Show that the distance of the centre of mass of the object from \(A B\) is \(\frac { 3 \mathrm { a } \left( 2 - \mathrm { k } ^ { 2 } \right) } { 2 ( 8 - 3 \mathrm { k } ) }\).
When the object is freely suspended from the point \(A\), the line \(A B\) makes an angle \(\theta\) with the downward vertical, where \(\tan \theta = \frac { 7 } { 18 }\).
Find the possible values of \(k\).
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Two uniform smooth spheres \(A\) and \(B\) of equal radii have masses \(m\) and \(\frac { 3 } { 2 } m\) respectively. 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 is along the line of centres, and \(B\) 's direction of motion makes an angle of \(60 ^ { \circ }\) with the line of centres (see diagram). The coefficient of restitution between the spheres is \(\frac { 2 } { 3 }\).
Find the angle through which the direction of motion of \(B\) is deflected by the collision.
Find the loss in the total kinetic energy of the system as a result of the collision.