Rod on inclined plane

4. \begin{figure}[h] \end{figure} Figure 1 shows a uniform rod \(A B\) of length 2 m and mass 6 kg inclined at an angle of \(30 ^ { \circ }\) to the horizontal with \(A\) on smooth horizontal ground and \(B\) supported by a rough peg. The rod is in limiting equilibrium and the coefficient of friction between \(B\) and the peg is \(\mu\).

1. Find, in terms of \(g\), the magnitude of the reactions at \(A\) and \(B\).
2. Show that \(\mu = \frac { 1 } { \sqrt { 3 } }\).

6 A uniform rod AB has length \(2 a\) and weight \(W\). The rod is in equilibrium in a horizontal position. The end A rests on a smooth plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The force exerted on AB by the plane is \(R\). The end B is attached to a light inextensible string inclined at an angle of \(\theta\) to AB as shown in Fig. 6. The rod and string are in the same vertical plane, which also contains the line of greatest slope of the plane on which A lies. The tension in the string is \(T\). \begin{figure}[h] \end{figure}

1. Add the forces \(R\) and \(T\) to the copy of Fig. 6 in the Printed Answer Booklet.
2. By taking moments about B , find an expression for \(R\) in terms of \(W\).
3. By resolving horizontally, show that \(6 T \cos \theta = W \sqrt { 3 }\).
4. By finding a second equation connecting \(T\) and \(\theta\), determine
   • the value of \(\theta\),
   • an expression for \(T\) in terms of \(W\).

4 Fig. 4 shows a non-uniform rigid plank AB of weight 900 N and length 2.5 m . The centre of mass of the plank is at G which is 2 m from A . The end A rests on rough horizontal ground and does not slip. The plank is held in equilibrium at \(20 ^ { \circ }\) above the horizontal by a force of \(T \mathrm {~N}\) applied at B at an angle of \(55 ^ { \circ }\) above the horizontal as shown in Fig. 4. \begin{figure}[h] \end{figure}

1. Show that \(T = 700\) (correct to 3 significant figures).
2. Determine the possible values of the coefficient of friction between the plank and the ground.

\includegraphics{figure_4} A uniform solid cone has base radius \(0.4 \text{ m}\) and height \(4.4 \text{ m}\). A uniform solid cylinder has radius \(0.4 \text{ m}\) and weight equal to the weight of the cone. An object is formed by attaching the cylinder to the cone so that the base of the cone and a circular face of the cylinder are in contact and their circumferences coincide. The object rests in equilibrium with its circular base on a plane inclined at an angle of \(20°\) to the horizontal (see diagram).

1. Calculate the least possible value of the coefficient of friction between the plane and the object. [2]
2. Calculate the greatest possible height of the cylinder. [4]

\includegraphics{figure_4} A uniform solid cone has base radius \(0.4\) m and height \(4.4\) m. A uniform solid cylinder has radius \(0.4\) m and weight equal to the weight of the cone. An object is formed by attaching the cylinder to the cone so that the base of the cone and a circular face of the cylinder are in contact and their circumferences coincide. The object rests in equilibrium with its circular base on a plane inclined at an angle of \(20°\) to the horizontal (see diagram).

1. Calculate the least possible value of the coefficient of friction between the plane and the object. [2]
2. Calculate the greatest possible height of the cylinder. [4]

\includegraphics{figure_11} A uniform rod \(AB\), of weight \(20 \text{N}\) and length \(2.8 \text{m}\), rests in equilibrium with the end \(A\) in contact with rough horizontal ground and the end \(B\) resting against a smooth wall inclined at \(55°\) to the horizontal. The rod, which rests in a vertical plane that is perpendicular to the wall, is inclined at \(30°\) to the horizontal (see diagram).

1. Show that the magnitude of the force acting on the rod at \(B\) is \(9.56 \text{N}\), correct to 3 significant figures. [3]
2. Determine the magnitude of the contact force between the rod and the ground. Give your answer correct to 3 significant figures. [5]