| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2012 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hooke's law and elastic energy |
| Type | Particle at midpoint of string between vertical fixed points |
| Difficulty | Challenging +1.2 This is a multi-part elastic string problem requiring energy conservation with variable extension in two configurations, plus finding maximum KE by considering force equilibrium. It demands careful geometric analysis and systematic application of Hooke's law and energy principles, making it moderately harder than average A-level mechanics questions but still within standard M2 scope. |
| Spec | 6.02g Hooke's law: T = k*x or T = lambda*x/l6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle |
| Answer | Marks | Guidance |
|---|---|---|
| (i) | \([a = 0.2 + 0.012t]\) | M1 |
| \([0.2 + 0.012t = 2.5 \times 0.2]]\) | M1 | For attempting to solve \(a(t) = 2.5a(0)\) |
| \(t = 25\) | A1 | 3 marks total (AG) |
| (ii) | \([s = 0.1t^2 + 0.002t^3 (+ C)]\) | M1 |
| \([s = 0.1 \times 625 + 0.002 \times 15625]\) | DM1 | For using limits 0 to 25 or evaluating \(s(t)\) with \(C = 0\) (which may be implied by its absence) |
| Displacement is 93.75 (accept 93.7 or 93.8) | A1 | 3 marks total |
(i) | $[a = 0.2 + 0.012t]$ | M1 | For differentiating to find $a(t)$ |
| $[0.2 + 0.012t = 2.5 \times 0.2]]$ | M1 | For attempting to solve $a(t) = 2.5a(0)$ |
| $t = 25$ | A1 | 3 marks total (AG) |
(ii) | $[s = 0.1t^2 + 0.002t^3 (+ C)]$ | M1 | For integrating to find $s(t)$ |
| $[s = 0.1 \times 625 + 0.002 \times 15625]$ | DM1 | For using limits 0 to 25 or evaluating $s(t)$ with $C = 0$ (which may be implied by its absence) |
| Displacement is 93.75 (accept 93.7 or 93.8) | A1 | 3 marks total |
---2 A light elastic string has natural length 4 m and modulus of elasticity 60 N . A particle $P$ of mass 0.6 kg is attached to the mid-point of the string. The ends of the string are attached to fixed points $A$ and $B$ which lie in the same vertical line with $A$ at a distance of 6 m above $B$. $P$ is projected vertically upwards from the point 2 m vertically above $B$. In the subsequent motion, $P$ comes to instantaneous rest at a distance of 2 m below $A$.\\
(i) Calculate the speed of projection of $P$.\\
(ii) Calculate the distance of $P$ from $A$ at an instant when $P$ has its greatest kinetic energy, and calculate this kinetic energy.
\hfill \mbox{\textit{CAIE M2 2012 Q2 [8]}}