| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9795/2 (Pre-U Further Mathematics Paper 2) |
| Year | 2016 |
| Session | June |
| Marks | 8 |
| Topic | Moments |
| Type | Ladder on smooth wall and rough ground |
| Difficulty | Standard +0.8 This is a two-part ladder equilibrium problem requiring moments, friction, and resolution of forces. Part (i) involves finding limiting equilibrium angle using moment equation about a strategic point, while part (ii) requires finding maximum distance before slipping occurs. The problem demands careful setup of multiple equations (moments, vertical/horizontal equilibrium, friction law) and algebraic manipulation, which is more challenging than routine statics questions but follows standard ladder problem methodology taught in mechanics courses. |
| Spec | 3.04b Equilibrium: zero resultant moment and force |
(i) $N = 80$
$M(X)$: $F\times5\sin\theta = 80\times2.5\cos\theta$
$F \leq 0.4N$
$\tan\theta \geq 1.25$
$\theta_{\min} = \mathbf{51.3°}$ or $51.4°$, $0.896$
- B1: Normal force at ground (can be implied)
- M1: Moments about any point, needs both cos and sin
- M1: Use $F \leq \mu N$ or $F = \mu N$
- M1: Solve equations to obtain $\tan\theta$
- A1: Correct answer, in range $[51.3, 51.4]$ or a.r.t. $0.896$
**[5]**
(ii) $F\times5\sin\theta = (80\times2.5 + 750d)\cos\theta$
Use $60°$ and $\mu$ to obtain
$d_{\max} = \mathbf{3.57}$
- M1*: Moments equation with variable $d$ $[F = 332]$; $[332\times5\times0.5 = 100\sqrt{3} + 337\sqrt{3}\,d]$
- depM1
- A1: Answer, a.r.t. 3.57 or 3.56
**[3]**10 A uniform ladder $A B$ of length 5 m and mass 8 kg is placed at an angle $\theta$ to the horizontal, with $A$ on rough horizontal ground and $B$ against a smooth vertical wall. The coefficient of friction between the ladder and the ground is 0.4 .\\
(i) By taking moments, find the smallest value of $\theta$ for which the ladder is in equilibrium.\\
(ii) A man of mass 75 kg stands on the ladder when $\theta = 60 ^ { \circ }$. Find the greatest distance from $A$ that he can stand without the ladder slipping.
\hfill \mbox{\textit{Pre-U Pre-U 9795/2 2016 Q10 [8]}}