| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2015 |
| Session | June |
| Marks | 4 |
| Topic | Simple Harmonic Motion |
| Type | Time to travel between positions |
| Difficulty | Standard +0.3 This is a standard SHM question requiring identification of equilibrium position, verification of SHM equation (F = -kx), calculation of period using ω = √(λ/ml), and finding time between positions using the standard SHM formula x = a cos(ωt). All steps are routine applications of well-practiced techniques with no novel insight required, making it slightly easier than average. |
| 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 |
**Question 8(i) and 8(ii)**
**(i)**
Hooke's law: $T = \dfrac{8x}{0.4} = 20x$ — B1 (Can be specific $x$)
Newton II: $\dfrac{1}{5}\ddot{x} = -20x$ — M1 (Needs general $x$, $-$ sign)
$\Rightarrow \ddot{x} = -100x$ which is SHM — A1 (A.e. exact f.)
Period $= \dfrac{2\pi}{10} = \dfrac{1}{5}\pi$ seconds — A1
**[4]**
**(ii)**
$-0.1 = 0.2\cos 10t$ — M1A1 (Or: Obtain $\dfrac{1}{60}\pi$)
$\Rightarrow 10t = \dfrac{2\pi}{3} \Rightarrow t = \dfrac{1}{15}\pi$ seconds — M1A1 (Add quarter period to get $\dfrac{1}{15}\pi$)
**[4]**8\\
\includegraphics[max width=\textwidth, alt={}, center]{86cc07e7-ea69-4480-96c8-82b818445199-4_182_803_264_671}
A light spring of modulus of elasticity 8 N and natural length 0.4 m has one end fixed to a smooth horizontal table at a fixed point $L$. A particle of mass 0.2 kg is attached to the other end of the spring and pulled out horizontally to a point $M$ on the table, so that the spring is extended by 0.2 m . The particle is then released from rest. The mid-point of $L M$ is $N$ and the point $O$ is on $L M$ such that $L O = 0.4 \mathrm {~m}$ (see diagram).\\
(i) Show that the particle moves in simple harmonic motion with centre $O$ and state the exact period of its motion.\\
(ii) Find the exact time taken for the particle to move directly from $M$ to $N$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2015 Q8 [4]}}