A particle $$\(P\)$$ of mass 0.05 kg is attached to one end of a light inextensible string of length 1 m . The other end of the string is attached to a fixed point $$\(O\)$$. A particle $$\(Q\)$$ of mass 0.04 kg is attached to one end of a second light inextensible string. The other end of this string is attached to $$\(P\)$$. The particle $$\(P\)$$ moves in a horizontal circle of radius 0.8 m with angular speed $$\(\omega \mathrm{rad} \mathrm{s}^{-1}\)$$. The particle $$\(Q\)$$ moves in a horizontal circle of radius 1.4 m also with angular speed $$\(\omega \mathrm{rad} \mathrm{s}^{-1}\)$$. The centres of the circles are vertically below $$\(O\)$$, and $$\(O, P\)$$ and $$\(Q\)$$ are always in the same vertical plane. The strings $$\(O P\)$$ and $$\(P Q\)$$ remain at constant angles $$\(\alpha\)$$ and $$\(\beta\)$$ respectively to the vertical (see diagram). Find the value of $$\(\omega\)$$. ........................................................................................................................................................ . ........................................................................................................................................................ . ........................................................................................................................................................ . ........................................................................................................................................................ . ........................................................................................................................................................ . ........................................................................................................................................................ . ........................................................................................................................................................ . ........................................................................................................................................................ . ........................................................................................................................................................ . ........................................................................................................................................................ . ........................................................................................................................................................ . ........................................................................................................................................................ . ........................................................................................................................................................ . ........................................................................................................................................................ .

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