| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2010 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hooke's law and elastic energy |
| Type | Particle at midpoint of string between two horizontal fixed points: vertical motion |
| Difficulty | Standard +0.3 This is a standard two-part elastic string problem requiring equilibrium analysis (resolving forces, Hooke's law) followed by energy conservation. While it involves multiple steps and careful geometry, the techniques are routine for M2 students with no novel insight required—slightly above average due to the calculation complexity and energy method application. |
| Spec | 6.02d Mechanical energy: KE and PE concepts6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle |
| Answer | Marks | Guidance |
|---|---|---|
| Modulus is 25N | \(M1, A1, M1, A1\) [4] | For resolving forces on P vertically. For using \(T = 2\lambda v/L\). AG |
| Answer | Marks | Guidance |
|---|---|---|
| EE when P is at \(M = 25 \times 0.8^2/(2 \times 4)\) | \(M1, A1, A1\) | For using \(EE = \lambda x^2/2L\) |
| Answer | Marks | Guidance |
|---|---|---|
| Speed is 4.90ms\(^{-1}\) | \(M1, A1, A1\) [6] | For using EE on release \(= mgh + \) EE when P is at \(M + \frac{1}{2}mv^2\) |
**(i)**
$[0.35g = 2T\frac{}{0.7/(2 \times 4)}]$
Tension is 6.25N
$[6.25 = 2 \times \frac{1}{4}]$
Modulus is 25N | $M1, A1, M1, A1$ [4] | For resolving forces on P vertically. For using $T = 2\lambda v/L$. AG
**(ii)**
EE on release $= 25 \times 2^2/(2 \times 4)$
EE when P is at $M = 25 \times 0.8^2/(2 \times 4)$ | $M1, A1, A1$ | For using $EE = \lambda x^2/2L$
$25 \times 2^2/(2 \times 4) = 0.35g \times 1.8 + 25 \times 0.8^2/(2 \times 4) + \frac{1}{2} \times 0.35v^2$
Speed is 4.90ms$^{-1}$ | $M1, A1, A1$ [6] | For using EE on release $= mgh + $ EE when P is at $M + \frac{1}{2}mv^2$
---6\\
\includegraphics[max width=\textwidth, alt={}, center]{ae809dfc-c5af-4c0a-9c88-009949d3e9f9-4_324_1267_794_440}
A particle $P$ of mass 0.35 kg is attached to the mid-point of a light elastic string of natural length 4 m . The ends of the string are attached to fixed points $A$ and $B$ which are 4.8 m apart at the same horizontal level. $P$ hangs in equilibrium at a point 0.7 m vertically below the mid-point $M$ of $A B$ (see diagram).\\
(i) Find the tension in the string and hence show that the modulus of elasticity of the string is 25 N .\\
$P$ is now held at rest at a point 1.8 m vertically below $M$, and is then released.\\
(ii) Find the speed with which $P$ passes through $M$.
\hfill \mbox{\textit{CAIE M2 2010 Q6 [10]}}