| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Session | Specimen |
| Marks | 5 |
| Topic | Simple Harmonic Motion |
| Type | Prove SHM and find period: vertical spring/string (single attachment) |
| Difficulty | Standard +0.8 This is a standard SHM spring problem requiring proof of SHM condition, finding mass from period, and relating acceleration to extension. While it involves multiple parts and requires careful application of Hooke's law and Newton's second law, the techniques are well-established for Further Maths students. The calculations are straightforward once the setup is understood, making it moderately above average difficulty but not requiring novel insight. |
| Spec | 6.06a Variable force: dv/dt or v*dv/dx methods |
**Question 3**
**(i)** Let $T$ be the tension in the spring and $e$ be the extension in the equilibrium position.
$T = mg$ and by Hooke's law $T = \dfrac{6.4e}{0.4} = 16e \Rightarrow 16e = mg \Rightarrow e = \frac{1}{16}mg$ B1
Let $T'$ be the tension in the spring when extended $x$ below equilibrium:
$T' = \dfrac{6.4(e+x)}{0.4}$ and applying Newton II:
$$mg - \frac{6.4(e+x)}{0.4} = m\ddot{x} \Rightarrow -16x = m\ddot{x} \Rightarrow \ddot{x} = -\frac{16}{m}x \Rightarrow \text{SHM}$$
M1A1A1 **[4 marks]**
**(ii)(a)** Period is $2\pi\sqrt{\dfrac{m}{16}} = 1.12$ M1A1
$$\Rightarrow m = \frac{4 \times 1.12^2}{\pi^2} = 0.508$$ A1 **[3 marks]**
**(b)** $\ddot{x} = 2 \Rightarrow -16x = 2 \times 0.508 \Rightarrow x = -0.0635...$ B1
$e = \dfrac{10 \times 0.508}{16} = 0.3177...$ B1
extension $= 0.3177 - 0.0635 = 0.254$ m B1 **[3 marks]**
(Or, using Newton II: $mg - 16y = 2m$, where $y$ is the required extension)3 A light spring, of natural length 0.4 m and modulus of elasticity 6.4 N , has one end $A$ attached to the ceiling of a room. A particle of mass $m \mathrm {~kg}$ is attached to the free end of the spring and hangs in equilibrium. The particle is displaced vertically downwards and released from rest. In the subsequent motion the particle does not reach the ceiling and air resistance may be neglected.\\
(i) Show that the particle oscillates in simple harmonic motion.\\
(ii) Given that the period of the motion is 1.12 s , find
\begin{enumerate}[label=(\alph*)]
\item the value of $m$, correct to 3 significant figures,
\item the extension of the spring when the particle has a downwards acceleration of $2 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.
\end{enumerate}
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 Q3 [5]}}