| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Simple Harmonic Motion |
| Type | Two springs/strings system equilibrium |
| Difficulty | Standard +0.8 This is a two-spring SHM problem requiring equilibrium analysis with Hooke's law, then proving SHM by finding the restoring force equation and deriving the period formula. It involves multiple steps: setting up force balance, algebraic manipulation, displacement from equilibrium, and applying SHM theory. More demanding than standard single-spring SHM but follows established M3 techniques without requiring exceptional insight. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x6.02h Elastic PE: 1/2 k x^2 |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(T_1 = \frac{s}{v} = \frac{\lambda(y-3a)}{3a}\) | M1 A1 | |
| \(T_2 = \frac{2\lambda(5a-y)}{2a}\) | A1 | |
| eqn. \(\therefore T_1 = T_2\), \(\frac{\lambda(y-3a)}{3a} = \frac{2\lambda(5a-y)}{2a}\) | M1 | |
| giving \(y - 3a = 3(5a-y)\) \(\therefore y = \frac{9}{2}a\) | A1 | |
| (b) \(m\ddot{x} = T_2 - T_1 = \frac{2\lambda[(a-x) - \lambda[\frac{1}{3}(a-3x) - (\frac{3}{4}a+x)]}\) | M2 A2 | |
| giving \(\ddot{x} = -\frac{4k}{3m}x\) \(\therefore\) SHM with \(\omega^2 = \frac{4k}{3m}\), \(\omega = 2\sqrt{\frac{k}{3m}}\) | M1 A2 | |
| period \(= \frac{2\pi}{\omega} = \pi\sqrt{\frac{3m}{k}}\) | M1 A1 | (14) |
(a) $T_1 = \frac{s}{v} = \frac{\lambda(y-3a)}{3a}$ | M1 A1 |
$T_2 = \frac{2\lambda(5a-y)}{2a}$ | A1 |
eqn. $\therefore T_1 = T_2$, $\frac{\lambda(y-3a)}{3a} = \frac{2\lambda(5a-y)}{2a}$ | M1 |
giving $y - 3a = 3(5a-y)$ $\therefore y = \frac{9}{2}a$ | A1 |
(b) $m\ddot{x} = T_2 - T_1 = \frac{2\lambda[(a-x) - \lambda[\frac{1}{3}(a-3x) - (\frac{3}{4}a+x)]}$ | M2 A2 |
giving $\ddot{x} = -\frac{4k}{3m}x$ $\therefore$ SHM with $\omega^2 = \frac{4k}{3m}$, $\omega = 2\sqrt{\frac{k}{3m}}$ | M1 A2 |
period $= \frac{2\pi}{\omega} = \pi\sqrt{\frac{3m}{k}}$ | M1 A1 | (14)6.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{cab238c9-f4e2-4637-a079-f74779548f49-4_206_977_201_470}
\captionsetup{labelformat=empty}
\caption{Fig. 2}
\end{center}
\end{figure}
Figure 2 shows a particle $P$ of mass $m$ which lies on a smooth horizontal table. It is attached to a point $A$ on the table by a light elastic spring of natural length $3 a$ and modulus of elasticity $\lambda$, and to a point $B$ on the table by a light elastic spring of natural length $2 a$ and modulus of elasticity $2 \lambda$. The distance between the points $A$ and $B$ is $7 a$.
\begin{enumerate}[label=(\alph*)]
\item Show that in equilibrium $A P = \frac { 9 } { 2 } a$.
The particle is released from rest at a point $Q$ where $Q$ lies on the line $A B$ and $A Q = 5 a$.
\item Prove that the subsequent motion of the particle is simple harmonic with a period of $\pi \sqrt { \frac { 3 m a } { \lambda } }$.\\
(9 marks)
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 Q6 [14]}}