Challenging +1.2 This is a standard vertical circular motion problem requiring energy conservation and Newton's second law at two symmetric positions. While it involves multiple steps (finding speeds at both positions, applying circular motion equations, and algebraic manipulation), the approach is methodical and follows a well-established template for this topic. The symmetry simplifies the algebra, and the constraint θ < cos⁻¹(2/3) ensures the bead remains in contact throughout. This is moderately above average difficulty due to the multi-step nature and need to handle two positions systematically, but it's a classic Further Maths mechanics question without requiring novel insight.
2 A small bead \(B\) of mass \(m\) is threaded on a smooth wire fixed in a vertical plane. The wire forms a circle of radius \(a\) and centre \(O\). The highest point of the circle is \(A\). The bead is slightly displaced from rest at \(A\). When angle \(A O B = \theta\), where \(\theta < \cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)\), the force exerted on the bead by the wire has magnitude \(R _ { 1 }\). When angle \(A O B = \pi + \theta\), the force exerted on the bead by the wire has magnitude \(R _ { 2 }\). Show that \(R _ { 2 } - R _ { 1 } = 4 m g\).
2 A small bead $B$ of mass $m$ is threaded on a smooth wire fixed in a vertical plane. The wire forms a circle of radius $a$ and centre $O$. The highest point of the circle is $A$. The bead is slightly displaced from rest at $A$. When angle $A O B = \theta$, where $\theta < \cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)$, the force exerted on the bead by the wire has magnitude $R _ { 1 }$. When angle $A O B = \pi + \theta$, the force exerted on the bead by the wire has magnitude $R _ { 2 }$. Show that $R _ { 2 } - R _ { 1 } = 4 m g$.
\hfill \mbox{\textit{CAIE FP2 2008 Q2 [8]}}