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Cylinder oscillations

A thin-walled half-tube of mass $m$ and radius $2r$ can roll without slipping on a horizontal surface. A cylinder of mass $2m$, radius $r$, with homogeneous mass density, can roll without slipping inside the half-tube. At instant $t = 0$, the half-tube is at rest, while the axis of the cylinder moves with a small horizontal speed $v_0$.

1 Calculate the angular displacement of the half-tube as a function of time.