| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2017 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Collision/impulse during SHM |
| Difficulty | Challenging +1.2 Part (a) is routine SHM calculation using standard formulas (ω from frequency, max acceleration = ω²a). Part (b) requires understanding that maximum speed occurs at the centre, then applying impulse-momentum with the new amplitude condition—this involves connecting SHM velocity formula with impulse, requiring more insight than typical textbook exercises but still within standard M3 scope. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x6.03e Impulse: by a force6.03f Impulse-momentum: relation |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(\frac{2\pi}{\omega} = 0.25\), \(\omega = 8\pi\) | M1, A1 | M1: Use \(\frac{2\pi}{\omega} = \frac{1}{4}\) or 4 to obtain value for \(\omega\); A1: Correct value exact or decimal |
| \(\text{max accel} = a\omega^2 = 0.25 \times 64\pi^2 = 16\pi^2\) \((158)\) m s\(^{-2}\) | dM1A1 (4) | dM1: Use \(a\omega^2\) with their \(\omega\) and \(a = 0.25\); A1: Correct max magnitude, exact or 158 or better |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Guidance |
| \(\omega_1 = 8\pi\), \(a_1 = 0.125\) accept \(\frac{0.25}{2}\) | B1ft | New \(\omega\) and amp, follow through their original \(\omega\) and amp |
| Max speed for new motion \(= 8\pi \times 0.125\) m s\(^{-1}\) \((= \pi)\) | M1(either) | One speed needed. Award M1 for either, using their \(\omega\) and amp. May obtain \(v\) or \(v^2\) |
| Max speed for original motion \(= 8\pi \times 0.25\) m s\(^{-1}\) \((= 2\pi)\) | A1ft(both) | Award A1 if both speeds correct, follow through their \(\omega\) and amp |
| \( | I | = 0.5(8\pi \times 0.25 + 8\pi \times 0.125) = 1.5\pi\) Ns \((= 4.7123\ldots\) accept 4.7 or better\()\) |
# Question 3(a):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $\frac{2\pi}{\omega} = 0.25$, $\omega = 8\pi$ | M1, A1 | M1: Use $\frac{2\pi}{\omega} = \frac{1}{4}$ or 4 to obtain value for $\omega$; A1: Correct value exact or decimal |
| $\text{max accel} = a\omega^2 = 0.25 \times 64\pi^2 = 16\pi^2$ $(158)$ m s$^{-2}$ | dM1A1 (4) | dM1: Use $a\omega^2$ with their $\omega$ and $a = 0.25$; A1: Correct max magnitude, exact or 158 or better |
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# Question 3(b):
| Working/Answer | Marks | Guidance |
|---|---|---|
| $\omega_1 = 8\pi$, $a_1 = 0.125$ accept $\frac{0.25}{2}$ | B1ft | New $\omega$ and amp, follow through their original $\omega$ and amp |
| Max speed for new motion $= 8\pi \times 0.125$ m s$^{-1}$ $(= \pi)$ | M1(either) | One speed needed. Award M1 for either, using their $\omega$ and amp. May obtain $v$ or $v^2$ |
| Max speed for original motion $= 8\pi \times 0.25$ m s$^{-1}$ $(= 2\pi)$ | A1ft(both) | Award A1 if both speeds correct, follow through their $\omega$ and amp |
| $|I| = 0.5(8\pi \times 0.25 + 8\pi \times 0.125) = 1.5\pi$ Ns $(= 4.7123\ldots$ accept 4.7 or better$)$ | dM1A1 (5) | dM1: Use impulse = change of momentum with their speeds (neither = 0). Allow if momenta subtracted as long as no incorrect formula seen; A1: Correct magnitude, exact or min 2 sf, must be positive |
---3. A particle $P$ of mass 0.5 kg moves in a straight line with simple harmonic motion, completing 4 oscillations per second. The particle comes to instantaneous rest at the fixed points $A$ and $B$, where $A B = 0.5 \mathrm {~m}$.
\begin{enumerate}[label=(\alph*)]
\item Find the maximum magnitude of the acceleration of $P$.
When $P$ is moving at its maximum speed it receives an impulse. The direction of this impulse is opposite to the direction in which $P$ is moving when it receives the impulse. The impulse causes $P$ to reverse its direction of motion but $P$ continues to move with simple harmonic motion. The centre and period of this new simple harmonic motion are the same as the centre and period of the original simple harmonic motion. The amplitude is now half the original amplitude.
\item Find the magnitude of the impulse.\\
\section*{II}
" ; O L
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2017 Q3 [9]}}