| Exam Board | OCR |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2004 |
| Session | June |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments of inertia |
| Type | Small oscillations period |
| Difficulty | Challenging +1.2 This compound pendulum problem requires applying the parallel axis theorem to find moment of inertia for a rectangular lamina about a corner, then using the standard formula for period of a compound pendulum. While it involves multiple steps and M4-level content (rigid body dynamics), the techniques are standard applications of learned formulas without requiring novel geometric insight or complex problem-solving. |
| Spec | 6.04d Integration: for centre of mass of laminas/solids6.05f Vertical circle: motion including free fall |
| Answer | Marks | Guidance |
|---|---|---|
| \(I = \frac{4}{3}m(\frac{3}{2}a)^2 + \frac{4}{3}m(2a)^2\) | B1, M1 | For either term; Use of perpendicular axes rule |
| \(= \frac{25}{3}ma^2\) | A1 | Total: 3 |
| OR \(I = \frac{1}{3}m\{(\frac{3}{2}a)^2 + (2a)^2\} + m(\frac{5}{2}a)^2\) | B1, M1, A1 | For \(\frac{1}{3}m\{(\frac{3}{2}a)^2 + (2a)^2\}\); Use of parallel axes rule |
| Answer | Marks | Guidance |
|---|---|---|
| Period is \(2\pi\sqrt{\frac{I}{mgh}}\) | M1 | or \((-) mgh\sin\theta = I\ddot{\theta}\) |
| \(= 2\pi\sqrt{\frac{\frac{25}{3}ma^2}{mg\frac{5}{2}a}}\) | A1 ft | or \(-mg(\frac{5}{2}a)\theta \approx \frac{25}{3}ma^2\ddot{\theta}\) |
| \(= 2\pi\sqrt{\frac{10a}{3g}}\) | A1 | Total: 3 |
# Question 2(i):
$I = \frac{4}{3}m(\frac{3}{2}a)^2 + \frac{4}{3}m(2a)^2$ | B1, M1 | For either term; Use of perpendicular axes rule
$= \frac{25}{3}ma^2$ | A1 | **Total: 3**
OR $I = \frac{1}{3}m\{(\frac{3}{2}a)^2 + (2a)^2\} + m(\frac{5}{2}a)^2$ | B1, M1, A1 | For $\frac{1}{3}m\{(\frac{3}{2}a)^2 + (2a)^2\}$; Use of parallel axes rule
## Question 2(ii):
Period is $2\pi\sqrt{\frac{I}{mgh}}$ | M1 | or $(-) mgh\sin\theta = I\ddot{\theta}$
$= 2\pi\sqrt{\frac{\frac{25}{3}ma^2}{mg\frac{5}{2}a}}$ | A1 ft | or $-mg(\frac{5}{2}a)\theta \approx \frac{25}{3}ma^2\ddot{\theta}$
$= 2\pi\sqrt{\frac{10a}{3g}}$ | A1 | **Total: 3**
---2 A uniform rectangular lamina has mass $m$ and sides of length $3 a$ and $4 a$, and rotates freely about a fixed horizontal axis. The axis is perpendicular to the lamina and passes through a corner. The lamina makes small oscillations in its own plane, as a compound pendulum.\\
(i) Find the moment of inertia of the lamina about the axis.\\
(ii) Find the approximate period of the small oscillations.
\hfill \mbox{\textit{OCR M4 2004 Q2 [6]}}