| Exam Board | CAIE |
|---|---|
| Module | FP2 (Further Pure Mathematics 2) |
| Year | 2016 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Time to travel between positions |
| Difficulty | Standard +0.3 This is a standard SHM problem requiring application of standard formulas (v² = ω²(a² - x²), max acceleration = ω²a, period T = 2π/ω) with straightforward geometry to find amplitude and centre. The multi-part structure and time calculation in part (iii) add some complexity, but all steps follow routine procedures without requiring novel insight or proof. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x |
| Answer | Marks |
|---|---|
| \(4^2 = \omega^2(3^2 - 1^2)\), \(\omega^2 = 2\) | M1 A1 |
| \(2 \times 3 = 6\) [m s\(^{-2}\)] (allow \(-6\)) | A1\(\checkmark\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(60 / (2\pi/\omega)\) [= \(60 / 4.443 = 13.5\)] | M1 | |
| \(13\) | A1 | allow M1 A0 for \(60/(\pi/\omega)\) [= 27] |
| Answer | Marks |
|---|---|
| \(\omega^{-1}\sin^{-1}(1) + \omega^{-1}\sin^{-1}\frac{1}{3}\) | M1 |
| or \(\frac{1}{4}T + \omega^{-1}\sin^{-1}\frac{1}{3}\) [= \(1.111 + 0.240\)] | |
| or \(\omega^{-1}\cos^{-1}(-\frac{1}{3})\) | A1 |
| or \(\frac{1}{2}T - \omega^{-1}\cos^{-1}\frac{1}{3}\) [= \(2.221 - 0.870\)] | A1 |
| \(= 1.91/\omega\); \(= 1.35\) [s] |
# Question 1:
## Part (i)
| $4^2 = \omega^2(3^2 - 1^2)$, $\omega^2 = 2$ | M1 A1 | |
|---|---|---|
| $2 \times 3 = 6$ [m s$^{-2}$] (allow $-6$) | A1$\checkmark$ | |
## Part (ii)
| $60 / (2\pi/\omega)$ [= $60 / 4.443 = 13.5$] | M1 | |
|---|---|---|
| $13$ | A1 | allow M1 A0 for $60/(\pi/\omega)$ [= 27] |
## Part (iii)
| $\omega^{-1}\sin^{-1}(1) + \omega^{-1}\sin^{-1}\frac{1}{3}$ | M1 | |
|---|---|---|
| or $\frac{1}{4}T + \omega^{-1}\sin^{-1}\frac{1}{3}$ [= $1.111 + 0.240$] | | |
| or $\omega^{-1}\cos^{-1}(-\frac{1}{3})$ | A1 | |
| or $\frac{1}{2}T - \omega^{-1}\cos^{-1}\frac{1}{3}$ [= $2.221 - 0.870$] | A1 | |
| $= 1.91/\omega$; $= 1.35$ [s] | | |
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\includegraphics[max width=\textwidth, alt={}, center]{184020e1-7ff2-4172-8d33-baff963afa76-2_125_641_262_751}
The point $C$ is on the fixed line $l$. Points $A$ and $B$ on $l$ are such that $A C = 4 \mathrm {~m}$ and $C B = 2 \mathrm {~m}$, with $C$ between $A$ and $B$. The point $M$ is the mid-point of $A B$ (see diagram). A particle $P$ of mass $m$ oscillates between $A$ and $B$ in simple harmonic motion. When $P$ is at $C$, its speed is $4 \mathrm {~m} \mathrm {~s} ^ { - 1 }$. Find\\
(i) the magnitude of the maximum acceleration of $P$,\\
(ii) the number of complete oscillations made by $P$ in one minute,\\
(iii) the time that $P$ takes to travel directly from $A$ to $C$.
\hfill \mbox{\textit{CAIE FP2 2016 Q1 [8]}}