# Rotation

A uniform ring of mass $$m$$ and radius $$R$$ is performing uniform pure rolling motion on a horizontal surface. The velocity of the center of the ring is $$V_0$$. If the kinetic energy of the semicircular arc $$AOB$$ is $$mv_0^2 \left(\frac{1}{\alpha} - \frac{\beta}{\pi} \right)$$, then find the value of $$\alpha + \beta$$.

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