| 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 time calculation in part (iii) uses the standard SHM position equation. All techniques are routine for Further Maths students with no novel problem-solving required, making it slightly easier than average. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(4^2 = \omega^2(3^2 - 1^2)\), \(\omega^2 = 2\) | M1 A1 | Find \(\omega^2\) from \(v^2 = \omega^2(a^2 - x^2)\) |
| \(2 \times 3 = 6\) \([\text{m s}^{-2}]\) (allow \(-6\)) | A1\(\checkmark\) | Find max. acceln. from \(\frac{d^2x}{dt^2} = -\omega^2 x\), \(\checkmark\) on \(\omega^2\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(60 \div (2\pi/\omega)\) \([= 60 / 4.443 = 13.5]\) | M1 | Find no. of oscillations from \(T = 2\pi/\omega\) |
| \(13\) | A1 | Allow M1 A0 for \(60/(\pi/\omega) [= 27]\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| \(\omega^{-1}\sin^{-1}(1) + \omega^{-1}\sin^{-1}\frac{1}{3}\) | M1 | Find time from \(A\) to \(C\) |
| \(or\ \frac{1}{4}T + \omega^{-1}\sin^{-1}\frac{1}{3} [= 1.111 + 0.240]\) | A1 | |
| \(or\ \omega^{-1}\cos^{-1}(-\frac{1}{3})\) | ||
| \(or\ \frac{1}{2}T - \omega^{-1}\cos^{-1}\frac{1}{3} [= 2.221 - 0.870]\) | ||
| \(= 1.91/\omega\ ;= 1.35\) [s] | A1 |
# Question 1:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $4^2 = \omega^2(3^2 - 1^2)$, $\omega^2 = 2$ | M1 A1 | Find $\omega^2$ from $v^2 = \omega^2(a^2 - x^2)$ |
| $2 \times 3 = 6$ $[\text{m s}^{-2}]$ (allow $-6$) | A1$\checkmark$ | Find max. acceln. from $\frac{d^2x}{dt^2} = -\omega^2 x$, $\checkmark$ on $\omega^2$ |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $60 \div (2\pi/\omega)$ $[= 60 / 4.443 = 13.5]$ | M1 | Find no. of oscillations from $T = 2\pi/\omega$ |
| $13$ | A1 | Allow M1 A0 for $60/(\pi/\omega) [= 27]$ |
## Part (iii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\omega^{-1}\sin^{-1}(1) + \omega^{-1}\sin^{-1}\frac{1}{3}$ | M1 | Find time from $A$ to $C$ |
| $or\ \frac{1}{4}T + \omega^{-1}\sin^{-1}\frac{1}{3} [= 1.111 + 0.240]$ | A1 | |
| $or\ \omega^{-1}\cos^{-1}(-\frac{1}{3})$ | | |
| $or\ \frac{1}{2}T - \omega^{-1}\cos^{-1}\frac{1}{3} [= 2.221 - 0.870]$ | | |
| $= 1.91/\omega\ ;= 1.35$ [s] | A1 | |
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\includegraphics[max width=\textwidth, alt={}, center]{58728f93-bfdb-4f76-a9b9-3a1d1592bfc9-2_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]}}