| Exam Board | WJEC |
|---|---|
| Module | Further Unit 6 (Further Unit 6) |
| Year | 2022 |
| Session | June |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Prove motion is SHM from equation |
| Difficulty | Standard +0.8 This is a multi-part SHM question requiring: (a) differentiation twice to verify SHM form and identify centre, converting sin+cos to R-form for amplitude/period; (b) using the SHM speed-displacement relationship v²=ω²(a²-x²); (c) solving simultaneous trigonometric equations. While systematic, it demands fluency across multiple SHM techniques and careful algebraic manipulation, placing it moderately above average difficulty for Further Maths mechanics. |
| Spec | 1.05n Harmonic form: a sin(x)+b cos(x) = R sin(x+alpha) etc4.10f Simple harmonic motion: x'' = -omega^2 x |
2. A particle $P$ moves along the $x$-axis such that its position $x$ metres, after $t$ seconds, is given by
$$x = \sin ( \pi t ) + \sqrt { 3 } \cos ( \pi t )$$
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Show that the motion of the particle $P$ is Simple Harmonic. State the value of $x$ at the centre of motion.
\item Show that the period of the motion of $P$ is 2 s and determine the amplitude.
Suppose that another particle $Q$ is introduced so that it also moves along the $x$-axis with Simple Harmonic Motion with centre of motion, $O$, and period equal to that of particle $P$. When $t = 0$, the particle $Q$ is at $O$ and when it is $2 \sqrt { 3 } \mathrm {~m}$ from $O$ its speed is $2 \pi \mathrm {~ms} ^ { - 1 }$.
\end{enumerate}\item Find the amplitude of particle $Q$.
\item Determine the time when particles $P$ and $Q$ first meet.
\end{enumerate}
\hfill \mbox{\textit{WJEC Further Unit 6 2022 Q2 [15]}}