| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Session | Specimen |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Find period from given information |
| Difficulty | Challenging +1.3 This is a standard M4 damped harmonic motion problem requiring force analysis to derive the differential equation, solving a second-order linear ODE with constant coefficients (underdamped case), and extracting the period. While it involves multiple steps and M4-level mechanics, the techniques are routine for this module with no novel insight required—harder than average A-level due to the advanced topic but standard within M4. |
| Spec | 4.10d Second order homogeneous: auxiliary equation method4.10g Damped oscillations: model and interpret |
An elastic string spring of modulus $2mg$ and natural length $l$ is fixed at one end. To the other end is attached a mass $m$ which is allowed to hang in equilibrium. The mass is then pulled vertically downwards through a distance $l$ and released from rest. The air resistance is modelled as having magnitude $2m\omega v$, where $v$ is the speed of the particle and $\omega = \sqrt{\frac{g}{l}}$. The particle is at distance $x$ from its equilibrium position at time $t$.
\begin{enumerate}[label=(\alph*)]
\item Show that $\frac{\mathrm{d}^2 x}{\mathrm{d} t^2} + 2\omega \frac{\mathrm{d} x}{\mathrm{d} t} + 2\omega^2 x = 0$.
[7]
\item Find the general solution of this differential equation.
[4]
\item Hence find the period of the damped harmonic motion.
[1]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M4 Q5 [12]}}