| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2012 |
| Session | June |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Potential energy with inextensible strings or gravity only |
| Difficulty | Challenging +1.8 This is an M4 (Further Maths Mechanics) question requiring potential energy formulation for a complex system with a rod, string over two pegs, and a hanging mass. Part (a) involves careful geometry to track string length changes and expressing PE in terms of one variable, requiring trigonometric manipulation to reach the given form. Part (b) requires differentiation, solving a trigonometric equation, and applying the second derivative test. While systematic, it demands strong geometric visualization, multi-step algebraic manipulation with trig identities, and careful bookkeeping—significantly above average A-level difficulty but standard for M4. |
| Spec | 3.04b Equilibrium: zero resultant moment and force6.02i Conservation of energy: mechanical energy principle6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| \(V = -Wa\cos 2\theta + \frac{1}{2}W\{3a - (L - 6a\cos\theta - 4a)\}\) | B1 M1 A1 | GPE of rod e.g. \(-W a\cos 2\theta\) GPE of the particle e.g. \(-\frac{1}{2}W\{3a - (L - 6a\cos\theta - 4a)\}\) Condone 3a term missing. Correct expression including the 3a (unless in the GPE for the rod) Accept aef e.g. \(\sqrt{18a^2(1+\cos 2\theta)}\) for \(6a\cos\theta\) |
| \(= -Wa\cos 2\theta + 3Wa\cos\theta + \left(\frac{7Wa}{2} - \frac{WL}{2}\right)\) | ||
| \(= Wa(3\cos\theta - \cos 2\theta) + \text{constant}\) | A1 | Obtain the given answer correctly |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dV}{d\theta} = Wa(-3\sin\theta + 2\sin 2\theta)\) | M1 A1 | Differentiate the given \(V\) wrt \(\theta\) correct |
| For equilibrium, \(Wa(-3\sin\theta + 2\sin 2\theta) = 0\) | For equilibrium, | |
| \(\sin\theta(4\cos\theta - 3) = 0\) | DM1 A1 | Set their derivative = 0. First answer |
| \(\Rightarrow \theta = 0\) or \(\theta = \cos^{-1}\left(\frac{3}{4}\right)\) | A1 | Second answer - ignore \(\theta = -\cos^{-1}\left(\frac{3}{4}\right)\). 0.72 rads or better |
| \(\frac{d^2V}{d\theta^2} = Wa(-3\cos\theta + 4\cos 2\theta)\) | M1 | Obtain the second derivative of \(V\) and substitute one of their values for \(\theta\) |
| \(\theta = 0\): \(\frac{d^2V}{d\theta^2} = Wa > 0 \Rightarrow\) stable | A1 | Correct working and conclusion for one value |
| \(\theta = \cos^{-1}\frac{3}{4}\): \(\frac{d^2V}{d\theta^2} = -\frac{7Wa}{4} < 0 \Rightarrow\) unstable | A1 | Correct working and reasoning for the second. ISW for work on \(-\cos^{-1}\left(\frac{3}{4}\right)\) |
**Part (a)**
| $V = -Wa\cos 2\theta + \frac{1}{2}W\{3a - (L - 6a\cos\theta - 4a)\}$ | B1 M1 A1 | GPE of rod e.g. $-W a\cos 2\theta$ GPE of the particle e.g. $-\frac{1}{2}W\{3a - (L - 6a\cos\theta - 4a)\}$ Condone 3a term missing. Correct expression including the 3a (unless in the GPE for the rod) Accept aef e.g. $\sqrt{18a^2(1+\cos 2\theta)}$ for $6a\cos\theta$ |
| $= -Wa\cos 2\theta + 3Wa\cos\theta + \left(\frac{7Wa}{2} - \frac{WL}{2}\right)$ | | |
| $= Wa(3\cos\theta - \cos 2\theta) + \text{constant}$ | A1 | Obtain the given answer correctly |
**(4) marks**(12)
**Part (b)**
| $\frac{dV}{d\theta} = Wa(-3\sin\theta + 2\sin 2\theta)$ | M1 A1 | Differentiate the given $V$ wrt $\theta$ correct |
| For equilibrium, $Wa(-3\sin\theta + 2\sin 2\theta) = 0$ | | For equilibrium, |
| $\sin\theta(4\cos\theta - 3) = 0$ | DM1 A1 | Set their derivative = 0. First answer |
| $\Rightarrow \theta = 0$ or $\theta = \cos^{-1}\left(\frac{3}{4}\right)$ | A1 | Second answer - ignore $\theta = -\cos^{-1}\left(\frac{3}{4}\right)$. 0.72 rads or better |
| $\frac{d^2V}{d\theta^2} = Wa(-3\cos\theta + 4\cos 2\theta)$ | M1 | Obtain the second derivative of $V$ and substitute one of their values for $\theta$ |
| $\theta = 0$: $\frac{d^2V}{d\theta^2} = Wa > 0 \Rightarrow$ stable | A1 | Correct working and conclusion for one value |
| $\theta = \cos^{-1}\frac{3}{4}$: $\frac{d^2V}{d\theta^2} = -\frac{7Wa}{4} < 0 \Rightarrow$ unstable | A1 | Correct working and reasoning for the second. ISW for work on $-\cos^{-1}\left(\frac{3}{4}\right)$ |
**(8) 12 marks total for Question 5**
---5.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{07536810-a589-4820-a330-78c35022eb68-10_977_1224_205_360}
\captionsetup{labelformat=empty}
\caption{Figure 1}
\end{center}
\end{figure}
A uniform rod $A B$, of length $4 a$ and weight $W$, is free to rotate in a vertical plane about a fixed smooth horizontal axis which passes through the point $C$ of the rod, where $A C = 3 a$. One end of a light inextensible string of length $L$, where $L > 10 a$, is attached to the end $A$ of the rod and passes over a small smooth fixed peg at $P$ and another small smooth fixed peg at $Q$. The point $Q$ lies in the same vertical plane as $P , A$ and $B$. The point $P$ is at a distance $3 a$ vertically above $C$ and $P Q$ is horizontal with $P Q = 4 a$. A particle of weight $\frac { 1 } { 2 } W$ is attached to the other end of the string and hangs vertically below $Q$. The rod is inclined at an angle $2 \theta$ to the vertical, where $- \pi < 2 \theta < \pi$, as shown in Figure 1.
\begin{enumerate}[label=(\alph*)]
\item Show that the potential energy of the system is
$$W a ( 3 \cos \theta - \cos 2 \theta ) + \text { constant }$$
\item Find the positions of equilibrium and determine their stability.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M4 2012 Q5 [12]}}