Bent Rod Moment

There is a thin uniform rod of mass MM in the xyxy-plane which is initially aligned with the xx-axis. The body of the rod begins at the origin x=0x=0 and ends at x=Lx = L.

Suppose that the portion of the rod up to x=ax = a, where a<La < L remains aligned with the xx-axis, and the portion from x=ax = a to x=Lx = L is bent upwards at a right angle so as to be perpendicular to the xyxy-plane.

The bent rod's moment of inertia with respect to the zz-axis can be expressed as Ma2αβMa3L.{M a^{2} - \dfrac{\alpha}{\beta} \dfrac{ Ma^{3}}{ L}} . If α\alpha and β\beta are coprime positive integers, determine α+β\alpha+\beta.

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