5 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle completes vertical circles with centre \(O\). The points \(A\) and \(B\) are on the path of \(P\), both on the same side of the vertical through \(O\). OA makes an angle \(\theta\) with the downward vertical through \(O\) and \(O B\) makes an angle \(\theta\) with the upward vertical through \(O\).
The speed of \(P\) when it is at \(A\) is \(u\) and the speed of \(P\) when it is at \(B\) is \(\sqrt { \mathrm { ag } }\). The tensions in the string at \(A\) and \(B\) are \(T _ { A }\) and \(T _ { B }\) respectively. It is given that \(T _ { A } = 7 T _ { B }\).
Find the value of \(\theta\) and find an expression for \(u\) in terms of \(a\) and \(g\).
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Two uniform smooth spheres \(A\) and \(B\) of equal radii each have mass \(m\). The two spheres are each moving with speed \(u\) on a horizontal surface when they collide. Immediately before the collision, A's direction of motion makes an angle \(\alpha\) with the line of centres, and \(B\) 's direction of motion makes an angle \(\beta\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 1 } { 3 }\) and \(2 \cos \beta = \cos \alpha\).
- Show that the direction of motion of \(A\) after the collision is perpendicular to the line of centres.
The total kinetic energy of the spheres after the collision is \(\frac { 3 } { 4 } \mathrm { mu } ^ { 2 }\). - Find the value of \(\alpha\).