| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2018 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Circular Motion 2 |
| Type | String through hole/bead on string |
| Difficulty | Standard +0.8 This is a multi-part circular motion problem requiring analysis of forces in two connected systems (particle on surface and hanging particle), consideration of elastic string extension conditions, and solving simultaneous equations involving tension, weight, elastic force, and centripetal acceleration. It requires more sophisticated problem-solving than standard single-particle circular motion questions, but uses well-established M2 techniques without requiring exceptional insight. |
| Spec | 6.02h Elastic PE: 1/2 k x^26.05c Horizontal circles: conical pendulum, banked tracks |
| Answer | Marks | Guidance |
|---|---|---|
| \(r [= 0.6 - (0.4 - 0.3)] = 0.5\) | B1 | |
| \(T = 0.3g\) | B1 | Resolve vertically for Q |
| \(0.2v^2 / 0.5 = 0.3g\) | M1 | Use Newton's Second Law horizontally for P |
| \(v = 2.74 \text{ ms}^{-1}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| \(r = 0.5 + e\) | B1 | \(e\) = extension of the string |
| \(T = \frac{15e}{0.3} = 50e\) | B1 | Use \(T = \frac{\lambda x}{l}\) |
| \(0.2 \times 8^2(5 + e) = 50e + 0.3g\) | M1 | Use Newton's Second Law horizontally with \(a = r\omega^2\) |
| \(e = \frac{(6.4 - 3)}{(50 - 12.8)}\) \((= 0.0914)\) | A1 | |
| \(HP = 0.591\) m | A1 |
## Question 6(i):
$r [= 0.6 - (0.4 - 0.3)] = 0.5$ | B1 |
$T = 0.3g$ | B1 | Resolve vertically for Q
$0.2v^2 / 0.5 = 0.3g$ | M1 | Use Newton's Second Law horizontally for P
$v = 2.74 \text{ ms}^{-1}$ | A1 |
**Total: 4 marks**
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## Question 6(ii):
$r = 0.5 + e$ | B1 | $e$ = extension of the string
$T = \frac{15e}{0.3} = 50e$ | B1 | Use $T = \frac{\lambda x}{l}$
$0.2 \times 8^2(5 + e) = 50e + 0.3g$ | M1 | Use Newton's Second Law horizontally with $a = r\omega^2$
$e = \frac{(6.4 - 3)}{(50 - 12.8)}$ $(= 0.0914)$ | A1 |
$HP = 0.591$ m | A1 |
**Total: 5 marks**
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\includegraphics[max width=\textwidth, alt={}, center]{8dda6c21-7cb5-43b6-9a34-485bdf4042c4-10_262_732_264_705}
A particle $P$ of mass 0.2 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a particle $Q$ of mass 0.3 kg . The string passes through a small hole $H$ in a smooth horizontal surface. A light elastic string of natural length 0.3 m and modulus of elasticity 15 N joins $Q$ to a fixed point $A$ which is 0.4 m vertically below $H$. The particle $P$ moves on the surface in a horizontal circle with centre $H$ (see diagram).\\
(i) Calculate the greatest possible speed of $P$ for which the elastic string is not extended.\\
(ii) Find the distance $H P$ given that the angular speed of $P$ is $8 \mathrm { rad } \mathrm { s } ^ { - 1 }$.\\
\hfill \mbox{\textit{CAIE M2 2018 Q6 [9]}}