| Exam Board | Edexcel |
|---|---|
| Module | M3 (Mechanics 3) |
| Year | 2016 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Hooke's law and elastic energy |
| Type | Particle at midpoint of string between two horizontal fixed points: vertical motion |
| Difficulty | Standard +0.8 This M3 question requires setting up equilibrium with elastic strings in 2D (resolving forces, using Hooke's law with extensions), then applying energy conservation with elastic potential energy. The geometry is non-trivial (finding lengths and extensions at two positions), and combining elastic EPE with gravitational PE requires careful bookkeeping. More demanding than standard M3 questions but follows established patterns for this topic. |
| Spec | 6.02i Conservation of energy: mechanical energy principle |
| \includegraphics[max width=\textwidth, alt={}]{ffe0bc72-3136-48d9-9d5b-4a364d134070-05_542_51_2026_1982} | VIIV SIHI NI JIIIM IONOO | VI4V SIHI NI JIIYM ION OO |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| \(R(\uparrow): \ 2T\sin\theta = mg\) | M1A1 | Resolve vertically; must have \(2T\); correct equation |
| \(\sin\theta = \frac{3}{5}\) or \(\cos\theta = \frac{4}{5}\) | B1 | A correct trig value for angle |
| \(T = \frac{5}{6}mg\) | ||
| \(T = \frac{\lambda \times 2a}{3a}\) | M1 | Hooke's law inc attempting extension in terms of \(a\) |
| \(\lambda = \frac{3}{2} \times \frac{5}{6}mg = \frac{5}{4}mg\) * | A1cso (5) | Obtain given value of \(\lambda\) from correct working |
| Answer | Marks | Guidance |
|---|---|---|
| Working | Marks | Guidance |
| EPE at \(D = \frac{\lambda a^2}{2 \times 3a}\), at \(C = \frac{\lambda \times 4a^2}{2 \times 3a}\) | B1 | Correct EPE at either start or finish; may use half strings |
| \(\frac{1}{2}mv^2 + \frac{4\lambda a^2}{6a} = \frac{\lambda a^2}{6a} + mg \times \frac{3}{2}a\) | M1A1 | Energy equation with correct number of terms; EPE terms of form \(k\frac{\lambda x^2}{l}\); fully correct equation |
| \(v^2 = \frac{7}{4}ag\) | dM1 | Solve to \(v^2 = ...\) |
| \(v = \frac{\sqrt{7ag}}{2}\) | A1 (5) [10] | Correct expression for \(v\), any equivalent form |
## Question 4:
### Part (a):
| Working | Marks | Guidance |
|---------|-------|----------|
| $R(\uparrow): \ 2T\sin\theta = mg$ | M1A1 | Resolve vertically; must have $2T$; correct equation |
| $\sin\theta = \frac{3}{5}$ or $\cos\theta = \frac{4}{5}$ | B1 | A correct trig value for angle |
| $T = \frac{5}{6}mg$ | | |
| $T = \frac{\lambda \times 2a}{3a}$ | M1 | Hooke's law inc attempting extension in terms of $a$ |
| $\lambda = \frac{3}{2} \times \frac{5}{6}mg = \frac{5}{4}mg$ * | A1cso (5) | Obtain given value of $\lambda$ from correct working |
### Part (b):
| Working | Marks | Guidance |
|---------|-------|----------|
| EPE at $D = \frac{\lambda a^2}{2 \times 3a}$, at $C = \frac{\lambda \times 4a^2}{2 \times 3a}$ | B1 | Correct EPE at either start or finish; may use half strings |
| $\frac{1}{2}mv^2 + \frac{4\lambda a^2}{6a} = \frac{\lambda a^2}{6a} + mg \times \frac{3}{2}a$ | M1A1 | Energy equation with correct number of terms; EPE terms of form $k\frac{\lambda x^2}{l}$; fully correct equation |
| $v^2 = \frac{7}{4}ag$ | dM1 | Solve to $v^2 = ...$ |
| $v = \frac{\sqrt{7ag}}{2}$ | A1 (5) [10] | Correct expression for $v$, any equivalent form |
---4. Fixed points $A$ and $B$ are on a horizontal ceiling, where $A B = 4 a$. A light elastic string has natural length $3 a$ and modulus of elasticity $\lambda$. One end of the string is attached to $A$ and the other end is attached to $B$. A particle $P$ of mass $m$ is attached to the midpoint of the string. The particle hangs freely in equilibrium at the point $C$, where $C$ is at a distance $\frac { 3 } { 2 } a$ vertically below the ceiling.
\begin{enumerate}[label=(\alph*)]
\item Show that $\lambda = \frac { 5 m g } { 4 }$\\
(5)
The point $D$ is the midpoint of $A B$. The particle is now raised vertically upwards to $D$, and released from rest.
\item Find the speed of $P$ as it passes through $C$.\\
\begin{center}
\begin{tabular}{|l|l|l|}
\hline
\includegraphics[max width=\textwidth, alt={}]{ffe0bc72-3136-48d9-9d5b-4a364d134070-05_542_51_2026_1982}
& VIIV SIHI NI JIIIM IONOO & VI4V SIHI NI JIIYM ION OO \\
\hline
\end{tabular}
\end{center}
\end{enumerate}
\hfill \mbox{\textit{Edexcel M3 2016 Q4 [10]}}