Challenging +1.2 This is a standard circular motion problem requiring application of energy conservation and the condition for completing a vertical circle (tension = 0 at top, v² = ag). The setup is typical for Further Mechanics, involving straightforward algebraic manipulation of well-known formulas across 5 marks, but requires careful bookkeeping of the energy equation and understanding of the 'just completes' condition.
\includegraphics{figure_2}
A particle \(P\) 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 particle \(P\) is held with the string taut and making an angle \(\theta\) with the downward vertical. The particle \(P\) is then projected with speed \(\frac{1}{2}\sqrt{5ag}\) perpendicular to the string and just completes a vertical circle (see diagram).
Find the value of \(\cos\theta\). [5]
\includegraphics{figure_2}
A particle $P$ 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 particle $P$ is held with the string taut and making an angle $\theta$ with the downward vertical. The particle $P$ is then projected with speed $\frac{1}{2}\sqrt{5ag}$ perpendicular to the string and just completes a vertical circle (see diagram).
Find the value of $\cos\theta$. [5]
\hfill \mbox{\textit{CAIE Further Paper 3 2020 Q2 [5]}}