Challenging +1.2 This is a two-part Further Maths mechanics question requiring energy conservation methods. Part (a) involves standard elastic energy with gravity on an incline—straightforward energy equation setup. Part (b) requires applying the condition for completing a vertical circle (tension = 0 at top) combined with energy conservation, which is a well-known technique but requires careful algebraic manipulation. Both parts are standard Further Maths fare with multiple steps but no novel insights needed.
1 A particle \(P\) of mass \(m\) is placed on a fixed smooth plane which is inclined at an angle \(\theta\) to the horizontal. A light spring, of natural length \(a\) and modulus of elasticity \(3 m g\), has one end attached to \(P\) and the other end attached to a fixed point \(O\) at the top of the plane. The spring lies along a line of greatest slope of the plane. The system is released from rest with the spring at its natural length.
Find, in terms of \(a\) and \(\theta\), an expression for the greatest extension of the spring in the subsequent motion.
\includegraphics[max width=\textwidth, alt={}, center]{1c53c407-25ea-43fc-a571-74ba1fffea8f-04_515_707_267_685}
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 \(\frac { 4 } { 5 } \sqrt { 5 a g }\) perpendicular to the string and just completes a vertical circle (see diagram).
Find the value of \(\cos \theta\).
1 A particle $P$ of mass $m$ is placed on a fixed smooth plane which is inclined at an angle $\theta$ to the horizontal. A light spring, of natural length $a$ and modulus of elasticity $3 m g$, has one end attached to $P$ and the other end attached to a fixed point $O$ at the top of the plane. The spring lies along a line of greatest slope of the plane. The system is released from rest with the spring at its natural length.
Find, in terms of $a$ and $\theta$, an expression for the greatest extension of the spring in the subsequent motion.\\
\includegraphics[max width=\textwidth, alt={}, center]{1c53c407-25ea-43fc-a571-74ba1fffea8f-04_515_707_267_685}
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 $\frac { 4 } { 5 } \sqrt { 5 a g }$ perpendicular to the string and just completes a vertical circle (see diagram).
Find the value of $\cos \theta$.\\
\hfill \mbox{\textit{CAIE Further Paper 3 2020 Q1 [3]}}