If we have a wheel which is decelerating due to friction with acceleration $-A$ and angular acceleration $-\alpha$, and we want to find the acceleration at any point, then we can simply find it using superposition of acceleration vectors.

My question is: If we want to write the equation governing centripetal force at any given point on the disc, what should be force equation for it? I mean how do we find what will be providing the centripetal force without first finding out the acceleration at that point? Is this approach even possible?

An example: How do we write the force equation at $\theta = 30°$ from the vertical axis at $r= \frac{R}{2}$? We can find the acceleration using superposition, but can we find it making a free body diagram?

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Your intuition is correct: you can't find the centripetal force from an analysis of only forces. The centripetal force is found from an analysis of motion. The problem is that one of the forces varies in order to meet the requirements of the motion: it is a force of constraint. Consider a test particle on the rim of the disk. There are two forces on it: gravity, and the force that holds it in place on the disk. The latter force is variable, and cannot be determined without knowing the motion.

A more familiar force of this type is the normal force on a book due to the table on which it sits. The magnitude of the normal force cannot be determined on its own. One must know the motion (typically: at rest) and all the other forces (gravity). Then the normal force can be calculated.

So too in this problem. The centripetal force is not available via an analysis of forces, rather it is found by analysis of motion. But you can find the force of constraint if you want. You must know all of the other forces (just gravity here) and the motion, which includes a centripetal acceleration.


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