Friction inequality derivation

2. \begin{figure}[h] \end{figure} A particle \(P\) of weight 40 N lies at rest in equilibrium on a fixed rough horizontal surface. A force of magnitude 20 N is applied to \(P\). The force acts at angle \(\theta\) to the horizontal, as shown in Figure 2. The coefficient of friction between \(P\) and the surface is \(\mu\). Given that the particle remains at rest, show that $$\mu \geqslant \frac { \cos \theta } { 2 + \sin \theta }$$ \includegraphics[max width=\textwidth, alt={}, center]{04b73f81-3316-4f26-ad98-a7be3a4b738f-07_119_167_2615_1777}

6. \begin{figure}[h] \end{figure} A particle of weight \(W\) newtons lies at rest on a rough horizontal surface, as shown in Figure 3.
A force of magnitude \(P\) newtons is applied to the particle.
The force acts at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 4 } { 3 }\) The coefficient of friction between the particle and the surface is \(\frac { 1 } { 4 }\) Given that the particle does not move, show that $$P \leqslant \frac { 5 W } { 8 }$$

\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]

\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,

1. show that \(\mu = 0.35\). [6]

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,

1. Find the range of possible values of \(k\). [7]

\includegraphics{figure_2} A ladder \(AB\), of mass \(m\) and length \(4a\), has one end \(A\) resting on rough horizontal ground. The other end \(B\) rests against a smooth vertical wall. A load of mass \(3m\) is fixed on the ladder at the point \(C\), where \(AC = a\). The ladder is modelled as a uniform rod in a vertical plane perpendicular to the wall and the load is modelled as a particle. The ladder rests in limiting equilibrium making an angle of 30° with the wall, as shown in Figure 2. Find the coefficient of friction between the ladder and the ground. [10]

A uniform plank of wood \(XY\), of mass 1.4 kg, rests with its upper end \(X\) against a rough vertical wall and its lower end \(Y\) on rough horizontal ground. The coefficient of friction between the plank and both the wall and the ground is \(\mu\). The plank is in limiting equilibrium at both ends and the vertical component of the force exerted on the plank by the ground has magnitude 12 N. Find the value of \(\mu\), to 2 decimal places. [8 marks]