| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2017 |
| Session | June |
| Marks | 9 |
| Topic | Hooke's law and elastic energy |
| Type | Particle at midpoint of string between two horizontal fixed points: horizontal surface motion |
| Difficulty | Challenging +1.2 This is a multi-part elastic strings problem requiring Hooke's law application, SHM recognition, and energy methods. While it involves several steps and careful bookkeeping of extensions/compressions at different positions, the techniques are standard for Further Maths mechanics: setting up equations of motion, identifying SHM parameters, and using energy conservation. The geometric setup requires attention but no novel insight beyond applying familiar principles systematically. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x6.02g Hooke's law: T = k*x or T = lambda*x/l6.02i Conservation of energy: mechanical energy principle |
Total \(4.52\) A1 Allow \(\dfrac{\sqrt{10}(4+\pi)}{5}\) oe
Total: 7 marks
**Part (i):**
$0.05\ddot{x} = -\dfrac{0.6}{1.2}x$ or $\ddot{x} = -10x$ **M1** Use $ma = -\lambda x/l$. If two non-zero tensions, M0.
$\ddot{x} = -10x$ **A1** Condone $a = -10x$
**Part (ii):**
$v = \omega\sqrt{(a^2 - x^2)} = \omega a$ **M1** Use $v = \omega\sqrt{(a^2 - x^2)}$
$= 0.316$ **A1** $\omega = \sqrt{10}$ and $a = 0.1 \Rightarrow v = \sqrt{10}/10$ or awrt 0.316
**Part (iii):**
One complete period: $2\pi/\omega$ **M1** Use $2\pi/\omega$ somewhere
$= 1.987$ s **A1** Correct time for SHM parts. Allow $2\pi/\sqrt{10}$
$2 \times 0.4$ m at constant speed $v$: **M1** Or one way: 0.4 m at constant speed $v$
$2.53$ s or $0.8\sqrt{10}$ o.e. **A1** 1.26s
Total $4.52$ **A1** Allow $\dfrac{\sqrt{10}(4+\pi)}{5}$ oe
Total: **7 marks**14\\
\includegraphics[max width=\textwidth, alt={}, center]{22640c3b-792f-4003-a4f8-78220efd73b0-5_86_1589_1297_278}
A particle of mass 0.05 kg is attached to two identical light elastic strings, each of natural length 1.2 m and modulus of elasticity 0.6 N . The other ends of the strings are attached to points $A$ and $E$ on a smooth horizontal table. The distance $A E$ is 2 m and points $B , C$ and $D$ lie between $A$ and $E$ so that $A B = 0.7 \mathrm {~m} , B C = 0.1 \mathrm {~m} , C D = 0.4 \mathrm {~m}$ and $D E = 0.8 \mathrm {~m}$ (see diagram). Initially the particle is held at $B$ and it is then released. In the subsequent motion the displacement of the particle from $C$, in the direction of $A$, is denoted by $x \mathrm {~m}$.\\
(i) Find the equation of motion for the particle when it is between $B$ and $C$.\\
(ii) Find the velocity of the particle when it is at $C$.\\
(iii) Find the total time that elapses before the particle first returns to $B$.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2017 Q14 [9]}}