| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2013 |
| Session | June |
| 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 multi-part SHM question requiring equilibrium analysis with two springs (Hooke's law with extensions in opposite directions), proving SHM by showing restoring force proportional to displacement, and finding time using SHM equations with given maximum speed. While systematic, it requires careful handling of multiple springs, sign conventions, and integration of several M3 concepts beyond routine single-spring problems. |
| Spec | 4.10f Simple harmonic motion: x'' = -omega^2 x6.02g Hooke's law: T = k*x or T = lambda*x/l6.02h Elastic PE: 1/2 k x^26.02i Conservation of energy: mechanical energy principle6.02j Conservation with elastics: springs and strings |
(a)
M1 for using Hooke's Law for each string, equating the two tensions and solving to find the extension in either string. The extensions should add to 1.5. The formula for Hooke's law must be correct, either shown explicitly in its general form or implicitly by the substitution.
A1 for a correct equation.
A1 for $e = 0.9$.
A1cso for 2.4 m.
**Alternative:** Find the ratio of the two extensions and divide 1.5 m in that ratio.
M1 complete method, A1 correct ratio, A1 extension in AO, A1 2.4 m.
(b)
M1 for an equation of motion for P. There must be a difference of two tensions. The acceleration can be $a$ or $\ddot{x}$ here and $x$ should be measured from the equilibrium position (O) unless a suitable substitution is made later. Mass can6.
\begin{figure}[h]
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\includegraphics[alt={},max width=\textwidth]{f6ab162c-8473-4464-ad62-87a359d85ab3-10_191_972_276_484}
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\caption{Figure 5}
\end{center}
\end{figure}
The points $A$ and $B$ are 3.75 m apart on a smooth horizontal floor. A particle $P$ has mass 0.8 kg . One end of a light elastic spring, of natural length 1.5 m and modulus of elasticity 24 N , is attached to $P$ and the other end is attached to $A$. The ends of another light elastic spring, of natural length 0.75 m and modulus of elasticity 18 N , are attached to $P$ and $B$. The particle $P$ rests in equilibrium at the point $O$, where $A O B$ is a straight line, as shown in Figure 5.
\begin{enumerate}[label=(\alph*)]
\item Show that $A O = 2.4 \mathrm {~m}$.
The point $C$ lies on the straight line $A O B$ between $O$ and $B$. The particle $P$ is held at $C$ and released from rest.
\item Show that $P$ moves with simple harmonic motion.
The maximum speed of $P$ is $\sqrt { } 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }$.
\item Find the time taken by $P$ to travel 0.3 m from $C$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2013 Q6 [14]}}