| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2009 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 2 |
| Type | Rotating disc/platform system |
| Difficulty | Challenging +1.2 This is a multi-part circular motion problem requiring geometric reasoning (relating speeds via radii), resolving forces in vertical and horizontal directions, and applying F=mrω². While it involves several steps and careful setup, the techniques are standard M2 content with no novel insights required. The 'show that' part provides scaffolding, making it moderately above average difficulty. |
| Spec | 3.03n Equilibrium in 2D: particle under forces6.05b Circular motion: v=r*omega and a=v^2/r6.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \([r = 0.8]\) | M1 | For using \(v_P/v_A = r/0.5\) |
| M1 | For using \(\sin\theta = (r - 0.5)/0.8\) | |
| \(\sin\theta = \frac{3}{8}\) | A1 | AG Subtotal: 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \([T\cos\theta = mg]\) | M1 | For resolving forces vertically |
| Tension is 4.31 N | A1 | Subtotal: 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \([T\sin\theta = m\omega^2 r]\) | M1 | For using Newton's second law and \(a = \omega^2 r\) |
| \(0.375T = 0.4 \times 0.8\omega^2\) | A1 | |
| \(\omega = 2.25\) | A1 | Subtotal: 3 |
## Question 5:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $[r = 0.8]$ | M1 | For using $v_P/v_A = r/0.5$ |
| | M1 | For using $\sin\theta = (r - 0.5)/0.8$ |
| $\sin\theta = \frac{3}{8}$ | A1 | AG **Subtotal: 3** |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $[T\cos\theta = mg]$ | M1 | For resolving forces vertically |
| Tension is 4.31 N | A1 | **Subtotal: 2** |
### Part (iii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $[T\sin\theta = m\omega^2 r]$ | M1 | For using Newton's second law and $a = \omega^2 r$ |
| $0.375T = 0.4 \times 0.8\omega^2$ | A1 | |
| $\omega = 2.25$ | A1 | **Subtotal: 3** | **Total: 8** |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{fe5c198d-5d05-4241-98f5-894ba92f7afe-3_593_828_1530_660}
A horizontal disc of radius 0.5 m is rotating with constant angular speed $\omega \mathrm { rad } \mathrm { s } ^ { - 1 }$ about a fixed vertical axis through its centre $O$. One end of a light inextensible string of length 0.8 m is attached to a point $A$ of the circumference of the disc. A particle $P$ of mass 0.4 kg is attached to the other end of the string. The string is taut and the system rotates so that the string is always in the same vertical plane as the radius $O A$ of the disc. The string makes a constant angle $\theta$ with the vertical (see diagram). The speed of $P$ is 1.6 times the speed of $A$.\\
(i) Show that $\sin \theta = \frac { 3 } { 8 }$.\\
(ii) Find the tension in the string.\\
(iii) Find the value of $\omega$.
\hfill \mbox{\textit{CAIE M2 2009 Q5 [8]}}