| Exam Board | CAIE |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2011 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hooke's law and elastic energy |
| Type | Vertical elastic string: released from rest at natural length or above (string initially slack) |
| Difficulty | Standard +0.3 This is a standard Hooke's law problem requiring equilibrium condition (T=mg), then energy conservation for parts (ii) and (iii). The steps are routine for M2: find extension using λx/l = mg, apply PE + EPE = KE, and use energy conservation again for maximum extension. All techniques are textbook applications with no novel insight required, making it slightly easier than average. |
| Spec | 6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle6.02j Conservation with elastics: springs and strings |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(0.25g = 20e/0.4\) | M1 | Uses \(T = 2\pi\sqrt{L}\) |
| \(OP ( = 0.05 + 0.4) = 0.45\) m | A1 | [2] |
| (ii) \(20 \times 0.05^2/(2 \times 0.4) + 0.25v^2/2 = 0.25g \times 0.45\) | M1 | |
| \(v = 2.92\) m s\(^{-1}\) | A1 | [3] |
| (iii) \(20(d - 0.4)^2/(2 \times 0.4) = 0.25gd\) | M1 | Hence \(d^2 - (0.8 + 0.1)d + 0.16 = 0\) |
| \(d = [0.9 \pm \sqrt{(0.9^2 - 4 \times 0.16)}]/2\) | M1 | Solves a 3 term quadratic equation |
| \(d = 0.656\) | A1 | [3] Ignore \(d = 0.244\) if seen |
**(i)** $0.25g = 20e/0.4$ | M1 | Uses $T = 2\pi\sqrt{L}$
$OP ( = 0.05 + 0.4) = 0.45$ m | A1 | [2]
**(ii)** $20 \times 0.05^2/(2 \times 0.4) + 0.25v^2/2 = 0.25g \times 0.45$ | M1 |
$v = 2.92$ m s$^{-1}$ | A1 | [3]
**(iii)** $20(d - 0.4)^2/(2 \times 0.4) = 0.25gd$ | M1 | Hence $d^2 - (0.8 + 0.1)d + 0.16 = 0$
$d = [0.9 \pm \sqrt{(0.9^2 - 4 \times 0.16)}]/2$ | M1 | Solves a 3 term quadratic equation
$d = 0.656$ | A1 | [3] Ignore $d = 0.244$ if seenOne end of a light elastic string of natural length 0.4 m and modulus of elasticity 20 N is attached to a fixed point $O$. The other end of the string is attached to a particle $P$ of mass 0.25 kg. $P$ hangs in equilibrium below $O$.
\begin{enumerate}[label=(\roman*)]
\item Calculate the distance $OP$. [2]
\end{enumerate}
The particle $P$ is raised, and is released from rest at $O$.
\begin{enumerate}[label=(\roman*)]
\setcounter{enumi}{1}
\item Calculate the speed of $P$ when it passes through the equilibrium position. [3]
\item Calculate the greatest value of the distance $OP$ in the subsequent motion. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE M2 2011 Q3 [8]}}