| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2016 |
| Session | November |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Advanced work-energy problems |
| Type | Particle in circular tube or on wire |
| Difficulty | Challenging +1.2 This is a multi-step energy conservation problem involving elastic potential energy, gravitational potential energy, and kinetic energy in circular motion. While it requires careful bookkeeping of energy terms and understanding when the string becomes slack (tension = 0), the approach is methodical and follows standard M2 techniques without requiring novel insight. The calculations are straightforward once the energy equation is set up correctly. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.05d Variable speed circles: energy methods |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(EE = 8(0.9\pi - 1.2)^2/(2 \times 1.2)\) | B1 | Initial \(EE = 8.83\) J |
| \(8.83 = 0.2g \times 0.9 + 0.2v^2/2 + 8(0.9\pi/2 - 1.2)^2/(2 \times 1.2)\) | M1, A1 | |
| \(v = 8.29\text{ ms}^{-1}\) | A1 | Total: 4 marks |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\theta = 1.2/0.9 = 4/3\text{ rad } (=76.4°)\) | B1 | |
| \(8.83 = 0.2g \times 0.9 + 0.2g \times 0.9\cos\theta + 0.2v^2/2\) | M1 | \(0.2 \times 8.29^2/2 = 0.2g \times 0.9\cos\theta + 0.2v^2/2\) |
| \(v = 8.13\text{ ms}^{-1}\) | A1 | Total: 3 marks |
## Question 6:
### Part (i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $EE = 8(0.9\pi - 1.2)^2/(2 \times 1.2)$ | B1 | Initial $EE = 8.83$ J |
| $8.83 = 0.2g \times 0.9 + 0.2v^2/2 + 8(0.9\pi/2 - 1.2)^2/(2 \times 1.2)$ | M1, A1 | |
| $v = 8.29\text{ ms}^{-1}$ | A1 | Total: 4 marks |
### Part (ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\theta = 1.2/0.9 = 4/3\text{ rad } (=76.4°)$ | B1 | |
| $8.83 = 0.2g \times 0.9 + 0.2g \times 0.9\cos\theta + 0.2v^2/2$ | M1 | $0.2 \times 8.29^2/2 = 0.2g \times 0.9\cos\theta + 0.2v^2/2$ |
| $v = 8.13\text{ ms}^{-1}$ | A1 | Total: 3 marks |
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\includegraphics[max width=\textwidth, alt={}, center]{0a80f46b-b37e-46ce-8907-9d10e4f62f6d-3_483_419_1800_863}
The diagram shows a smooth narrow tube formed into a fixed vertical circle with centre $O$ and radius 0.9 m . A light elastic string with modulus of elasticity 8 N and natural length 1.2 m has one end attached to the highest point $A$ on the inside of the tube. The other end of the string is attached to a particle $P$ of mass 0.2 kg . The particle is released from rest at the lowest point on the inside of the tube. By considering energy, calculate\\
(i) the speed of $P$ when it is at the same horizontal level as $O$,\\
(ii) the speed of $P$ at the instant when the string becomes slack.
\hfill \mbox{\textit{CAIE M2 2016 Q6 [7]}}