| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 15 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Prove SHM and find period: vertical spring/string (single attachment) |
| Difficulty | Standard +0.8 This is a substantial M3 question requiring proof of SHM from first principles, derivation of period formula, and algebraic manipulation involving two equilibrium positions. While the SHM setup is standard for M3, the multi-part structure with 15 marks total and the final 'show that' requiring careful algebraic work with two different systems elevates it above average difficulty. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x6.02h Elastic PE: 1/2 k x^2 |
| Answer | Marks |
|---|---|
| (a)(i) At displ. \(x\): \(T - mg = -mx\) | M1 A1 B1 B1 |
| \(T = \frac{\lambda}{l}(e+x)\) | |
| \(\frac{\lambda}{l}(e+x) - mg = -mx\) | |
| \(mx = -\frac{\lambda}{l}x\) Hence SHM about C | M1 A1 |
| (ii) Period \(= 2\pi\sqrt{\frac{lm}{\lambda}}\) | B1; B1 |
| (iii) String must not go slack | B1; B1 |
| (b)(i) \(m\) becomes \(m + M\), so \(T_1 = 2\pi\sqrt{\frac{(m+M)l}{\lambda}}\) | M1 A1 |
| (ii) \(T_1^2 - T^2 = \frac{4\pi^2}{λ}[l(m+M) - lm] = \frac{4\pi^2 lM}{\lambda}\) | M1 A1 |
| At \(D\): \((m+M)g = \frac{\lambda}{l}(e+d)\), so \(\lambda d = Mg\) | |
| \(T_1^2 - T^2 = \frac{4\pi^2 \lambda d}{g}\) | M1 A1 A1 |
**(a)(i)** At displ. $x$: $T - mg = -mx$ | M1 A1 B1 B1 |
$T = \frac{\lambda}{l}(e+x)$ | |
$\frac{\lambda}{l}(e+x) - mg = -mx$ | |
$mx = -\frac{\lambda}{l}x$ Hence SHM about C | M1 A1 |
**(ii)** Period $= 2\pi\sqrt{\frac{lm}{\lambda}}$ | B1; B1 |
**(iii)** String must not go slack | B1; B1 |
**(b)(i)** $m$ becomes $m + M$, so $T_1 = 2\pi\sqrt{\frac{(m+M)l}{\lambda}}$ | M1 A1 |
**(ii)** $T_1^2 - T^2 = \frac{4\pi^2}{λ}[l(m+M) - lm] = \frac{4\pi^2 lM}{\lambda}$ | M1 A1 |
At $D$: $(m+M)g = \frac{\lambda}{l}(e+d)$, so $\lambda d = Mg$ | |
$T_1^2 - T^2 = \frac{4\pi^2 \lambda d}{g}$ | M1 A1 A1 |
**Total: 15 marks**A particle of mass $m$ kg is attached to one end of an elastic string of natural length $l$ m and modulus of elasticity $\lambda$ N. The other end of the string is attached to a fixed point $O$. The particle hangs in equilibrium at a point $C$.
\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Prove that if the particle is slightly displaced in a vertical direction, it will perform simple harmonic motion about $C$. [6 marks]
\item Find the period, $T$ s, of the motion in terms of $l$, $m$ and $\lambda$. [1 mark]
\item Explain the significance of the term 'slightly' as used in (i) above. [1 mark]
\end{enumerate}
\end{enumerate}
When an additional mass $M$ is attached to the particle, it is found that the system oscillates about a point $D$, at a distance $d$ below $C$, with period $T_1$ s.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item \begin{enumerate}[label=(\roman*)]
\item Write down an expression for $T_1$ in terms of $l$, $m$, $M$ and $\lambda$. [2 marks]
\item Hence show that $T_1^2 - T^2 = \frac{4\pi^2 d}{g}$. [5 marks]
\end{enumerate}
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q7 [15]}}