| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Find amplitude of SHM |
| Difficulty | Standard +0.3 This is a standard SHM verification question requiring differentiation twice to show acceleration is proportional to negative displacement (routine M3 technique), followed by straightforward application of initial conditions and the amplitude formula √(p²+q²). The question is slightly above average difficulty due to the algebraic manipulation required, but follows a well-practiced template with no novel insight needed. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x |
| Answer | Marks |
|---|---|
| (a) \(x = p\sin\omega t + q\cos\omega t\) | M1 A1 |
| \(\dot{x} = p\omega\cos\omega t - q\omega\sin\omega t\) | M1 A1 |
| \(\ddot{x} = -p\omega^2\sin\omega t - q\omega^2\cos\omega t = -\omega^2(p\sin\omega t + q\cos\omega t) = -\omega^2 x\) | M1 A1 |
| Acceleration proportional to displacement and directed towards \(O\), so SHM | B1 |
| (b) Maximum speed at \(O\), so \(15 = a\omega = 3a\) | M1 A1 |
| \(a = 5\) m | M1 A1 |
| Total: 7 marks |
(a) $x = p\sin\omega t + q\cos\omega t$ | M1 A1 |
$\dot{x} = p\omega\cos\omega t - q\omega\sin\omega t$ | M1 A1 |
$\ddot{x} = -p\omega^2\sin\omega t - q\omega^2\cos\omega t = -\omega^2(p\sin\omega t + q\cos\omega t) = -\omega^2 x$ | M1 A1 |
Acceleration proportional to displacement and directed towards $O$, so SHM | B1 |
(b) Maximum speed at $O$, so $15 = a\omega = 3a$ | M1 A1 |
$a = 5$ m | M1 A1 |
| **Total: 7 marks** |A particle moves along a straight line in such a way that its displacement $x$ m from a fixed point $O$ on the line, at time $t$ seconds after it leaves $O$, is given by $x = p \sin \omega t + q \cos \omega t$ where $p$, $q$ and $\omega$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Show that the motion of the particle is simple harmonic. [5 marks]
\item If the particle leaves $O$ with speed 15 ms$^{-1}$, and $\omega = 3$, find the amplitude of the motion. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q2 [7]}}