Equilibrium on slope with force at angle to slope

4 A particle of mass 20 kg is on a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. A force of magnitude 25 N , acting at an angle of \(20 ^ { \circ }\) above a line of greatest slope of the plane, is used to prevent the particle from sliding down the plane. The coefficient of friction between the particle and the plane is \(\mu\).

1. Complete the diagram below to show all the forces acting on the particle. \includegraphics[max width=\textwidth, alt={}, center]{87b42689-791c-4f4e-a36e-bfae3191ca11-06_495_615_543_726}
2. Find the least possible value of \(\mu\).

5 \includegraphics[max width=\textwidth, alt={}, center]{fd2fbf13-912c-46c5-a470-306b2269aa0b-3_394_531_260_806} A block of mass 2.5 kg is placed on a plane which is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The block is kept in equilibrium by a light string making an angle of \(20 ^ { \circ }\) above a line of greatest slope. The tension in the string is \(T \mathrm {~N}\), as shown in the diagram. The coefficient of friction between the block and plane is \(\frac { 1 } { 4 }\). The block is in limiting equilibrium and is about to move up the plane. Find the value of \(T\).

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2.7 kg lies on a rough plane inclined at \(40 ^ { \circ }\) to the horizontal. The particle is held in equilibrium by a force of magnitude 15 N acting at an angle of \(50 ^ { \circ }\) to the plane, as shown in Figure 4. The force acts in a vertical plane containing a line of greatest slope of the plane. The particle is in equilibrium and is on the point of sliding down the plane. Find

1. the magnitude of the normal reaction of the plane on \(P\),
2. the coefficient of friction between \(P\) and the plane. The force of magnitude 15 N is removed.
3. Determine whether \(P\) moves, justifying your answer.

3. \begin{figure}[h] \end{figure} Figure 2 shows a ball of mass 3 kg lying on a smooth plane inclined at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 3 } { 5 }\). The ball is held in equilibrium by a force of magnitude \(P\) newtons, which acts at an angle of \(10 ^ { \circ }\) to the line of greatest slope of the plane.

1. Suggest a suitable model for the ball. Giving your answers correct to 1 decimal place,
2. find the value of \(P\),
3. find the magnitude of the reaction between the ball and the plane.

\includegraphics{figure_1} A box of mass 2 kg is held in equilibrium on a fixed rough inclined plane by a rope. The rope lies in a vertical plane containing a line of greatest slope of the inclined plane. The rope is inclined to the plane at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\), and the plane is at an angle of \(30°\) to the horizontal, as shown in Figure 1. The coefficient of friction between the box and the inclined plane is \(\frac{1}{2}\) and the box is on the point of slipping up the plane. By modelling the box as a particle and the rope as a light inextensible string, find the tension in the rope. [8]

\includegraphics{figure_3} A small parcel of mass \(2\) kg moves on a rough plane inclined at an angle of \(30°\) to the horizontal. The parcel is pulled up a line of greatest slope of the plane by means of a light rope which it attached to it. The rope makes an angle of \(30°\) with the plane, as shown in Fig. 3. The coefficient of friction between the parcel and the plane is \(0.4\). Given that the tension in the rope is \(24\) N,

1. find, to 2 significant figures, the acceleration of the parcel. [8]

The rope now breaks. The parcel slows down and comes to rest.

1. Show that, when the parcel comes to this position of rest, it immediately starts to move down the plane again. [4]
2. Find, to 2 significant figures, the acceleration of the parcel as it moves down the plane after it has come to this position of instantaneous rest. [3]

A particle \(P\), of mass \(m\), is in contact with a rough plane inclined at 30° to the horizontal as shown. A light string is attached to \(P\) and makes an angle of 30° with the plane. When the tension in this string has magnitude \(kmg\), \(P\) is just on the point of moving up the plane. \includegraphics{figure_7}

1. Show that \(\mu\), the coefficient of friction between \(P\) and the plane, is \(\frac{k\sqrt{3} - 1}{\sqrt{3} - k}\). [7 marks]
2. Given further that \(k = \frac{3\sqrt{3}}{7}\), deduce that \(\mu = \frac{\sqrt{3}}{6}\). [3 marks]

The string is now removed.

1. Determine whether \(P\) will move down the plane and, if it does, find its acceleration. [5 marks]
2. Give a reason why the way in which \(P\) is shown in the diagram might not be consistent with the modelling assumptions that have been made. [1 mark]

A block \(B\) of weight \(10\) N is projected down a line of greatest slope of a plane inclined at an angle of \(20°\) to the horizontal. \(B\) travels down the plane at constant speed.

1. 1. Find the components perpendicular and parallel to the plane of the contact force between \(B\) and the plane. [2]
   2. Hence show that the coefficient of friction is \(0.364\), correct to \(3\) significant figures. [2]
2. \includegraphics{figure_6} \(B\) is in limiting equilibrium when acted on by a force of \(T\) N directed towards the plane at an angle of \(45°\) to a line of greatest slope (see diagram). Given that the frictional force on \(B\) acts down the plane, find \(T\). [7]