| Exam Board | OCR |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2004 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Small oscillations: rigid body compound pendulum |
| Difficulty | Standard +0.3 This is a straightforward rotational dynamics problem requiring standard formulas (I = 2mr²/5 for a sphere, τ = Iα, kinematic equations, and work-energy). All steps are routine applications of memorized results with no conceptual challenges or novel problem-solving required, making it slightly easier than average. |
| Spec | 6.02b Calculate work: constant force, resolved component6.05a Angular velocity: definitions |
| Answer | Marks | Guidance |
|---|---|---|
| \(I = \frac{2}{5} \times 14 \times 0.25^2 \quad (= 0.35)\) | B1 | |
| \(4.2 = I\alpha\) | M1 | |
| \(\alpha = 12 \text{ rad s}^{-2}\) | A1 | Total: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\theta = \omega_1 t + \frac{1}{2}\alpha t^2\); \(\quad 7500 = \omega_1 \times 30 + \frac{1}{2} \times 12 \times 30^2\) | M1 | |
| \(\omega_1 = 70 \text{ rad s}^{-1}\) | A1 ft | Ft \(250 - 15\alpha\) Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\omega_2 = \omega_1 + \alpha t\); \(\quad \omega_2 = 70 + 12 \times 30\) | M1 | |
| \(\omega_2 = 430 \text{ rad s}^{-1}\) | A1 ft | Ft \(250 + 15\alpha\) Total: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Work done is \(L\theta = 4.2 \times 7500\) | M1 | Or \(\frac{1}{2}I(\omega_2^2 - \omega_1^2)\) |
| \(= 31500 \text{ J}\) | A1 | Total: 2 |
# Question 4(i):
$I = \frac{2}{5} \times 14 \times 0.25^2 \quad (= 0.35)$ | B1 |
$4.2 = I\alpha$ | M1 |
$\alpha = 12 \text{ rad s}^{-2}$ | A1 | **Total: 3**
## Question 4(ii):
$\theta = \omega_1 t + \frac{1}{2}\alpha t^2$; $\quad 7500 = \omega_1 \times 30 + \frac{1}{2} \times 12 \times 30^2$ | M1 |
$\omega_1 = 70 \text{ rad s}^{-1}$ | A1 ft | Ft $250 - 15\alpha$ **Total: 2**
## Question 4(iii):
$\omega_2 = \omega_1 + \alpha t$; $\quad \omega_2 = 70 + 12 \times 30$ | M1 |
$\omega_2 = 430 \text{ rad s}^{-1}$ | A1 ft | Ft $250 + 15\alpha$ **Total: 2**
## Question 4(iv):
Work done is $L\theta = 4.2 \times 7500$ | M1 | Or $\frac{1}{2}I(\omega_2^2 - \omega_1^2)$
$= 31500 \text{ J}$ | A1 | **Total: 2**
---4 A uniform solid sphere, of mass 14 kg and radius 0.25 m , is rotating about a fixed axis which is a diameter of the sphere. A couple of constant moment 4.2 Nm about the axis, acting in the direction of rotation, is applied to the sphere.\\
(i) Find the angular acceleration of the sphere.
During a time interval of 30 seconds the sphere rotates through 7500 radians.\\
(ii) Find the angular speed of the sphere at the start of the time interval.\\
(iii) Find the angular speed of the sphere at the end of the time interval.\\
(iv) Find the work done by the couple during the time interval.
\hfill \mbox{\textit{OCR M4 2004 Q4 [9]}}