\includegraphics{figure_14}
One end of a light inextensible string is attached to a particle \(A\) of mass \(2\text{kg}\). The other end of the string is attached to a second particle \(B\) of mass \(3\text{kg}\). Particle \(A\) is in contact with a smooth plane inclined at \(30°\) to the horizontal and particle \(B\) is in contact with a rough horizontal plane.
A second light inextensible string is attached to \(B\). The other end of this second string is attached to a third particle \(C\) of mass \(4\text{kg}\). Particle \(C\) is in contact with a smooth plane \(\Pi\) inclined at an angle of \(60°\) to the horizontal.
Both strings are taut and pass over small smooth pulleys that are at the tops of the inclined planes. The parts of the strings from \(A\) to the pulley, and from \(C\) to the pulley, are parallel to lines of greatest slope of the corresponding planes (see diagram).
The coefficient of friction between \(B\) and the horizontal plane is \(\mu\). The system is released from rest and in the subsequent motion \(C\) moves down \(\Pi\) with acceleration \(a\text{ms}^{-2}\).
- By considering an equation involving \(\mu\), \(a\) and \(g\) show that \(a < \frac{5}{9}g(2\sqrt{3} - 1)\).
[7]
- Given that \(a = \frac{1}{5}g\), determine the magnitude of the contact force between \(B\) and the horizontal plane. Give your answer correct to 3 significant figures.
[4]