| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Maximum acceleration in SHM |
| Difficulty | Standard +0.3 This is a standard M3 SHM question with springs. Part (a) is routine equilibrium using Hooke's law. Parts (b-d) follow textbook procedures: showing F ∝ -x for SHM, finding period from ω = √(k/m), and calculating maximum acceleration from ω²a. The two-spring setup adds minor complexity but the method is entirely standard for M3, making it slightly easier than average overall. |
| 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 | Guidance |
|---|---|---|
| (a) \(24g = 2T = 2\frac{2}{3}(0.3)\) | M1 A1 | |
| \(\frac{\lambda}{l} = \frac{24g \times 0.8}{2 \times 0.3} = 392\) | ||
| \(\lambda = 392l\) | ||
| (b) At dist. \(x\) from \(A\), \(mg - 2\lambda(0.3 + x) = mx\) | M1 A1 A1 | |
| \(x = -\frac{2\lambda}{m} = -\frac{98}{3}x\) | A1 A1 | |
| Hence S.H.M. with centre \(A\) | ||
| (c) \(\omega^2 = \frac{98}{3} = 32.7\) | M1 A1 A1 | |
| Freq. \(= \frac{2\pi}{\omega} = \frac{\sqrt{32.7}}{2\pi} = 0.91 \text{ osc.s}^{-1}\) | ||
| (d) Max. acc. \(= \omega^2(0.2) = 6.54 \text{ ms}^{-2}\) | M1 | Total: 12 marks |
**(a)** $24g = 2T = 2\frac{2}{3}(0.3)$ | M1 A1 |
$\frac{\lambda}{l} = \frac{24g \times 0.8}{2 \times 0.3} = 392$ | |
$\lambda = 392l$ | |
**(b)** At dist. $x$ from $A$, $mg - 2\lambda(0.3 + x) = mx$ | M1 A1 A1 |
$x = -\frac{2\lambda}{m} = -\frac{98}{3}x$ | A1 A1 |
Hence S.H.M. with centre $A$ | |
**(c)** $\omega^2 = \frac{98}{3} = 32.7$ | M1 A1 A1 |
Freq. $= \frac{2\pi}{\omega} = \frac{\sqrt{32.7}}{2\pi} = 0.91 \text{ osc.s}^{-1}$ | |
**(d)** Max. acc. $= \omega^2(0.2) = 6.54 \text{ ms}^{-2}$ | M1 | **Total: 12 marks**The figure shows a swing consisting of two identical vertical light springs attached symmetrically to a light horizontal cross-bar and supported from a strong fixed horizontal beam. When a mass of 24 kg is placed at the mid-point of the cross-bar, both springs extend by 30 cm to the position $A$, as shown.
\includegraphics{figure_6}
Each spring has natural length $l$ m and modulus of elasticity $\lambda$ N.
\begin{enumerate}[label=(\alph*)]
\item Show that $\lambda = 392l$. [2 marks]
\end{enumerate}
The 24 kg mass is left on the bar and the bar is then displaced downwards by a further 20 cm.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Prove that the system comprising the bar and the mass now performs simple harmonic motion with the centre of oscillation at the level $A$. [5 marks]
\item Calculate the number of oscillations made per second in this motion. [3 marks]
\item Find the maximum acceleration which the mass experiences during the motion. [2 marks]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q6 [12]}}