulley.
Then relate the two accelerations to one another and solve the system of three equations and three unknowns.
This problem commonly comes up in introductory physics courses including high school physics, AP Physics 1, AP Physics C and serves as the foundation for problems found on the JEE and A Level Exams.

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Hello sir..i watched your videos..that's great.
I have always doubt in my mind which describe here:
Suppose there is a Rod in horizontal plane and it is not hinged with any thing free to move..and suppose now someone gives an impulsive force at some distance from its C.G ...and then it starts moving in the direction of applied Impulse while rotating around C.G of the rod ..
My question is why it rotates about its CG..can u explain this please.
Please don't give such reasons that it rotates about its CG bcoz CG has lowest Moment of inertia OR all mass is concentrated at CG point..
Please provide some genuine reason behind this