| Exam Board | Edexcel |
|---|---|
| Module | M5 (Mechanics 5) |
| Session | Specimen |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Particular solution with initial conditions |
| Difficulty | Challenging +1.3 This is a second-order vector differential equation requiring complementary function (SHM solution) and particular integral (with resonance consideration since forcing frequency ω=1 differs from natural frequency ω=3), then applying two initial conditions to find constants. While mechanically lengthy (11 marks), it follows a standard procedure for M5 students who have learned forced oscillations, making it moderately challenging but not requiring novel insight. |
| Spec | 1.10h Vectors in kinematics: uniform acceleration in vector form4.10e Second order non-homogeneous: complementary + particular integral4.10f Simple harmonic motion: x'' = -omega^2 x |
| Answer | Marks | Guidance |
|---|---|---|
| \(m^2 + 9 = 0 \Rightarrow m = \pm3i\) | M1 | |
| \(r = A\sin 3t + B\cos 3t\) | A1 | |
| Let \(r = p\sin \hat{n}t\) | ||
| \(\dot{r} = p\cos \hat{n}t\) | M1 A1 | |
| \(\ddot{r} = -p\sin \hat{n}t\) | ||
| \(-p\sin \hat{n}t + 9p\sin \hat{n}t = 8\sin \hat{n}t\) | M1 | |
| \(\Rightarrow p = 1\) | A1 | |
| \(r = A\sin 3t + B\cos 3t + \sin \hat{n}t\) | M1 | |
| At \(t = 0\): \(0 = B\) | A1 | |
| \(\dot{r} = 3A\cos 3t + \cos \hat{n}t + \cos \hat{n}t\) | M1 | |
| At \(t = 0\): \(\hat{n} + 3\hat{j} = 3A + \hat{n} \Rightarrow A = \hat{j}\) | A1 | |
| \(\therefore r = \sin \hat{n}t + \sin 3\hat{j}t\) | A1 | (11 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\sin t = \sin 3t = 0\) | M1 | |
| \(\Rightarrow t = \pi\) | A1 | (2 marks) |
Total: 13 marks
## Part (a)
$m^2 + 9 = 0 \Rightarrow m = \pm3i$ | M1 |
$r = A\sin 3t + B\cos 3t$ | A1 |
Let $r = p\sin \hat{n}t$ | |
$\dot{r} = p\cos \hat{n}t$ | M1 A1 |
$\ddot{r} = -p\sin \hat{n}t$ | |
$-p\sin \hat{n}t + 9p\sin \hat{n}t = 8\sin \hat{n}t$ | M1 |
$\Rightarrow p = 1$ | A1 |
$r = A\sin 3t + B\cos 3t + \sin \hat{n}t$ | M1 |
At $t = 0$: $0 = B$ | A1 |
$\dot{r} = 3A\cos 3t + \cos \hat{n}t + \cos \hat{n}t$ | M1 |
At $t = 0$: $\hat{n} + 3\hat{j} = 3A + \hat{n} \Rightarrow A = \hat{j}$ | A1 |
$\therefore r = \sin \hat{n}t + \sin 3\hat{j}t$ | A1 | (11 marks)
## Part (b)
$\sin t = \sin 3t = 0$ | M1 |
$\Rightarrow t = \pi$ | A1 | (2 marks)
Total: 13 marksA particle $P$ moves in the $x$-$y$ plane and has position vector $\mathbf{r}$ metres relative to a fixed origin $O$ at time $t$ s. Given that $\mathbf{r}$ satisfies the vector differential equation
$$\frac{d^2\mathbf{r}}{dt^2} + 9\mathbf{r} = 8\sin t \mathbf{i}$$
and that when $t = 0$ s, $P$ is at $O$ and moving with velocity $(\mathbf{i} + 3\mathbf{j})$ m s$^{-1}$,
\begin{enumerate}[label=(\alph*)]
\item find $\mathbf{r}$ at time $t$. [11]
\item Hence find when $P$ next returns to $O$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M5 Q8 [13]}}