Particle on inclined plane - force parallel to slope

5. \begin{figure}[h] \end{figure} A particle of mass \(m\) rests in equilibrium on a fixed rough plane under the action of a force of magnitude \(X\). The force acts up a line of greatest slope of the plane, as shown in Figure 3. The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\) The coefficient of friction between the particle and the plane is \(\mu\).

• When \(X = 2 P\), the particle is on the point of sliding up the plane.
• When \(X = P\), the particle is on the point of sliding down the plane.

Find the value of \(\mu\).

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

A brick \(P\) of mass \(m\) is placed on the plane.
The coefficient of friction between \(P\) and the plane is \(\mu\) Brick \(P\) is in equilibrium and on the point of sliding down the plane.
Brick \(P\) is modelled as a particle.
Using the model,

1. find, in terms of \(m\) and \(g\), the magnitude of the normal reaction of the plane on brick \(P\)
2. show that \(\mu = \frac { 3 } { 4 }\) For parts (c) and (d), you are not required to do any further calculations.
   Brick \(P\) is now removed from the plane and a much heavier brick \(Q\) is placed on the plane. The coefficient of friction between \(Q\) and the plane is also \(\frac { 3 } { 4 }\)
3. Explain briefly why brick \(Q\) will remain at rest on the plane. Brick \(Q\) is now projected with speed \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down a line of greatest slope of the plane.
   Brick \(Q\) is modelled as a particle.
   Using the model,
4. describe the motion of brick \(Q\), giving a reason for your answer.

3 A box weighing 130 N is on a rough plane inclined at \(12 ^ { \circ }\) to the horizontal.
The box is held at rest on the plane by the action of a force of magnitude 70 N acting up the plane in a direction 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 70 N is removed.
2. Determine whether or not the box remains in equilibrium.

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2.5 kg rests in equilibrium on a rough plane under the action of a force of magnitude \(X\) newtons acting up a line of greatest slope of the plane, as shown in Figure 3. The plane is inclined at \(20 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4 . The particle is in limiting equilibrium and is on the point of moving up the plane. Calculate

1. the normal reaction of the plane on \(P\),
2. the value of \(X\). The force of magnitude \(X\) newtons is now removed.
3. Show that \(P\) remains in equilibrium on the plane.

9 \includegraphics[max width=\textwidth, alt={}, center]{f0c32e07-f3a0-4d58-bd00-c266177ceaac-4_430_565_260_790} The diagram shows a block of wood, weighing 100 N , at rest on a rough plane inclined at \(35 ^ { \circ }\) to the horizontal. The coefficient of friction between the block and the plane is 0.2 . A force of \(P \mathrm {~N}\) acts on the block up the slope.

1. Find the maximum possible value of the friction acting on the block.
2. Given that the block is on the point of moving up the slope, find \(P\).
3. Given that the block is on the point of moving down the slope, find \(P\).

12

1. Whilst a helicopter is hovering, the floor of its cargo hold maintains an angle of \(30 ^ { \circ }\) to the horizontal. There is a box of mass 20 kg on the floor. If the box is just on the point of sliding, show by resolving forces that the coefficient of friction between the box and the floor is \(\frac { 1 } { \sqrt { 3 } }\).
2. The helicopter ascends at a constant acceleration 0.5 g . If the cargo hold is now maintained at \(10 ^ { \circ }\) to the horizontal, determine the frictional force and the normal reaction between the box and the floor.

A particle of mass \(3\text{ kg}\) is on a rough plane inclined at an angle of \(20°\) to the horizontal. A force of magnitude \(P\text{ N}\) acting parallel to a line of greatest slope of the plane is used to keep the particle in equilibrium. The coefficient of friction between the particle and the plane is \(0.35\). Show that the least possible value of \(P\) is \(0.394\), correct to 3 significant figures, and find the greatest possible value of \(P\). [6]

\includegraphics{figure_3} A particle of mass \(0.6\) kg is placed on a rough plane which is inclined at an angle of \(21°\) to the horizontal. The particle is kept in equilibrium by a force of magnitude \(P\) N acting parallel to a line of greatest slope of the plane, as shown in the diagram. The coefficient of friction between the particle and the plane is \(0.3\). Show that the least possible value of \(P\) is \(0.470\), correct to \(3\) significant figures, and find the greatest possible value of \(P\). [6]

A block of mass 3 kg is at rest on a rough plane inclined at 60° to the horizontal. A force of magnitude 15 N acting up a line of greatest slope of the plane is just sufficient to prevent the block from sliding down the plane.

1. Find the coefficient of friction between the block and the plane. [5]

The force of magnitude 15 N is now replaced by a force of magnitude \(X\) N acting up the line of greatest slope.

1. Find the greatest value of \(X\) for which the block does not move. [2]

\includegraphics{figure_2} A particle \(P\), of mass 2 kg, lies on a rough plane inclined at an angle of 30° to the horizontal. A force \(H\), whose line of action is parallel to the line of greatest slope of the plane, is applied to the particle as shown in Figure 2. The coefficient of friction between the particle and the plane is \(\frac{1}{\sqrt{3}}\). Given that the particle is on the point of moving up the plane,

1. draw a diagram showing all the forces acting on the particle, [2 marks]
2. show that the ratio of the magnitude of the frictional force to the magnitude of \(H\) is equal to \(1 : 2\) [7 marks]

The force \(H\) is now removed but \(P\) remains at rest.

1. Use the principle of friction to explain how this is possible. [2 marks]