| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Year | 2006 |
| Session | January |
| Marks | 13 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Friction |
| Type | Friction inequality derivation |
| Difficulty | Standard +0.8 This is a two-part statics problem requiring moments about two points, resolution of forces, and understanding of limiting equilibrium. Part (a) is a standard 'show that' requiring systematic application of equilibrium conditions with friction. Part (b) requires finding a range by considering two limiting cases (slipping and toppling), which demands deeper conceptual understanding and more sophisticated problem-solving than typical M2 questions. |
| Spec | 3.03m Equilibrium: sum of resolved forces = 03.03u Static equilibrium: on rough surfaces3.04a Calculate moments: about a point3.04b Equilibrium: zero resultant moment and force |
| Answer | Marks |
|---|---|
| Diagram showing forces | B1 |
| \(R = 5W\) | B1 |
| \(M(B): 4W\cos\theta + W.2a\cos\theta + \mu R4a\sin\theta = R.4a\cos\theta\) | M1 A1 |
| Having enough equations & solving them for \(\mu\) | M1 |
| \(\mu = 0.35\) | A1 |
| Answer | Marks |
|---|---|
| \(S = (5 + k)W\) | B1 |
| Use of \(F = 0.35S\) or \(F \leq 0.35S\) | M1 |
| \(M(B): kW4a\cos\theta + W.2a\cos\theta + Fa\sin\theta = S.4a\cos\theta\) | M1 A1 |
| Having enough equations & solving them for \(k\) | M1 |
| \(k = \frac{10}{9}\) awrt 1.42 | A1 |
| \(k \square \frac{10}{9}\) ft their \(k\), accept \(>\) and decimals | A1ft |
## (a)
Diagram showing forces | B1 |
$R = 5W$ | B1 |
$M(B): 4W\cos\theta + W.2a\cos\theta + \mu R4a\sin\theta = R.4a\cos\theta$ | M1 A1 |
Having enough equations & solving them for $\mu$ | M1 |
$\mu = 0.35$ | A1 |
## (b)
$S = (5 + k)W$ | B1 |
Use of $F = 0.35S$ or $F \leq 0.35S$ | M1 |
$M(B): kW4a\cos\theta + W.2a\cos\theta + Fa\sin\theta = S.4a\cos\theta$ | M1 A1 |
Having enough equations & solving them for $k$ | M1 |
$k = \frac{10}{9}$ awrt 1.42 | A1 |
$k \square \frac{10}{9}$ ft their $k$, accept $>$ and decimals | A1ft |
---\includegraphics{figure_2}
A ladder $AB$, of weight $W$ and length $4a$, has one end $A$ on rough horizontal ground. The coefficient of friction between the ladder and the ground is $\mu$. The other end $B$ rests against a smooth vertical wall. The ladder makes an angle $\theta$ with the horizontal, where $\tan \theta = 2$. A load of weight $4W$ is placed at the point $C$ on the ladder, where $AC = 3a$, as shown in Figure 2. The ladder is modelled as a uniform rod which is in a vertical plane perpendicular to the wall. The load is modelled as a particle. Given that the system is in limiting equilibrium,
\begin{enumerate}[label=(\alph*)]
\item show that $\mu = 0.35$. [6]
\end{enumerate}
A second load of weight $kW$ is now placed on the ladder at $A$. The load of weight $4W$ is removed from $C$ and placed on the ladder at $B$. The ladder is modelled as a uniform rod which is in a vertical plane perpendicular to the wall. The loads are modelled as particles. Given that the ladder and the loads are in equilibrium,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the range of possible values of $k$. [7]
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 2006 Q6 [13]}}