| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2016 |
| Session | June |
| Marks | 14 |
| Topic | Simple Harmonic Motion |
| Type | Complete motion cycle with slack phase |
| Difficulty | Challenging +1.2 This is a standard SHM question with elastic strings requiring multiple techniques (equilibrium, energy conservation, SHM equation, period calculation) across a slack-taut transition. While it involves several parts and careful handling of the slack phase, the methods are well-established for Further Maths students and follow predictable patterns without requiring novel insight. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle |
(i) $T - mg = 0 \Rightarrow \dfrac{3}{0.5}\times e = 0.3g$
$\Rightarrow e = \mathbf{0.5}$
- M1: Use N2 for equilibrium
- A1: Equilibrium extension 0.5, a.r.t. 0.500
**[2]**
(ii) GPE lost $=$ EPE gained:
$3(0.5+x) = \tfrac{1}{2}\dfrac{3}{0.5}x^2\ [= 3x^2]$
Solve to get $x = \dfrac{1+\sqrt{3}}{2} = \mathbf{1.37}$
- M1*: Use cons of energy, need variable on both sides, e.g. $3(1+h) = 3(h+\tfrac{1}{2})^2$ or $3h = 3(h-\tfrac{1}{2})^2$
- depM1: Obtain and solve quadratic equation
- A1: Correct equation, add/subtract 0.5 etc if necessary
- A1: Solve to get a.r.t. 1.37 *only*, allow surds
**[4]**
(iii) $0.3\ddot{x} = 0.3g - \dfrac{3}{0.5}(x+e)$
$\ddot{x} = -20x$
- M1: Use N2 including extension
- A1: All correct, check signs
- B1: Obtain correct value of $\omega^2$, no wrong working
**[3]**
(iv) $x = \dfrac{\sqrt{3}}{2}\cos\sqrt{20}\,t$
$x = 0.5$ at $\omega t = 0.9553$ or $5.3279$
$t = 0.2136$ or $1.1913$
Difference $= \mathbf{0.978}$
- M1: Use $x = a\cos\omega t$ or $a\sin\omega t$, allow $a=1$
- A1ft: $a$ correct ft ($=$ their (ii) $-$ their (i)), $\omega$ from (iii)
- M1: Equate to $(\pm)e$ and use correct trig method, e.g. $\dfrac{2}{\sqrt{20}}\cos^{-1}\!\left(-\dfrac{1}{\sqrt{3}}\right)$ or $\dfrac{1}{\sqrt{20}}\!\left(\pi + 2\sin^{-1}\!\left(\dfrac{1}{\sqrt{3}}\right)\right)$
- A1: One correct value of $t$
- A1: Correct final answer $[0.9777]$
**[5]**14 One end of a light elastic string of natural length 0.5 m and modulus of elasticity 3 N is attached to a ceiling at a point $P$. A particle of mass 0.3 kg is attached to the other end of the string.\\
(i) Find the extension of the string when the particle hangs vertically in equilibrium.
The particle is released from rest at $P$ so that it falls vertically. Find\\
(ii) the maximum extension of the string,\\
(iii) the equation of motion for the particle when the string is stretched, in terms of the displacement $x \mathrm {~m}$ below the equilibrium position,\\
(iv) the time between the string first becoming stretched and next becoming unstretched again.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2016 Q14 [14]}}