Find the percentage loss in the total kinetic energy of the spheres as a result of this collision.
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A bead of mass \(m\) moves on a smooth circular wire, with centre \(O\) and radius \(a\), in a vertical plane. The bead has speed \(\mathrm { v } _ { \mathrm { A } }\) when it is at the point \(A\) where \(O A\) makes an angle \(\alpha\) with the downward vertical through \(O\), and \(\cos \alpha = \frac { 3 } { 5 }\). Subsequently the bead has speed \(\mathrm { v } _ { \mathrm { B } }\) at the point \(B\), where \(O B\) makes an angle \(\theta\) with the upward vertical through \(O\). Angle \(A O B\) is a right angle (see diagram). The reaction of the wire on the bead at \(B\) is in the direction \(O B\) and has magnitude equal to \(\frac { 1 } { 6 }\) of the magnitude of the reaction when the bead is at \(A\).
Find, in terms of \(m\) and \(g\), the magnitude of the reaction at \(B\).
Given that \(\mathrm { V } _ { \mathrm { A } } = \sqrt { \mathrm { kag } }\), find the value of \(k\).