5.
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\(A B\) and \(B C\) are two fixed smooth vertical barriers on a smooth horizontal surface, with angle \(A B C = 60 ^ { \circ }\). A particle of mass \(m\) is moving with speed \(u\) on the surface. The particle strikes \(A B\) at an angle \(\theta\) with \(A B\). It then strikes \(B C\) and rebounds at an angle \(\beta\) with \(B C\) (see diagram). The coefficient of restitution between the particle and each barrier is \(e\) and \(\tan \theta = 2\). The kinetic energy of the particle after the first collision is \(40 \%\) of its kinetic energy before the first collision.

1. Find the value of \(e\).
2. Find the size of angle \(\beta\).
   [0pt] [Question 5 Continued] \section*{6.}
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   A particle \(P\) of mass 0.05 kg is attached to one end of a light inextensible string of length 1 m . The other end of the string is attached to a fixed point \(O\). A particle \(Q\) of mass 0.04 kg is attached to one end of a second light inextensible string. The other end of this string is attached to \(P\). The particle \(P\) moves in a horizontal circle of radius 0.8 m with angular speed \(\omega \operatorname { rad~s } ^ { - 1 }\). The particle \(Q\) moves in a horizontal circle of radius 1.4 m also with angular speed \(\omega \mathrm { rad } \mathrm { s } ^ { - 1 }\). The centres of the circles are vertically below \(O\), and \(O , P\) and \(Q\) are always in the same vertical plane. The strings \(O P\) and \(P Q\) remain at constant angles \(\alpha\) and \(\beta\) respectively to the vertical (see diagram).
3. Find the tension in the string \(O P\).
4. Find the value of \(\omega\).
5. Find the value of \(\beta\).
   [0pt] [Question 6 Continued]