| Exam Board | Edexcel |
|---|---|
| Module | M5 (Mechanics 5) |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Topic | Moments of inertia |
| Type | Impulse and angular speed |
| Difficulty | Challenging +1.8 This M5 question requires integration to derive moment of inertia using parallel strips (a standard but non-trivial technique), then applies impulse-momentum and energy conservation to rotational motion. Part (a) is a guided proof with 6 marks requiring careful setup of integration. Part (b) involves multiple steps: finding angular velocity from impulse, identifying the geometry of rotation, and using energy conservation with a 120° rotation. The combination of deriving MOI by integration and multi-step rotational dynamics with energy methods makes this substantially harder than average, though it follows established M5 patterns. |
| Spec | 6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle6.04d Integration: for centre of mass of laminas/solids |
\includegraphics{figure_4}
**Figure 1**
A uniform lamina of mass $M$ is in the shape of a right-angled triangle $OAB$. The angle $OAB$ is $90°$, $OA = a$ and $AB = 2a$, as shown in Figure 1.
\begin{enumerate}[label=(\alph*)]
\item Prove, using integration, that the moment of inertia of the lamina $OAB$ about the edge $OA$ is $\frac{8}{3}Ma^2$.
(You may assume without proof that the moment of inertia of a uniform rod of mass $m$ and length $2l$ about an axis through one end and perpendicular to the rod is $\frac{4}{3}ml^2$.)
[6]
\end{enumerate}
The lamina $OAB$ is free to rotate about a fixed smooth horizontal axis along the edge $OA$ and hangs at rest with $B$ vertically below $A$. The lamina is then given a horizontal impulse of magnitude $J$. The impulse is applied to the lamina at the point $B$, in a direction which is perpendicular to the plane of the lamina. Given that the lamina first comes to instantaneous rest after rotating through an angle of $120°$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find an expression for $J$, in terms of $M$, $a$ and $g$.
[7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M5 Q4 [13]}}