| Exam Board | Edexcel |
|---|---|
| Module | M2 (Mechanics 2) |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Moments |
| Type | Ladder on smooth wall and rough ground |
| Difficulty | Standard +0.3 This is a standard ladder equilibrium problem requiring resolution of forces, friction law (F = μR), and taking moments about a point. While it involves multiple steps (modeling, forces, moments, solving), it follows a well-established textbook method with no novel insight required. The given tan θ = 2 simplifies trigonometry. Slightly easier than average due to its routine nature. |
| Spec | 3.03t Coefficient of friction: F <= mu*R model3.03u Static equilibrium: on rough surfaces3.04a Calculate moments: about a point3.04b Equilibrium: zero resultant moment and force |
| Answer | Marks | Guidance |
|---|---|---|
| (a) (i) uniform rod | B1 | |
| (ii) particle | B1 | |
| (b) Resolve \(\uparrow\): \(R - 15g - 75g = 0\) \(\therefore R = 90g\) | M1 | |
| Resolve \(\rightarrow\): \(\mu R - S = 0\) \(\therefore S = 30g\) | M1 A1 | |
| Mom. about \(B\): \(S.8\sin\theta - 15g.4\cos\theta - 75g.d\cos\theta = 0\) | M1 A1 | |
| \(8S\tan\theta - 60g = 75gd\) | M1 | |
| \(d = \frac{420g}{75g} = 5.6\) \(\therefore AP = 8 - 5.6 = 2.4 \text{ m}\) | M1 A1 | (10 marks) |
**(a)** (i) uniform rod | B1 |
(ii) particle | B1 |
**(b)** Resolve $\uparrow$: $R - 15g - 75g = 0$ $\therefore R = 90g$ | M1 |
Resolve $\rightarrow$: $\mu R - S = 0$ $\therefore S = 30g$ | M1 A1 |
Mom. about $B$: $S.8\sin\theta - 15g.4\cos\theta - 75g.d\cos\theta = 0$ | M1 A1 |
$8S\tan\theta - 60g = 75gd$ | M1 |
$d = \frac{420g}{75g} = 5.6$ $\therefore AP = 8 - 5.6 = 2.4 \text{ m}$ | M1 A1 | (10 marks)3.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{f0e751be-f095-4a56-8ee9-8433cc4873e9-2_424_360_1155_648}
\captionsetup{labelformat=empty}
\caption{Fig. 1}
\end{center}
\end{figure}
Figure 1 shows a uniform ladder of mass 15 kg and length 8 m which rests against a smooth vertical wall at $A$ with its lower end on rough horizontal ground at $B$. The coefficient of friction between the ladder and the ground is $\frac { 1 } { 3 }$ and the ladder is inclined at an angle $\theta$ to the horizontal, where $\tan \theta = 2$.
A man of mass 75 kg ascends the ladder until he reaches a point $P$. The ladder is then on the point of slipping.
\begin{enumerate}[label=(\alph*)]
\item Write down suitable models for
\begin{enumerate}[label=(\roman*)]
\item the ladder,
\item the man.
\end{enumerate}\item Find the distance $A P$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel M2 Q3 [10]}}