| Exam Board | Edexcel |
|---|---|
| Module | M5 (Mechanics 5) |
| Year | 2004 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments of inertia |
| Type | Force at pivot/axis |
| Difficulty | Standard +0.8 This M5 question requires finding moment of inertia for a rectangle about an edge (standard formula application), then using energy conservation to find angular speed (moderate setup), and finally calculating reaction forces involving both weight and centripetal acceleration. The multi-step nature, combination of rotational dynamics concepts, and force analysis at a specific position make this harder than average, though the techniques are systematic for Further Maths M5 students. |
| Spec | 3.04a Calculate moments: about a point6.02i Conservation of energy: mechanical energy principle6.04c Composite bodies: centre of mass |
| Answer | Marks | Guidance |
|---|---|---|
| \(I_{PQ} = \frac{4}{3}m(3a)^2 = 12ma^2\) | M1 A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Energy: \(\frac{1}{2} \times 12ma^2 \times \dot{\theta}^2 = mg \times 3a\) | M1 A1 ft | |
| \(\Rightarrow \dot{\theta} = \sqrt{\left(\frac{g}{2a}\right)}\) (*) | A1 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| \(R(\uparrow): Y - mg = m \times 3a\dot{\theta}^2\) | M1 A1 | |
| \(Y = mg + m \times 3a \times \frac{g}{2a} = \frac{5}{2}mg\) | M1 A1 | (4 marks) |
## Question 3:
### Part (a)
$I_{PQ} = \frac{4}{3}m(3a)^2 = 12ma^2$ | M1 A1 | (2 marks)
### Part (b)
Energy: $\frac{1}{2} \times 12ma^2 \times \dot{\theta}^2 = mg \times 3a$ | M1 A1 ft |
$\Rightarrow \dot{\theta} = \sqrt{\left(\frac{g}{2a}\right)}$ (*) | A1 | (3 marks)
### Part (c)
$R(\uparrow): Y - mg = m \times 3a\dot{\theta}^2$ | M1 A1 |
$Y = mg + m \times 3a \times \frac{g}{2a} = \frac{5}{2}mg$ | M1 A1 | (4 marks)
---3. A uniform lamina of mass $m$ is in the shape of a rectangle $P Q R S$, where $P Q = 8 a$ and $Q R = 6 a$.
\begin{enumerate}[label=(\alph*)]
\item Find the moment of inertia of the lamina about the edge $P Q$.
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\includegraphics[alt={},max width=\textwidth]{26fef791-e6fb-45a8-89e9-16c4b4a1a4e4-3_336_772_528_642}
\end{center}
\end{figure}
The flap on a letterbox is modelled as such a lamina. The flap is free to rotate about an axis along its horizontal edge $P Q$, as shown in Fig. 1. The flap is released from rest in a horizontal position. It then swings down into a vertical position.
\item Show that the angular speed of the flap as it reaches the vertical position is $\sqrt { \left( \frac { g } { 2 a } \right) }$.
\item Find the magnitude of the vertical component of the resultant force of the axis $P Q$ on the flap, as it reaches the vertical position.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M5 2004 Q3 [9]}}