Challenging +1.8 This is a challenging Further Maths mechanics problem requiring students to set up equilibrium equations (forces and moments) for a ladder with friction at both contact points, then manipulate the resulting equations to derive a specific trigonometric identity. It demands careful geometric reasoning, systematic application of friction laws at limiting equilibrium, and algebraic manipulation of multiple simultaneous equations involving trigonometric functions. While the individual techniques are standard, the combination and the 'show that' format requiring work towards a specific result elevates this above typical A-level questions.
8 A ladder has length 4 metres and mass 20 kg
The ladder rests in equilibrium with one end on a horizontal surface and the ladder resting on the top of a vertical wall.
In this position the ladder is on the point of slipping.
The top of the wall is 1.5 metres above the horizontal surface.
The angle between the ladder and the horizontal surface is \(\alpha\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-14_362_863_804_593}
The coefficient of friction between the ladder and the wall is 0.5
The coefficient of friction between the ladder and the ground is also 0.5
Show that
$$\cos \alpha \sin ^ { 2 } \alpha = \frac { 3 } { 10 }$$
stating clearly any assumptions you make.
\includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-16_2490_1735_219_139}
AG Obtains the required expression from a complete argument
Total: 8 marks
## Question 8:
| Answer/Working | Mark | Guidance |
|---|---|---|
| The ladder is uniform. | B1 (AO 3.3) | States that the ladder is assumed to be uniform |
| Resolving horizontally: $F_2 \cos\alpha + F_1 = S\sin\alpha$ | M1 (AO 3.3) | Resolves horizontally or parallel to the ladder to obtain an equation |
| Resolving vertically: $R + S\cos\alpha + F_2\sin\alpha = 20g$ | M1 (AO 3.3) | Resolves vertically or perpendicular to the ladder to obtain an equation |
| Two correct force equations obtained | A1 (AO 1.1b) | Obtains two correct force equations |
| Taking moments about the base of the ladder: $20g \times 2\cos\alpha = S \times \dfrac{1.5}{\sin\alpha}$ | M1 (AO 3.3) | Takes moments about the base of the ladder, or any other point, using their distance from the base of the ladder to the top of the wall |
| Correct moments equation obtained | A1 (AO 1.1b) | Obtains a correct moments equation |
| Using $F = \mu R$: $R = 20g - S\cos\alpha - \dfrac{1}{2}S\sin\alpha$ | M1 (AO 3.4) | Uses $F = \mu R$ and obtains equations in $\alpha$ and one force by eliminating three forces |
| $\dfrac{S}{2}\cos\alpha + 10g - \dfrac{S}{2}\cos\alpha - \dfrac{S}{4}\sin\alpha = S\sin\alpha$ | | |
| $S = \dfrac{8g}{\sin\alpha}$ | | |
| $40g\cos\alpha = \dfrac{12g}{\sin^2\alpha}$ | | |
| $\cos\alpha\sin^2\alpha = \dfrac{3}{10}$ | R1 (AO 2.1) | **AG** Obtains the required expression from a complete argument |
**Total: 8 marks**
8 A ladder has length 4 metres and mass 20 kg
The ladder rests in equilibrium with one end on a horizontal surface and the ladder resting on the top of a vertical wall.
In this position the ladder is on the point of slipping.\\
The top of the wall is 1.5 metres above the horizontal surface.\\
The angle between the ladder and the horizontal surface is $\alpha$, as shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-14_362_863_804_593}
The coefficient of friction between the ladder and the wall is 0.5\\
The coefficient of friction between the ladder and the ground is also 0.5\\
Show that
$$\cos \alpha \sin ^ { 2 } \alpha = \frac { 3 } { 10 }$$
stating clearly any assumptions you make.\\
\includegraphics[max width=\textwidth, alt={}, center]{b0d0c552-71cb-4e5a-b545-de8a9052def0-16_2490_1735_219_139}
\hfill \mbox{\textit{AQA Further Paper 3 Mechanics 2020 Q8 [8]}}