3.03m Equilibrium: sum of resolved forces = 0

\includegraphics{figure_10} Particles \(A\) and \(B\) of masses \(2m\) and \(m\), respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley \(P\). The particle \(A\) rests in equilibrium on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\alpha \leqslant 45°\) and \(B\) is above the plane. The vertical plane defined by \(APB\) contains a line of greatest slope of the plane, and \(PA\) is inclined at angle \(2\alpha\) to the horizontal (see diagram).

1. Show that the normal reaction R between A and the plane is mg(2\(\cos\alpha - \sin\alpha\)). [3]
2. Show that R \(\geqslant \frac{1}{2}mg\sqrt{2}\). [3]

The coefficient of friction between \(A\) and the plane is \(\mu\). The particle is about to slip down the plane.

1. Show that \(0.5 < \tan\alpha \leqslant 1\). [3]
2. Express \(\mu\) as a function of \(\tan\alpha\) and deduce its maximum value as \(\alpha\) varies. [3]

\includegraphics{figure_10} Particles \(A\) and \(B\) of masses \(2m\) and \(m\), respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley \(P\). The particle \(A\) rests in equilibrium on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\alpha < 45°\) and \(B\) is above the plane. The vertical plane defined by \(APB\) contains a line of greatest slope of the plane, and \(PA\) is inclined at angle \(2\alpha\) to the horizontal (see diagram).

1. Show that the normal reaction R between \(A\) and the plane is mg(2\(\cos\alpha - \sin\alpha\)). [3]
2. Show that R \(\geqslant \frac{1}{2}mg\sqrt{2}\). [3]

The coefficient of friction between \(A\) and the plane is \(\mu\). The particle is about to slip down the plane.

1. Show that \(0.5 < \tan\alpha < 1\). [3]
2. Express \(\mu\) as a function of \(\tan\alpha\) and deduce its maximum value as \(\alpha\) varies. [3]

A particle \(P\) of mass \(1.5\) kg is placed on a smooth horizontal table. The particle is initially at the origin of a \(2\)-dimensional vector system defined by perpendicular unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) in the plane of the table. The particle is subject to three forces of magnitudes \(10\) N, \(12\) N and \(F\) N, acting in the directions of the vectors \(3\mathbf{i} + 4\mathbf{j}\), \(-\mathbf{j}\) and \(-\cos \theta \mathbf{i} + \sin \theta \mathbf{j}\) respectively, and no others.

1. Given that the system is in equilibrium, determine \(F\) and \(\theta\). [6]

The force of magnitude \(12\) N is replaced by one of magnitude \(4\) N, but in the opposite direction. The particle is initially at rest.

1. Find the position vector of the particle \(3\) seconds later. [5]

4 \includegraphics[max width=\textwidth, alt={}, center]{c1aae41e-530c-4db4-8959-8afe223c4dbc-3_563_572_258_785} A uniform circular disc, with centre \(O\) and weight \(W\), rests in equilibrium on a horizontal floor and against a vertical wall. The plane of the disc is vertical and perpendicular to the wall. The disc is in contact with the floor at \(A\) and with the wall at \(B\). A force of magnitude \(P\) acts tangentially on the disc at the point \(C\) on the edge of the disc, where the radius \(O C\) makes an angle \(\theta\) with the upward vertical, and \(\tan \theta = \frac { 4 } { 3 }\) (see diagram). The coefficient of friction between the disc and the floor and between the disc and the wall is \(\frac { 1 } { 2 }\). Show that the sum of the magnitudes of the frictional forces at \(A\) and \(B\) is equal to \(P\). Given that the equilibrium is limiting at both \(A\) and \(B\),

1. show that \(P = \frac { 15 } { 34 } \mathrm {~W}\),
2. find the ratio of the magnitude of the normal reaction at \(A\) to the magnitude of the normal reaction at \(B\).