| Exam Board | AQA |
|---|---|
| Module | Further Paper 1 (Further Paper 1) |
| Year | 2020 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Two springs/strings system equilibrium |
| Difficulty | Standard +0.8 This is a standard Further Maths SHM problem requiring students to find equilibrium position, verify SHM conditions (F ∝ -x), then apply energy conservation or SHM formulas. While it involves two strings and requires careful bookkeeping of extensions, the solution follows a well-established template taught in FM mechanics courses. The multi-step nature and algebraic manipulation place it above average difficulty but within the expected range for Further Maths. |
| 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 |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| Total extension of both strings is \(3l\) | B1 (2.2a) | |
| Expression for tension in one string in terms of \(c\) and extension | B1 (3.4) | |
| Forms two-term force equation at equilibrium | M1 (3.4) | |
| \(3 = y_{AB} + y_{BC}\); \(5cy_{AB} = cy_{BC}\); \(6y_{AB} = 3\); \(y_{AB} = \frac{1}{2}\), \(y_{BC} = \frac{5}{2}\) | A1 (1.1b) | Two correct equilibrium extensions |
| Forms equation of motion with general displacement, at least one correct extension FT their equilibrium extensions: \(5c\left(\frac{1}{2} - x\right) - c\left(\frac{5}{2} + x\right) = 3c\ddot{x}\) | M1 (3.4) | |
| \(\ddot{x} = -2x\); of form \(\ddot{x} = -\omega^2 x\), therefore SHM | A1F (1.1b) | Correct equation of motion FT equilibrium extensions |
| Simplifies equation to form \(\ddot{x} = -kx\) | M1 (1.1a) | May use \(a\), \(\frac{dv}{dt}\) or other correct symbol for acceleration |
| Correctly concludes SHM with clear reason from equation, e.g. comparison with \(\ddot{x} = -\omega^2 x\) | R1F (2.1) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Mark | Guidance |
| \(\omega = \sqrt{2}\) | B1F (1.1b) | Correct value for \(\omega\) FT final equation in (a) |
| \(A = \frac{1}{3}\) | B1 (3.1b) | Correct amplitude |
| \(v^2 = 2\left(\frac{1}{3^2} - \frac{1}{4^2}\right)\) | M1 (3.1b) | Correct complete method to find speed |
| \(v = 0.31\ \text{ms}^{-1}\) | A1F (3.2a) | Correct speed with correct units FT \(\omega\), accurate to 2 or more sf |
## Question 13(a):
| Working/Answer | Mark | Guidance |
|---|---|---|
| Total extension of both strings is $3l$ | B1 (2.2a) | |
| Expression for tension in one string in terms of $c$ and extension | B1 (3.4) | |
| Forms two-term force equation at equilibrium | M1 (3.4) | |
| $3 = y_{AB} + y_{BC}$; $5cy_{AB} = cy_{BC}$; $6y_{AB} = 3$; $y_{AB} = \frac{1}{2}$, $y_{BC} = \frac{5}{2}$ | A1 (1.1b) | Two correct equilibrium extensions |
| Forms equation of motion with general displacement, at least one correct extension FT their equilibrium extensions: $5c\left(\frac{1}{2} - x\right) - c\left(\frac{5}{2} + x\right) = 3c\ddot{x}$ | M1 (3.4) | |
| $\ddot{x} = -2x$; of form $\ddot{x} = -\omega^2 x$, therefore SHM | A1F (1.1b) | Correct equation of motion FT equilibrium extensions |
| Simplifies equation to form $\ddot{x} = -kx$ | M1 (1.1a) | May use $a$, $\frac{dv}{dt}$ or other correct symbol for acceleration |
| Correctly concludes SHM with clear reason from equation, e.g. comparison with $\ddot{x} = -\omega^2 x$ | R1F (2.1) | |
## Question 13(b):
| Working/Answer | Mark | Guidance |
|---|---|---|
| $\omega = \sqrt{2}$ | B1F (1.1b) | Correct value for $\omega$ FT final equation in (a) |
| $A = \frac{1}{3}$ | B1 (3.1b) | Correct amplitude |
| $v^2 = 2\left(\frac{1}{3^2} - \frac{1}{4^2}\right)$ | M1 (3.1b) | Correct complete method to find speed |
| $v = 0.31\ \text{ms}^{-1}$ | A1F (3.2a) | Correct speed with correct units FT $\omega$, accurate to 2 or more sf |
**Total: 12 marks**
---13 Two light elastic strings each have one end attached to a particle $B$ of mass $3 c \mathrm {~kg}$, which rests on a smooth horizontal table.
The other ends of the strings are attached to the fixed points $A$ and $C$, which are 8 metres apart.\\
$A B C$ is a horizontal line.\\
\includegraphics[max width=\textwidth, alt={}, center]{44e22a98-6424-4fb1-8a37-c965773cb7b6-26_92_910_635_566}
String $A B$ has a natural length of 4 metres and a stiffness of $5 c$ newtons per metre.\\
String $B C$ has a natural length of 1 metre and a stiffness of $c$ newtons per metre.\\
The particle is pulled a distance of $\frac { 1 } { 3 }$ metre from its equilibrium position towards $A$, and released from rest.
13
\begin{enumerate}[label=(\alph*)]
\item Show that the particle moves with simple harmonic motion.\\
13
\item Find the speed of the particle when it is at a point $P$, a distance $\frac { 1 } { 4 }$ metre from the equilibrium position. Give your answer to two significant figures.\\[0pt]
[4 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 1 2020 Q13 [12]}}