\includegraphics{figure_1} A uniform ladder \(AB\), of mass \(m\) and length \(2a\), has one end \(A\) on rough horizontal ground. The other end \(B\) rests against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall. The ladder makes an angle \(\alpha\) with the horizontal, where \(\tan \alpha = \frac{4}{3}\). A child of mass \(2m\) stands on the ladder at \(C\) where \(AC = \frac{1}{2}a\), as shown in Fig. 1. The ladder and the child are in equilibrium. By modelling the ladder as a rod and the child as a particle, calculate the least possible value of the coefficient of friction between the ladder and the ground. [9]

**Main Solution:**
$R = 3mg$ | B1 |

**Moments about B:**
$mga \cos \alpha + 2mg \times \frac{1}{3}a \cos \alpha + Fr \times 2a \sin \alpha = R \times 2a \cos \alpha$ | M1 A2 1.0 |

**Solving to $Fr = \frac{3}{4}mg$:** | M1 A1 |

**Friction constraint:**
$Fr \leq \mu R \Rightarrow \frac{3}{4}mg \leq \mu \times 3mg$ | M1 |

$\mu \geq \frac{1}{4}$ (least value is $\frac{1}{4}$) | M1 A1 | **(9 marks)**

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\includegraphics{figure_1}

A uniform ladder $AB$, of mass $m$ and length $2a$, has one end $A$ on rough horizontal ground. The other end $B$ rests against a smooth vertical wall. The ladder is in a vertical plane perpendicular to the wall. The ladder makes an angle $\alpha$ with the horizontal, where $\tan \alpha = \frac{4}{3}$. A child of mass $2m$ stands on the ladder at $C$ where $AC = \frac{1}{2}a$, as shown in Fig. 1. The ladder and the child are in equilibrium.

By modelling the ladder as a rod and the child as a particle, calculate the least possible value of the coefficient of friction between the ladder and the ground. [9]

\hfill \mbox{\textit{Edexcel M2 2003 Q3 [9]}}