| Exam Board | Edexcel |
|---|---|
| Module | M5 (Mechanics 5) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments of inertia |
| Type | Composite body MI calculation |
| Difficulty | Challenging +1.2 This is a standard M5 compound pendulum question requiring moment of inertia calculations (using parallel axis theorem), finding the center of mass, and applying SHM theory. While it involves multiple steps and careful bookkeeping of masses and distances, the techniques are routine for Further Maths mechanics students with no novel problem-solving required beyond applying learned formulas systematically. |
| Spec | 6.04c Composite bodies: centre of mass6.04d Integration: for centre of mass of laminas/solids6.05f Vertical circle: motion including free fall |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| MI of element \(= 2\pi\rho r\,\delta r \times r^2\) | M1 | |
| \(m = \pi\rho a^2\) | ||
| \(I = \frac{2m}{a^2}\int_0^a r^3\,\mathrm{d}r\) | M1 | |
| \(= \frac{2m}{a^2}\left[\frac{r^4}{4}\right]_0^a = \frac{1}{2}ma^2\) | A1 | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(I = I_\text{rod} + I_\text{disc} = \frac{4}{3}\times 3m\times a^2 + \frac{1}{2}\times 4m\left(\frac{a}{2}\right)^2 + 4m\left(\frac{3a}{2}\right)^2\) | B1, M1 | |
| \(= 4ma^2 + \frac{ma^2}{2} + 9ma^2\) | ||
| \(= \frac{27}{2}ma^2\) | A1 | (3) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(\frac{27}{2}ma^2\ddot{\theta} = -3mg\times a\sin\theta - 4mg\times\frac{3a}{2}\sin\theta\) | M1 A2, 1, 0 | |
| \(= -9mga\sin\theta\) | ||
| \(\ddot{\theta} = -\frac{2g}{3a}\sin\theta\) | ||
| Small oscillations \(\Rightarrow \sin\theta \approx \theta\) | M1 | |
| \(\Rightarrow \ddot{\theta} = -\frac{2g}{3a}\theta\) which is SHM | A1 | |
| \(T = 2\pi\sqrt{\frac{3a}{2g}}\) | A1 | (6) (12) |
# Question 6:
## Part (a)
| Answer/Working | Marks | Guidance |
|---|---|---|
| MI of element $= 2\pi\rho r\,\delta r \times r^2$ | M1 | |
| $m = \pi\rho a^2$ | | |
| $I = \frac{2m}{a^2}\int_0^a r^3\,\mathrm{d}r$ | M1 | |
| $= \frac{2m}{a^2}\left[\frac{r^4}{4}\right]_0^a = \frac{1}{2}ma^2$ | A1 | (3) |
## Part (b)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $I = I_\text{rod} + I_\text{disc} = \frac{4}{3}\times 3m\times a^2 + \frac{1}{2}\times 4m\left(\frac{a}{2}\right)^2 + 4m\left(\frac{3a}{2}\right)^2$ | B1, M1 | |
| $= 4ma^2 + \frac{ma^2}{2} + 9ma^2$ | | |
| $= \frac{27}{2}ma^2$ | A1 | (3) |
## Part (c)
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\frac{27}{2}ma^2\ddot{\theta} = -3mg\times a\sin\theta - 4mg\times\frac{3a}{2}\sin\theta$ | M1 A2, 1, 0 | |
| $= -9mga\sin\theta$ | | |
| $\ddot{\theta} = -\frac{2g}{3a}\sin\theta$ | | |
| Small oscillations $\Rightarrow \sin\theta \approx \theta$ | M1 | |
| $\Rightarrow \ddot{\theta} = -\frac{2g}{3a}\theta$ which is SHM | A1 | |
| $T = 2\pi\sqrt{\frac{3a}{2g}}$ | A1 | (6) **(12)** |
---6. (a) Show by integration that the moment of inertia of a uniform disc, of mass $m$ and radius $a$, about an axis through the centre of disc and perpendicular to the plane of the disc is $\frac { 1 } { 2 } m a ^ { 2 }$.\\
(3 marks)
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{4e874199-105a-460f-af7c-da0ef1603933-4_887_591_997_812}
\end{center}
\end{figure}
A uniform rod $A B$ has mass $3 m$ and length $2 a$. A uniform disc, of mass $4 m$ and radius $\frac { 1 } { 2 } a$, is attached to the rod with the centre of the disc lying on the rod a distance $\frac { 3 } { 2 } a$ from $A$. The rod lies in the plane of the disc, as shown in Fig. 1. The disc and rod together form a pendulum which is free to rotate about a fixed smooth horizontal axis $L$ which passes through $A$ and is perpendicular to the plane of the pendulum.\\
(b) Show that the moment of inertia of the pendulum about $L$ is $\frac { 27 } { 2 } m a ^ { 2 }$.
The pendulum makes small oscillations about its position of stable equilibrium.\\
(c) Show that the motion of the pendulum is approximately simple harmonic, and find the period of the oscillations.\\
(6 marks)\\
\hfill \mbox{\textit{Edexcel M5 Q6 [12]}}