| Exam Board | OCR |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2006 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Potential energy with elastic strings/springs |
| Difficulty | Challenging +1.8 This is a challenging M4 equilibrium problem requiring energy methods with multiple components (rod, particle, elastic string). Students must express total energy in terms of θ, differentiate to find equilibrium, and analyze second derivative for stability. The geometry is non-trivial (relating string extension to θ, finding heights of center of mass and particle), and the algebraic manipulation is substantial. However, the method is standard for M4 and clearly signposted, preventing it from reaching the highest difficulty levels. |
| Spec | 6.02i Conservation of energy: mechanical energy principle6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| \(RC = 2a \cos \theta\) | B1 | |
| \(EPE = \frac{5mg}{2a}(2a \cos \theta)^2\) | M1 | |
| \(GPE = mga \sin 2\theta + 2mg(2a \sin 2\theta)\) | M1 | |
| \(V = 10mga \cos^2 \theta + 5mga \sin 2\theta\) | A1 | |
| \(\frac{dV}{d\theta} = -20mga \cos \theta \sin \theta + 10mga \cos 2\theta\) | B1 | Correct differentiation of \(\cos^2 \theta\) (or \(\cos 2\theta\)) and \(\sin 2\theta\) |
| \(= -10mga \sin 2\theta + 10mga \cos 2\theta\) | For using \(\frac{dV}{d\theta} = 0\) | |
| For equilibrium: \(10mga(\cos 2\theta - \sin 2\theta) = 0\) | M1 | |
| \(\tan 2\theta = 1\) | ||
| \(\theta = \frac{1}{8}\pi\) | A1 | Accept \(22.1°\) ± \(0.393\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{d^2V}{d\theta^2} = -20mga \cos 2\theta - 20mga \sin 2\theta\) | B1 ft | |
| When \(\theta = \frac{1}{8}\pi\), \(\frac{d^2V}{d\theta^2} = (-20\sqrt{2}mga) < 0\) | M1 | Determining the sign of \(V^*\) |
| Hence the equilibrium is unstable | A1 | Correctly shown |
| Answer | Marks | Guidance |
|---|---|---|
| OR Other method for determining whether \(V\) has a maximum or a minimum | M1 | Correct determination |
| Correctly shown | A1 ft | |
| Equilibrium is unstable | A1 |
## Part (i)
$RC = 2a \cos \theta$ | B1 |
$EPE = \frac{5mg}{2a}(2a \cos \theta)^2$ | M1 |
$GPE = mga \sin 2\theta + 2mg(2a \sin 2\theta)$ | M1 |
$V = 10mga \cos^2 \theta + 5mga \sin 2\theta$ | A1 |
$\frac{dV}{d\theta} = -20mga \cos \theta \sin \theta + 10mga \cos 2\theta$ | B1 | Correct differentiation of $\cos^2 \theta$ (or $\cos 2\theta$) and $\sin 2\theta$
$= -10mga \sin 2\theta + 10mga \cos 2\theta$ | | For using $\frac{dV}{d\theta} = 0$
For equilibrium: $10mga(\cos 2\theta - \sin 2\theta) = 0$ | M1 |
$\tan 2\theta = 1$ | |
$\theta = \frac{1}{8}\pi$ | A1 | Accept $22.1°$ ± $0.393$
**Total: 7 marks**
## Part (ii)
$\frac{d^2V}{d\theta^2} = -20mga \cos 2\theta - 20mga \sin 2\theta$ | B1 ft |
When $\theta = \frac{1}{8}\pi$, $\frac{d^2V}{d\theta^2} = (-20\sqrt{2}mga) < 0$ | M1 | Determining the sign of $V^*$
Hence the equilibrium is unstable | A1 | Correctly shown
**Total: 3 marks**
**OR** Other method for determining whether $V$ has a maximum or a minimum | M1 | Correct determination
Correctly shown | A1 ft |
Equilibrium is unstable | A1 |
---\includegraphics{figure_4}
A uniform rod $AB$, of mass $m$ and length $2a$, is freely hinged to a fixed point at $A$. A particle of mass $2m$ is attached to the rod at $B$. A light elastic string, with natural length $a$ and modulus of elasticity $5mg$, passes through a fixed smooth ring $R$. One end of the string is fixed to $A$ and the other end is fixed to the mid-point $C$ of $AB$. The ring $R$ is at the same horizontal level as $A$, and is at a distance $a$ from $A$. The rod $AB$ and the ring $R$ are in a vertical plane, and $RC$ is at an angle $\theta$ above the horizontal, where $0 < \theta < \frac{1}{2}\pi$, so that the acute angle between $AB$ and the horizontal is $2\theta$ (see diagram).
\begin{enumerate}[label=(\roman*)]
\item By considering the energy of the system, find the value of $\theta$ for which the system is in equilibrium. [7]
\item Determine whether this position of equilibrium is stable or unstable. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR M4 2006 Q4 [10]}}