My question is inspired by problems of the following kind:
To solve such problems, what one essentially does is mathematically 'segment' the rod/body at a distance x from the pivot, and then write the force equations on the segment(s). In this case:
Here, the force equation for the segment on the right would be: $$T=\frac{M}{L}(L-x)\omega^2\cdot\frac{L-x}{2}$$
I found that this idea of segmentation raises some issues. For example, I wish to find the 'tension' at a point P in a uniformly accelerated thin rod (thickness$\to 0$) as shown below:
(The question seems sensible because I guess internal forces are responsible for all of the rod moving along with the CM.)
Segmenting the rod at P: Internal forces must have components perpendicular to the rod to ensure that the segments move in the direction of the net force. Since the rod is thin and there is no acceleration along the rod, there shouldn't be any 'shearing' forces either.
Here's the issue: This mathematical cut makes no sense to me, because it gives rise to seemingly unbalanced torques about the COMs of each new segment obtained. If the net force F were applied off-centre, for example, there would be a net angular acceleration, but the angular acceleration calculated for each segment would be completely at odds with $\alpha_{net}$.
So, a few questions:
1) Is this method valid at all? Can parts of a single object be isolated this way? Why/Why not?
2) What's going on in the case of the thin rod? (The weird torques)