Assume a homogenous hemisphere of radius R, that can rotate around a horizontal axis, inside the homogenous gravitational field of Earth (g). If we define a coordinate system, starting from point O(0,0,0) as shown in the figure below, then the axis of rotation which is parallel to y'y and lies on the plane z=0, has an abscissa x=ε (it passes through point A(ε,0,0))
• We want to find what should the distance ε from O be, in order for the initial angular acceleration of the hemisphere is the maximum possible (assume that initially the hemisphere is horizontal).
• If θ is the angle of the axis x′x (as shown in figure) and the vertical, then prove that for small angles θ, the hemisphere oscillates harmonically. We want to find the value of ε, so that the period of oscillation is minimum.
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