| Exam Board | Edexcel |
|---|---|
| Module | M5 (Mechanics 5) |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Topic | Circular Motion 2 |
| Type | Angular speed and period |
| Difficulty | Challenging +1.3 This is a standard M5 moment of inertia question requiring integration setup for part (a), application of perpendicular axis theorem and energy methods for part (b), and small angle approximation for SHM in part (c). While it involves multiple steps and careful coordinate geometry, the techniques are all standard for this module with no novel insights required. The integration is straightforward once the geometry is established, making it moderately above average difficulty. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x6.04d Integration: for centre of mass of laminas/solids6.04e Rigid body equilibrium: coplanar forces |
A uniform lamina $ABC$ of mass $m$ is in the shape of an isosceles triangle with $AB = AC = 5a$ and $BC = 8a$.
\begin{enumerate}[label=(\alph*)]
\item Show, using integration, that the moment of inertia of the lamina about an axis through $A$, parallel to $BC$, is $\frac{9}{2}ma^2$.
[6]
\end{enumerate}
The foot of the perpendicular from $A$ to $BC$ is $D$. The lamina is free to rotate in a vertical plane about a fixed smooth horizontal axis which passes through $D$ and is perpendicular to the plane of the lamina. The lamina is released from rest when $DA$ makes an angle $\alpha$ with the downward vertical. It is given that the moment of inertia of the lamina about an axis through $D$, perpendicular to $BC$ and in the plane of the lamina, is $\frac{8}{3}ma^2$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the angular acceleration of the lamina when $DA$ makes an angle $\theta$ with the downward vertical.
[8]
\end{enumerate}
Given that $\alpha$ is small,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item find an approximate value for the period of oscillation of the lamina about the vertical.
[2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M5 Q3 [16]}}