3.03u Static equilibrium: on rough surfaces

\includegraphics{figure_10} A uniform rod \(AB\), of weight \(W\) N and length \(2a\) m, rests with the end \(A\) on a rough horizontal table. A small object of weight \(2W\) N is attached to the rod at \(B\). The rod is maintained in equilibrium at an angle of \(30°\) to the horizontal by a force acting at \(B\) in a direction perpendicular to the rod in the same vertical plane as the rod (see diagram).

1. Find the least possible value of the coefficient of friction between the rod and the table. [7]
2. Given that the magnitude of the contact force at \(A\) is \(\sqrt{39}\) N, find the value of \(W\). [2]

A uniform ladder \(AB\), of weight \(150\text{N}\) and length \(4\text{m}\), rests in equilibrium with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth vertical wall. The ladder is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = 3\). A man of weight \(750\text{N}\) is standing on the ladder at a distance \(x\text{m}\) from \(A\).

1. Show that the magnitude of the frictional force exerted by the ground on the ladder is \(\frac{75}{2}(2 + 5x)\text{N}\). [4]

The coefficient of friction between the ladder and the ground is \(\frac{1}{4}\).

1. Find the greatest value of \(x\) for which equilibrium is possible. [3]

A body of mass 20 kg is on a rough plane inclined at angle \(\alpha\) to the horizontal. The body is held at rest on the plane by the action of a force of magnitude \(P\) N. The force is acting up the plane in a direction parallel to a line of greatest slope of the plane. The coefficient of friction between the body and the plane is \(\mu\).

1. When \(P = 100\), the body is on the point of sliding down the plane. Show that \(g \sin \alpha = g\mu \cos \alpha + 5\). [4]
2. When \(P\) is increased to 150, the body is on the point of sliding up the plane. Use this, and your answer to part (a), to find an expression for \(\alpha\) in terms of \(g\). [3]

A uniform ladder \(AB\) of mass 35 kg and length 7 m rests with its end \(A\) on rough horizontal ground and its end \(B\) against a rough vertical wall. The ladder is inclined at an angle of \(45°\) to the horizontal. A man of mass 70 kg is standing on the ladder at a point \(C\), which is \(x\) metres from \(A\). The coefficient of friction between the ladder and the wall is \(\frac{1}{4}\) and the coefficient of friction between the ladder and the ground is \(\frac{1}{2}\). The system is in limiting equilibrium. Find \(x\). [8]

Two particles, \(A\) and \(B\), each of mass 1 kg are connected by a light inextensible string. Particle \(A\) is at rest on a slope inclined at 30° to the horizontal. The string passes over a small smooth pulley at the top of the slope and particle \(B\) hangs freely, as shown in the diagram. \includegraphics{figure_11}

1. 1. In the case when the slope is smooth, draw a fully labelled diagram to show the forces acting on the particles. Hence find the acceleration of the particles and the tension in the string. [7]
   2. Write down the direction of the resultant force exerted by the string on the pulley. [1]
2. In fact the contact between particle \(A\) and the slope is rough. The coefficient of friction between \(A\) and the slope is \(\mu\). The system is in equilibrium. Find the set of possible values of \(\mu\). [5]

\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]

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 \(B\).