Particle just remains at rest

4 A particle moves up a line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\cos \alpha = 0.96\) and \(\sin \alpha = 0.28\).

1. Given that the normal component of the contact force acting on the particle has magnitude 1.2 N , find the mass of the particle.
2. Given also that the frictional component of the contact force acting on the particle has magnitude 0.4 N , find the deceleration of the particle. The particle comes to rest on reaching the point \(X\).
3. Determine whether the particle remains at \(X\) or whether it starts to move down the plane.

4 A box of mass 30 kg is at rest on a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.1\), acted on by a force of magnitude 40 N . The force acts upwards and parallel to a line of greatest slope of the plane. The box is on the point of slipping up the plane.

1. Find the coefficient of friction between the box and the plane. The force of magnitude 40 N is removed.
2. Determine, giving a reason, whether or not the box remains in equilibrium.

1. A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\).

A particle of mass \(m\) is placed on the plane and then projected up a line of greatest slope of the plane. The coefficient of friction between the particle and the plane is \(\mu\).
The particle moves up the plane with a constant deceleration of \(\frac { 4 } { 5 } \mathrm {~g}\).

1. Find the value of \(\mu\). The particle comes to rest at the point \(A\) on the plane.
2. Determine whether the particle will remain at \(A\), carefully justifying your answer.