| Exam Board | AQA |
|---|---|
| Module | Further Paper 2 (Further Paper 2) |
| Session | Specimen |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Difficulty | Challenging +1.2 This is a standard SHM problem requiring recognition of the differential equation, finding ω, applying initial conditions, and using amplitude to find minimum distance. While it involves multiple steps and careful setup, it follows a well-practiced procedure for Further Maths students with no novel insights required. |
| Spec | 1.08k Separable differential equations: dy/dx = f(x)g(y)6.02i Conservation of energy: mechanical energy principle6.05f Vertical circle: motion including free fall |
| Answer | Marks | Guidance |
|---|---|---|
| 7(a) | Models the motion of the ball by | |
| forming an equation of motion | AO3.1b | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| model for displacement | AO3.1a | M1 |
| Answer | Marks | Guidance |
|---|---|---|
| constant | AO1.1a | M1 |
| Obtains correct value for constant | AO1.1b | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| minimum distance from P | AO3.2a | A1F |
| (b) | Identifies a correct limitation of the |
| Answer | Marks | Guidance |
|---|---|---|
| effect due to air | AO3.5b | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| reasoned inference. | AO2.2b | R1 |
| Total | 7 | |
| Q | Marking Instructions | AO |
Question 7:
--- 7(a) ---
7(a) | Models the motion of the ball by
forming an equation of motion | AO3.1b | M1 | d2x
m (12.5mx)2
dt2
d2x
25x
dt2
x Asin(5t)
x 5Acos(5t)
when t 0, x 0.75 so
0.755A
A0.15
Hence
x0.15sin(5t)
Max displacement = 0.15 metres
from O, whensin(5t)1 , so
minimum distance from P is 0.75
metres
Uses SHM equations to form
model for displacement | AO3.1a | M1
Uses initial condition to find the
constant | AO1.1a | M1
Obtains correct value for constant | AO1.1b | A1
Interprets ‘their’ value to find
minimum distance from P | AO3.2a | A1F
(b) | Identifies a correct limitation of the
model for example friction between
ball and the surface or damping
effect due to air | AO3.5b | B1 | It is unlikely that the surface is
perfectly smooth so friction will be
acting. The ball will be likely to
travel a smaller distance before
coming to rest and the minimum
distance of the ball from P may
actually be greater than that
calculated in part (a).
Correctly infers whether the
distance is too big or too small
based on the limitation they have
identified. Accept any well-
reasoned inference. | AO2.2b | R1
Total | 7
Q | Marking Instructions | AO | Marks | Typical SolutionA small, hollow, plastic ball, of mass $m$ kg is at rest at a point $O$ on a polished horizontal surface. The ball is attached to two identical springs. The other ends of the springs are attached to the points $P$ and $Q$ which are 1.8 metres apart on a straight line through $O$.
The ball is struck so that it moves away from $O$, towards $P$ with a speed of 0.75 m s$^{-1}$.
As the ball moves, its displacement from $O$ is $x$ metres at time $t$ seconds after the motion starts.
The force that each of the springs applies to the ball is $12.5mx$ newtons towards $O$.
The ball is to be modelled as a particle. The surface is assumed to be smooth and it is assumed that the forces applied to the ball by the springs are the only horizontal forces acting on the ball.
\begin{enumerate}[label=(\alph*)]
\item Find the minimum distance of the ball from $P$, in the subsequent motion.
[5 marks]
\end{enumerate}
\hfill \mbox{\textit{AQA Further Paper 2 Q7 [5]}}