| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Maximum speed in SHM |
| Difficulty | Standard +0.8 This M3 question requires multiple techniques: Hooke's law, proving SHM from first principles, energy methods, and solving SHM equations with specific positions. While the individual steps are standard for M3, the multi-part structure requiring proof of SHM, period calculation, and time-to-position calculations makes it moderately challenging, though still within typical M3 scope. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle |
| Answer | Marks |
|---|---|
| (a) \(mg = \frac{4}{0.8} \times 0.7 = 0.5 \times 9.8\) | M1 A1 A1 |
| \(\lambda = 4.9 \times \frac{0.8}{0.7} = 5.6\) N | |
| (b) \((0.5 \times 9.8) - \frac{4.9}{0.8}(0.7 + x) = 0.5x\) | M1 A1 |
| \(4.9 - 4.9 - 7x = 0.5x\) | |
| \(x = -14x\), of form \(x = n^2x\) with \(n^2 = 14\), so simple harmonic | A1 A1 |
| (c) Period \(= \frac{2\pi}{\sqrt{14}} = 1.68\) s | A1 |
| Maximum speed \(= a n = 0.5\sqrt{14} = 1.87\) ms\(^{-1}\) | B1 M1 A1 |
| (d) \(x = 0.5\cos nt\) | B1 M1 |
| \(0.25 = 0.5\cos nt\) | |
| \(\cos nt = 0.5\) | |
| \(nt = \frac{\pi}{3}\) | A1 A1 |
| \(t = 0.28\) s |
**(a)** $mg = \frac{4}{0.8} \times 0.7 = 0.5 \times 9.8$ | M1 A1 A1 |
$\lambda = 4.9 \times \frac{0.8}{0.7} = 5.6$ N | |
**(b)** $(0.5 \times 9.8) - \frac{4.9}{0.8}(0.7 + x) = 0.5x$ | M1 A1 |
$4.9 - 4.9 - 7x = 0.5x$ | |
$x = -14x$, of form $x = n^2x$ with $n^2 = 14$, so simple harmonic | A1 A1 |
**(c)** Period $= \frac{2\pi}{\sqrt{14}} = 1.68$ s | A1 |
Maximum speed $= a n = 0.5\sqrt{14} = 1.87$ ms$^{-1}$ | B1 M1 A1 |
**(d)** $x = 0.5\cos nt$ | B1 M1 |
$0.25 = 0.5\cos nt$ | |
$\cos nt = 0.5$ | |
$nt = \frac{\pi}{3}$ | A1 A1 |
$t = 0.28$ s | |
**Total: 15 marks**A light elastic string, of natural length 0·8 m, has one end fastened to a fixed point $O$. The other end of the string is attached to a particle $P$ of mass 0·5 kg. When $P$ hangs in equilibrium, the length of the string is 1·5 m.
\begin{enumerate}[label=(\alph*)]
\item Calculate the modulus of elasticity of the string. [3 marks]
\end{enumerate}
$P$ is displaced to a point 0·5 m vertically below its equilibrium position and released from rest.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Show that the subsequent motion of $P$ is simple harmonic, with period 1·68 s. [5 marks]
\item Calculate the maximum speed of $P$ during its motion. [3 marks]
\item Show that the time taken for $P$ to first reach a distance 0·25 m from the point of release is 0·28 s, to 2 significant figures. [4 marks]
\end{enumerate]
\hfill \mbox{\textit{Edexcel M3 Q6 [15]}}