Particle on inclined plane - force at angle to slope

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 0.5 kg is at rest on a rough plane which is inclined to the horizontal at \(30 ^ { \circ }\). The particle is held in equilibrium by a force of magnitude 8 N , acting at an angle of \(40 ^ { \circ }\) to the plane, as shown in Figure 2. The line of action of the force lies in the vertical plane containing \(P\) and a line of greatest slope of the plane. The coefficient of friction between \(P\) and the plane is \(\mu\). Given that \(P\) is on the point of sliding up the plane, find the value of \(\mu\).

\includegraphics{figure_5} A block of mass \(12\text{kg}\) is placed on a plane which is inclined at an angle of \(24°\) to the horizontal. A light string, making an angle of \(36°\) above a line of greatest slope, is attached to the block. The tension in the string is \(65\text{N}\) (see diagram). The coefficient of friction between the block and plane is \(\mu\). The block is in limiting equilibrium and is on the point of sliding up the plane. Find \(\mu\). [6]

\includegraphics{figure_5} A particle of mass 0.8 kg lies on a rough plane which is inclined at an angle of \(28°\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(T\) N. This force acts at an angle of \(35°\) above a line of greatest slope of the plane (see diagram). The coefficient of friction between the particle and the plane is 0.2. Find the least and greatest possible values of \(T\). [8]

\includegraphics{figure_5} A particle of mass \(0.12\) kg is placed on a plane which is inclined at an angle of \(40°\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(P\) N acting up the plane at an angle of \(30°\) above a line of greatest slope, as shown in the diagram. The coefficient of friction between the particle and the plane is \(0.32\). Find the set of possible values of \(P\). [8]

\includegraphics{figure_2} A particle of mass 0.4 kg is held at rest on a fixed rough plane by a horizontal force of magnitude \(P\) newtons. The force acts in the vertical plane containing the line of greatest slope of the inclined plane which passes through the particle. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\), as shown in Figure 2. The coefficient of friction between the particle and the plane is \(\frac{1}{3}\). Given that the particle is on the point of sliding up the plane, find

1. the magnitude of the normal reaction between the particle and the plane, [5]
2. the value of \(P\). [5]

\includegraphics{figure_1} A box of mass 1.5 kg is placed on a plane which is inclined at an angle of 30° to the horizontal. The coefficient of friction between the box and plane is \(\frac{1}{4}\). The box is kept in equilibrium by a light string which lies in a vertical plane containing a line of greatest slope of the plane. The string makes an angle of 20° with the plane, as shown in Fig. 2. The box is in limiting equilibrium and is about to move up the plane. The tension in the string is \(T\) newtons. The box is modelled as a particle. Find the value of \(T\). [10]

\includegraphics{figure_2} A box of mass \(6 \text{ kg}\) lies on a rough plane inclined at an angle of \(30°\) to the horizontal. The box is held in equilibrium by means of a horizontal force of magnitude \(P\) newtons, as shown in Fig. 2. The line of action of the force is in the same vertical plane as a line of greatest slope of the plane. The coefficient of friction between the box and the plane is \(0.4\). The box is modelled as a particle. Given that the box is in limiting equilibrium and on the point of moving up the plane, find,

1. the normal reaction exerted on the box by the plane, [4]
2. the value of \(P\). [3]

The horizontal force is removed.

1. Show that the box will now start to move down the plane. [5]

\includegraphics{figure_1} A particle of weight \(W\) newtons is held in equilibrium on a rough inclined plane by a horizontal force of magnitude 4 N. The force acts in a vertical plane containing a line of greatest slope of the inclined plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\), as shown in Figure 1. The coefficient of friction between the particle and the plane is \(\frac{1}{2}\). Given that the particle is on the point of sliding down the plane,

1. show that the magnitude of the normal reaction between the particle and the plane is 20 N,
2. find the value of \(W\). [9]