A ball $($the yellow one$)$ is bouncing inside a semicircular sheet, the semicircular sheet sets on two wheels.

The ball has a mass of $m_b$ located at the center of the ball, and mass moment of inertial about center $O$ equal to $I_{bO}.$

The semicircular sheet has mass moment of inertia about center $O$ equal to $I_{so}.$ Note that the sheets mass will not be causidered in the solution for sake of easiness.

The sheet is always touching the two wheels. When it rotate it will not slip on the wheels. A harmonic moment is applied to the sheet to make it oscillate, and hence the ball will bounce.

Use Lagrange's equation to determine the equation of motion of the system