4 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 . Particle P is projected so that it moves in complete vertical circles with centre O ; there is no air resistance. A and B are two points on the circle, situated on opposite sides of the vertical through O . The lines OA and OB make angles \(\alpha\) and \(\beta\) with the upward vertical as shown in Fig. 4.
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\caption{Fig. 4}
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The speed of P at A is \(\sqrt { \frac { 17 a g } { 3 } }\). The speed of P at B is \(\sqrt { 5 a g }\) and \(\cos \beta = \frac { 2 } { 3 }\).
- Show that \(\cos \alpha = \frac { 1 } { 3 }\).
On one occasion, when P is at its lowest point and moving in a clockwise direction, it collides with a stationary particle Q . The two particles coalesce and the combined particle continues to move in the same vertical circle. When this combined particle reaches the point A , the string becomes slack.
- Show that when the string becomes slack, the speed of the combined particle is \(\sqrt { \frac { a g } { 3 } }\).
The mass of the particle Q is \(k m\).
- Find the value of \(k\).
- Find, in terms of \(m\) and \(g\), the instantaneous change in the tension in the string as a result of the collision.