| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2017 |
| Session | June |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Second order differential equations |
| Type | Particular solution with initial conditions |
| Difficulty | Standard +0.8 This is a damped harmonic motion problem requiring derivation of the differential equation from forces, solving a second-order DE with complex roots (underdamped case), applying initial conditions, and finding when velocity equals zero. While methodical, it involves multiple substantial steps across mechanics and differential equations, placing it moderately above average difficulty for A-level. |
| Spec | 4.10g Damped oscillations: model and interpret6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle |
6. A particle $P$ of mass 0.2 kg is suspended from a fixed point by a light elastic spring. The spring has natural length 0.8 m and modulus of elasticity 7 N . At time $t = 0$ the particle is released from rest from a point 0.2 metres vertically below its equilibrium position. The motion of $P$ is resisted by a force of magnitude $2 v$ newtons, where $v \mathrm {~ms} ^ { - 1 }$ is the speed of $P$. At time $t$ seconds, $P$ is $x$ metres below its equilibrium position.
\begin{enumerate}[label=(\alph*)]
\item Show that $\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 10 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 43.75 x = 0$
\item Find $x$ in terms of $t$.
\item Find the value of $t$ when $P$ first comes to instantaneous rest.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M4 2017 Q6 [13]}}