Suppose we are studying the case of uniform circular motion. The analysis of such motion is usually done using Newton's second law as $ma=m v^2/r$. Even when the motion isn't purely circular, like the case for central force problems, we still use familiar Newton's second law in our analysis. Moreover, the object in question is treated as a point mass.
However, the motion of a body in an orbit can be considered to be a special case of rotational mechanics. Suppose we have a ball spinning about its axis. We analyze this using the rotational analog of Newton's second law - using torques and moment of inertia - $\tau=I\alpha=I d^2\theta/dt^2$. This equation basically tells us how fast would a body spin if we apply a torque to it.
When a body spin about an axis, the different mass elements that constitute the entire body, can be thought of as moving in an orbit about the axis. The necessary centripetal acceleration is provided by the forces that make sure that the body doesn't deform - electrostatic forces I suppose. The torque, therefore, is the manifestation of where we are applying the force on our extended body. Similarly, the angular momentum is describing the momentum of individual mass elements away from the axis.
Is it possible in principle, to analyze the rotational motion of an object purely using force, mass, momentum, etc, instead of torque, angular momentum, and moment of inertia? Something along the lines of finding the forces on each of these small particles, and then integrating over the entire body, and solving the equations of motion to describe how the body as a whole rotates ? I'm sure, during any such analysis, we would be forced to re-invent moment of inertia and torque somewhere in the derivation. However, in principle, is it possible to describe the rotation of a body by using (starting with) $F=ma$, or rather $F=mv^2/r$ ?
In the case of point particles, we do use Newton's laws directly to analyze motion. In the case of extended objects, we use the rotational analogs of our linear concepts. Is it theoretically possible (definitely impractical), to describe the motion of rigid extended bodies, by starting with $F=ma$, and then deriving all the rotational analogs from this starting point?