| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2004 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Ladder or rod with friction at both contacts |
| Difficulty | Standard +0.8 This is a non-standard ladder equilibrium problem requiring resolution of forces in two directions, taking moments about a strategic point, and working with the friction inequality. The presence of the string at an intermediate point (not at an end) adds complexity beyond typical ladder problems, requiring careful geometric reasoning with tan θ = 2. The multi-part structure with a 'show that' proof for the friction coefficient elevates this above routine M2 questions. |
| Spec | 3.03r Friction: concept and vector form6.04a Centre of mass: gravitational effect6.04e Rigid body equilibrium: coplanar forces |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(M(B), N 2a \cos \theta = W a \cos \theta + \frac{1}{4} W \frac{3a}{2} \sin \theta\) | M1 A2 (−1 e.e.) | dep. M1 A1 (5) |
| \(N = \frac{7W}{8}\) | ||
| (b) \(R = \frac{1}{4} W\); \(F + N = W\) | B1; B1 | |
| \(F \leq \mu R\) or \(F = \mu R\) | M1 | |
| \(\frac{1}{2} \leq \mu\) (exact) | A1 c.s.o. | (5) |
| (c) It does not bend or has negligible thickness | B1 | (1) |
| (10 marks) |
**(a)** $M(B), N 2a \cos \theta = W a \cos \theta + \frac{1}{4} W \frac{3a}{2} \sin \theta$ | M1 A2 (−1 e.e.) | dep. M1 A1 (5)
$N = \frac{7W}{8}$ | |
**(b)** $R = \frac{1}{4} W$; $F + N = W$ | B1; B1 |
$F \leq \mu R$ or $F = \mu R$ | M1 |
$\frac{1}{2} \leq \mu$ (exact) | A1 c.s.o. | (5)
**(c)** It does not bend or has negligible thickness | B1 | (1)
| | | **(10 marks)** |
---4.
\begin{figure}[h]
\begin{center}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\includegraphics[alt={},max width=\textwidth]{fe64e6f1-e36b-465d-a41c-ac834439623b-4_889_741_370_639}
\end{center}
\end{figure}
A uniform ladder, of weight $W$ and length $2 a$, rests in equilibrium with one end $A$ on a smooth horizontal floor and the other end $B$ on a rough vertical wall. The ladder is in a vertical plane perpendicular to the wall. The coefficient of friction between the wall and the ladder is $\mu$. The ladder makes an angle $\theta$ with the floor, where $\tan \theta = 2$. A horizontal light inextensible string $C D$ is attached to the ladder at the point $C$, where $A C = \frac { 1 } { 2 } a$. The string is attached to the wall at the point $D$, with $B D$ vertical, as shown in Fig. 2. The tension in the string is $\frac { 1 } { 4 } W$. By modelling the ladder as a rod,
\begin{enumerate}[label=(\alph*)]
\item find the magnitude of the force of the floor on the ladder,
\item show that $\mu \geqslant \frac { 1 } { 2 }$.
\item State how you have used the modelling assumption that the ladder is a rod.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2004 Q4 [10]}}