CAIE FP2 2019 June — Question 11 EITHER

Exam BoardCAIE
ModuleFP2 (Further Pure Mathematics 2)
Year2019
SessionJune
PaperDownload PDF ↗
TopicSimple Harmonic Motion
TypeEnergy methods in SHM
DifficultyChallenging +1.2 This is a standard Further Maths SHM energy problem requiring Hooke's law, energy conservation, and SHM period formulas. Part (i) is straightforward energy bookkeeping, part (ii) uses standard F=-kx to establish SHM, and part (iii) requires finding when KE = 0.75(initial KE) using energy methods and inverse trig. While multi-step, all techniques are routine for FM students with no novel insight required.
Spec4.10f Simple harmonic motion: x'' = -omega^2 x6.02i Conservation of energy: mechanical energy principle
A light spring has natural length \(a\) and modulus of elasticity \(k m g\). The spring lies on a smooth horizontal surface with one end attached to a fixed point \(O\). A particle \(P\) of mass \(m\) is attached to the other end of the spring. The system is in equilibrium with \(O P = a\). The particle is projected towards \(O\) with speed \(u\) and comes to instantaneous rest when \(O P = \frac { 3 } { 4 } a\).
  1. Use an energy method to show that \(k = \frac { 16 u ^ { 2 } } { a g }\).
  2. Show that \(P\) performs simple harmonic motion and find the period of this motion, giving your answer in terms of \(u\) and \(a\).
  3. Find, in terms of \(u\) and \(a\), the time that elapses before \(P\) first loses \(25 \%\) of its initial kinetic energy.
A light spring has natural length $a$ and modulus of elasticity $k m g$. The spring lies on a smooth horizontal surface with one end attached to a fixed point $O$. A particle $P$ of mass $m$ is attached to the other end of the spring. The system is in equilibrium with $O P = a$. The particle is projected towards $O$ with speed $u$ and comes to instantaneous rest when $O P = \frac { 3 } { 4 } a$.\\
(i) Use an energy method to show that $k = \frac { 16 u ^ { 2 } } { a g }$.\\

(ii) Show that $P$ performs simple harmonic motion and find the period of this motion, giving your answer in terms of $u$ and $a$.\\

(iii) Find, in terms of $u$ and $a$, the time that elapses before $P$ first loses $25 \%$ of its initial kinetic energy.\\

\hfill \mbox{\textit{CAIE FP2 2019 Q11 EITHER}}
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