A small bead is threaded onto a smooth circular hoop, of radius \(r \mathrm {~m}\), fixed in a vertical plane. It is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) from the lowest point of the hoop. Find \(u\) in terms of \(g\) and \(r\) if
the bead just reaches the highest point of the hoop,
the reaction on the bead is zero when it is at the highest point of the hoop.
An ornamental tower is made from a solid right circular cylinder of mass \(M\) and height \(h\) by removing three identical cylindrical sections, each of height \(\frac { h } { 8 }\), equally spaced above a base of height \(\frac { h } { 4 }\), as shown. The tower is held in position by light, thin vertical strips \(A B\) and \(C D\).
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Find the distance of the centre of mass of the tower from its horizontal base.