| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2004 |
| Session | June |
| Marks | 16 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Collision/impulse during SHM |
| Difficulty | Challenging +1.3 This is a substantial multi-part SHM question requiring derivation of period, speed calculations, time-to-position analysis, and collision with subsequent SHM analysis. While it involves several techniques (SHM equations, energy conservation, impulse-momentum, coalescence), each part follows standard M3 procedures without requiring novel insight. The collision aspect adds complexity beyond basic SHM but remains a textbook application. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x4.10g Damped oscillations: model and interpret6.02h Elastic PE: 1/2 k x^26.02j Conservation with elastics: springs and strings6.03c Momentum in 2D: vector form6.03f Impulse-momentum: relation |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \((-)\dfrac{21.6x}{2} = 0.3\ddot{x}\) | M1 A1 | |
| \(-36x = \ddot{x}\) | M1 | |
| S.H.M., period \(= \dfrac{2\pi}{\sqrt{36}} = \dfrac{\pi}{3}\) | A1 c.s.o | (4 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| At \(A\): \(v = a\omega = 1.5\times 6 = 9\ \text{ms}^{-1}\) | M1 A1 | (2 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(x = a\cos\omega t\) | ||
| \(0.75 = 1.5\cos 6t\) | M1 | |
| \(\dfrac{\pi}{3} = 6t \Rightarrow t = \dfrac{\pi}{18}\) (no decimals) | M1 A1 | (3 marks) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \((-)\dfrac{21.6x}{2} = 0.5\ddot{x}\) | M1 A1 | |
| \(-21.6x = \ddot{x} \Rightarrow\) S.H.M., \(\omega = \sqrt{21.6}\) | A1 | |
| At collision CLM: \(0.3\times 9 = 0.5v \Rightarrow v = 5.4\) | M1 A1 ft | |
| \(a\times\sqrt{21.6} = 5.4\) | M1 | |
| \(a = 1.16\ \text{m}\ (3\text{SF})\) | A1 | (7 marks); (16 marks) |
## Question 7:
### Part (a):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $(-)\dfrac{21.6x}{2} = 0.3\ddot{x}$ | M1 A1 | |
| $-36x = \ddot{x}$ | M1 | |
| S.H.M., period $= \dfrac{2\pi}{\sqrt{36}} = \dfrac{\pi}{3}$ | A1 c.s.o | (4 marks) |
### Part (b):
| Working/Answer | Marks | Guidance |
|---|---|---|
| At $A$: $v = a\omega = 1.5\times 6 = 9\ \text{ms}^{-1}$ | M1 A1 | (2 marks) |
### Part (c):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $x = a\cos\omega t$ | | |
| $0.75 = 1.5\cos 6t$ | M1 | |
| $\dfrac{\pi}{3} = 6t \Rightarrow t = \dfrac{\pi}{18}$ (no decimals) | M1 A1 | (3 marks) |
### Part (d):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $(-)\dfrac{21.6x}{2} = 0.5\ddot{x}$ | M1 A1 | |
| $-21.6x = \ddot{x} \Rightarrow$ S.H.M., $\omega = \sqrt{21.6}$ | A1 | |
| At collision CLM: $0.3\times 9 = 0.5v \Rightarrow v = 5.4$ | M1 A1 ft | |
| $a\times\sqrt{21.6} = 5.4$ | M1 | |
| $a = 1.16\ \text{m}\ (3\text{SF})$ | A1 | (7 marks); **(16 marks)** |7. A particle $P$ of mass 0.3 kg is attached to one end of a light elastic spring. The other end of the spring is attached to a fixed point $O$ on a smooth horizontal table. The spring has natural length 2 m and modulus of elasticity 21.6 N . The particle $P$ is placed on the table at the point $A$, where $O A = 2 \mathrm {~m}$. The particle $P$ is now pulled away from $O$ to the point $B$, where $O A B$ is a straight line with $O B = 3.5 \mathrm {~m}$. It is then released from rest.
\begin{enumerate}[label=(\alph*)]
\item Prove that $P$ moves with simple harmonic motion of period $\frac { \pi } { 3 } \mathrm {~s}$.
\item Find the speed of $P$ when it reaches $A$.
The point $C$ is the mid-point of $A B$.
\item Find, in terms of $\pi$, the time taken for $P$ to reach $C$ for the first time.
Later in the motion, $P$ collides with a particle $Q$ of mass 0.2 kg which is at rest at $A$.\\
After the impact, $P$ and $Q$ coalesce to form a single particle $R$.
\item Show that $R$ also moves with simple harmonic motion and find the amplitude of this motion.
END
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2004 Q7 [16]}}