Challenging +1.8 This is a challenging Further Maths circular motion problem requiring energy conservation and centripetal force equations at multiple points, with geometric constraints linking two positions. Students must set up simultaneous equations involving tensions, speeds, and angles, then eliminate variables systematically—significantly harder than standard single-position circular motion questions but follows established FM2 techniques.
3 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 \(O\). The path of the particle is a complete vertical circle with centre \(O\). When \(P\) is at its lowest point, its speed is \(u\). When \(P\) is at the point \(A\), the tension in the string is \(T\) and the string makes an angle \(\theta\) with the downward vertical, where \(\cos \theta = \frac { 3 } { 5 }\). When \(P\) is at the point \(B\), above the level of \(O\), the tension in the string is \(\frac { 1 } { 8 } T\) and angle \(B O A = 90 ^ { \circ }\). Find \(u\) in terms of \(a\) and \(g\).
3 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 $O$. The path of the particle is a complete vertical circle with centre $O$. When $P$ is at its lowest point, its speed is $u$. When $P$ is at the point $A$, the tension in the string is $T$ and the string makes an angle $\theta$ with the downward vertical, where $\cos \theta = \frac { 3 } { 5 }$. When $P$ is at the point $B$, above the level of $O$, the tension in the string is $\frac { 1 } { 8 } T$ and angle $B O A = 90 ^ { \circ }$. Find $u$ in terms of $a$ and $g$.
\hfill \mbox{\textit{CAIE FP2 2013 Q3 [9]}}