5 A particle \(P\) of mass 2 kg moving on a horizontal straight line has displacement \(x \mathrm {~m}\) from a fixed point \(O\) on the line and velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) s. The only horizontal force acting on \(P\) is a variable force \(F \mathrm {~N}\) which can be expressed as a function of \(t\). It is given that $$\frac { v } { x } = \frac { 3 - t } { 1 + t }$$ and when \(t = 0 , x = 5\).

1. Find an expression for \(x\) in terms of \(t\).
   \includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-08_2718_38_106_2009}
   \includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-09_2723_33_99_21}
2. Find the magnitude of \(F\) when \(t = 3\).
   \includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-10_559_1257_255_445} 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 \(O P\).
   \includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-10_2716_38_109_2012}
   \includegraphics[max width=\textwidth, alt={}, center]{1eb2ef87-e858-41c1-8e96-75fb2222b57a-11_2725_35_99_20}
4. Find the value of \(\omega\).
5. Find the value of \(\beta\).