| Exam Board | Edexcel |
|---|---|
| Module | M5 (Mechanics 5) |
| Year | 2011 |
| Session | June |
| Marks | 17 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 2 |
| Type | Angular speed and period |
| Difficulty | Challenging +1.3 This is a standard M5 compound pendulum problem requiring moment of inertia calculations using parallel axis theorem, energy conservation for angular motion, and numerical approximation. While it involves multiple steps and careful bookkeeping of the composite system (rod + disc), the techniques are all standard for this module with no novel insights required. The 17-mark allocation reflects thoroughness rather than exceptional difficulty. |
| Spec | 6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces6.05f Vertical circle: motion including free fall |
| Answer | Marks |
|---|---|
| \(I_{DISC} = \frac{ma^2}{4} + m(2a)^2 = \frac{17ma^2}{4}\) | M1 A1 |
| \(I_{ROD} = \frac{3m(2a)^2}{3} = 4ma^2\) | B1 |
| \(I_{PENDULUM} = \frac{17ma^2}{4} + 4ma^2 = \frac{33ma^2}{4}\) | M1 A1 |
Total: 5 marks
| Answer | Marks |
|---|---|
| \(3mga(\cos\theta - \cos\alpha) + mg \cdot 2a(\cos\theta - \cos\alpha) = \frac{1}{2}\frac{33ma^2}{4}\dot{\theta}^2\) | M1 A2 |
| \(\frac{40g(\cos\theta - \cos\alpha)}{33a} = \dot{\theta}^2 *\) | A1 |
Total: 4 marks
| Answer | Marks |
|---|---|
| \(2\ddot{\theta} = -\frac{40g}{33a}\sin\theta\dot{\theta}\) | M1 A1 |
| \(\ddot{\theta} = -\frac{20g}{33a}\sin\theta\) | A1 |
Total: 3 marks
| Answer | Marks | Guidance |
|---|---|---|
| For small \(\theta\), \(\ddot{\theta} = -\frac{20g}{33a}\theta\) i.e. SHM | M1 | |
| \(\omega = \sqrt{\frac{20g}{33a}} = \sqrt{\frac{20g}{33x_1^2}} = 7\) | A1 | |
| \(\theta = \alpha\cos\omega t\) | M1 | |
| \(\dot{\theta} = -\alpha\omega\sin\omega t\) | M1 | |
| \(= -7\frac{\pi}{20}\sin 1.4\) | ||
| \( | \dot{\theta} | = 1.08\) rad s\(^{-1}\) (3SF) |
Total: 5 marks + 17 marks overall
## (a)
$I_{DISC} = \frac{ma^2}{4} + m(2a)^2 = \frac{17ma^2}{4}$ | M1 A1 |
$I_{ROD} = \frac{3m(2a)^2}{3} = 4ma^2$ | B1 |
$I_{PENDULUM} = \frac{17ma^2}{4} + 4ma^2 = \frac{33ma^2}{4}$ | M1 A1 |
Total: 5 marks
## (b)
$3mga(\cos\theta - \cos\alpha) + mg \cdot 2a(\cos\theta - \cos\alpha) = \frac{1}{2}\frac{33ma^2}{4}\dot{\theta}^2$ | M1 A2 |
$\frac{40g(\cos\theta - \cos\alpha)}{33a} = \dot{\theta}^2 *$ | A1 |
Total: 4 marks
## (c)
$2\ddot{\theta} = -\frac{40g}{33a}\sin\theta\dot{\theta}$ | M1 A1 |
$\ddot{\theta} = -\frac{20g}{33a}\sin\theta$ | A1 |
Total: 3 marks
## (d)
For small $\theta$, $\ddot{\theta} = -\frac{20g}{33a}\theta$ i.e. SHM | M1 |
$\omega = \sqrt{\frac{20g}{33a}} = \sqrt{\frac{20g}{33x_1^2}} = 7$ | A1 |
$\theta = \alpha\cos\omega t$ | M1 |
$\dot{\theta} = -\alpha\omega\sin\omega t$ | M1 |
$= -7\frac{\pi}{20}\sin 1.4$ | |
$|\dot{\theta}| = 1.08$ rad s$^{-1}$ (3SF) | A1 |
Total: 5 marks + 17 marks overallA pendulum consists of a uniform rod $PQ$, of mass $3m$ and length $2a$, which is rigidly fixed at its end $Q$ to the centre of a uniform circular disc of mass $m$ and radius $a$. The rod is perpendicular to the plane of the disc. The pendulum is free to rotate about a fixed smooth horizontal axis $L$ which passes through the end $P$ of the rod and is perpendicular to the rod.
\begin{enumerate}[label=(\alph*)]
\item Show that the moment of inertia of the pendulum about $L$ is $\frac{33}{4}ma^2$.
[5]
\end{enumerate}
The pendulum is released from rest in the position where $PQ$ makes an angle $\alpha$ with the downward vertical. At time $t$, $PQ$ makes an angle $\theta$ with the downward vertical.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that the angular speed, $\dot{\theta}$, of the pendulum satisfies
$$\dot{\theta}^2 = \frac{40g(\cos\theta - \cos\alpha)}{33a}$$
[4]
\item Hence, or otherwise, find the angular acceleration of the pendulum.
[3]
\end{enumerate}
Given that $\alpha = \frac{\pi}{20}$ and that $PQ$ has length $\frac{8}{33}$ m,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item find, to 3 significant figures, an approximate value for the angular speed of the pendulum $0.2$ s after it has been released from rest.
[5]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M5 2011 Q8 [17]}}