String at angle to slope

3 \includegraphics[max width=\textwidth, alt={}, center]{2bb3c9bb-60f0-440d-a148-b4db3478ca31-2_241_535_1247_806} A particle \(P\) of mass 0.5 kg rests on a rough plane inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\). A force of magnitude 0.6 N , acting upwards on \(P\) at angle \(\alpha\) from a line of greatest slope of the plane, is just sufficient to prevent \(P\) sliding down the plane (see diagram). Find

1. the normal component of the contact force on \(P\),
2. the frictional component of the contact force on \(P\),
3. the coefficient of friction between \(P\) and the plane.

1 \includegraphics[max width=\textwidth, alt={}, center]{79b90ef5-ef3a-4c59-b662-d0fbfba813ca-2_346_583_255_781} A small block of weight 5.1 N rests on a smooth plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 8 } { 17 }\). The block is held in equilibrium by means of a light inextensible string. The string makes an angle \(\beta\) above the line of greatest slope on which the block rests, where \(\sin \beta = \frac { 7 } { 25 }\) (see diagram). Find the tension in the string.

7. \begin{figure}[h] \end{figure} A package of mass 4 kg lies on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. The package is held in equilibrium by a force of magnitude 45 N acting at an angle of \(50 ^ { \circ }\) to the plane, as shown in Figure 3. The force is acting in a vertical plane through a line of greatest slope of the plane. The package is in equilibrium on the point of moving up the plane. The package is modelled as a particle. Find

1. the magnitude of the normal reaction of the plane on the package,
2. the coefficient of friction between the plane and the package.

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is held at rest in equilibrium on a rough plane by a constant force of magnitude 40 N . The direction of the force is inclined to the plane at an angle of \(30 ^ { \circ }\). The plane is inclined to the horizontal at an angle of \(20 ^ { \circ }\), 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 \(\mu\).

4. \begin{figure}[h] \end{figure} A fixed rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\) A small box of mass \(m\) is at rest on the plane. A force of magnitude \(k m g\), where \(k\) is a constant, is applied to the box. The line of action of the force is at angle \(\alpha\) to the line of greatest slope of the plane through the box, as shown in Figure 1, and lies in the same vertical plane as this line of greatest slope. The coefficient of friction between the box and the plane is \(\mu\). The box is on the point of slipping up the plane. By modelling the box as a particle, find \(k\) in terms of \(\mu\).

8. \begin{figure}[h] \end{figure} A parcel of mass 2 kg is pulled up a rough inclined plane by the action of a constant force. The force has magnitude 18 N and acts at an angle of \(40 ^ { \circ }\) to the plane.
The line of action of the force lies in a vertical plane containing a line of greatest slope of the inclined plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal, as shown in Figure 5.
The coefficient of friction between the plane and the parcel is 0.3
The parcel is modelled as a particle \(P\)

1. Find the acceleration of \(P\) The points \(A\) and \(B\) lie on a line of greatest slope of the plane, where \(A B = 5 \mathrm {~m}\) and \(B\) is above \(A\). Particle \(P\) passes through \(A\) with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(A B\).
2. Find the speed of \(P\) as it passes through \(B\). The force of 18 N is removed at the instant \(P\) passes through \(B\). As a result, \(P\) comes to rest at the point \(C\).
3. Determine whether \(P\) will remain at rest at \(C\). You must show all stages of your working clearly.

4. \begin{figure}[h] \end{figure} A particle of weight \(W\) lies on a rough plane.The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\) .The coefficient of friction between the particle and the plane is \(\frac { 1 } { 2 }\) The particle is held in equilibrium by a force of magnitude 1.2 W .The force makes an angle \(\theta\) with the plane,where \(0 < \theta < \pi\) ,and acts in a vertical plane containing a line of greatest slope of the plane,as shown in Figure 2.

1. Find the value of \(\theta\) for which there is no frictional force acting on the particle. The minimum value of \(\theta\) for the particle to remain in equilibrium is \(\beta\)
2. Show that $$\beta = \arccos \left( \frac { \sqrt { 5 } } { 3 } \right) - \arctan \left( \frac { 1 } { 2 } \right)$$
3. Find the range of values of \(\theta\) for which the particle remains in equilibrium with the frictional force acting up the plane.

8 A rough slope is inclined at an angle of \(10 ^ { \circ }\) to the horizontal. A particle of mass 6 kg is on the slope. A string is attached to the particle and is at an angle of \(30 ^ { \circ }\) to the slope. The tension in the string is 20 N . The diagram shows the slope, the particle and the string. \includegraphics[max width=\textwidth, alt={}, center]{7e0585ea-062a-487c-8e39-37a4ed414ff8-6_259_684_518_676} The particle moves up the slope with an acceleration of \(0.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Draw a diagram to show the forces acting on the particle.
2. Show that the magnitude of the normal reaction force is 47.9 N , correct to three significant figures.
3. Find the coefficient of friction between the particle and the slope.

\includegraphics{figure_6} A block \(B\), of mass 2 kg, lies on a rough inclined plane sloping at \(30°\) to the horizontal. A light rope, inclined at an angle of \(20°\) above a line of greatest slope, is attached to \(B\). The tension in the rope is \(T\) N. There is a friction force of \(F\) N acting on \(B\) (see diagram). The coefficient of friction between \(B\) and the plane is \(\mu\).

1. It is given that \(F = 5\) and that the acceleration of \(B\) up the plane is \(1.2\,\text{m}\,\text{s}^{-2}\).
   1. Find the value of \(T\). [3]
   2. Find the value of \(\mu\). [3]
2. It is given instead that \(\mu = 0.8\) and \(T = 15\). Determine whether \(B\) will move up the plane. [3]

\includegraphics{figure_4} A block of mass 8 kg is placed on a rough plane which is inclined at an angle of 18° to the horizontal. The block is pulled up the plane by a light string that makes an angle of 26° above a line of greatest slope. The tension in the string is \(T\) N (see diagram). The coefficient of friction between the block and plane is 0.65.

1. The acceleration of the block is 0.2 m s\(^{-2}\). Find \(T\). [7]
2. The block is initially at rest. Find the distance travelled by the block during the fourth second of motion. [2]

\includegraphics{figure_3} A particle \(P\) of mass \(8 \text{ kg}\) is on a smooth plane inclined at an angle of \(30°\) to the horizontal. A force of magnitude \(100 \text{ N}\), making an angle of \(\theta°\) with a line of greatest slope and lying in the vertical plane containing the line of greatest slope, acts on \(P\) (see diagram).

1. Given that \(P\) is in equilibrium, show that \(\theta = 66.4\), correct to \(1\) decimal place, and find the normal reaction between the plane and \(P\). [4]
2. Given instead that \(\theta = 30\), find the acceleration of \(P\). [2]

\includegraphics{figure_3} Figure 3 shows a block of mass 25 kg held in equilibrium on a plane inclined at an angle of 35° to the horizontal by means of a string which is at an angle of 15° to the line of greatest slope of the plane. In an initial model of the situation, the plane is assumed to be smooth. Giving your answers correct to 3 significant figures,

1. show that the tension in the string is 145 N. [3 marks]
2. find the magnitude of the reaction between the plane and the block. [4 marks]

In a more refined model, the plane is assumed to be rough. Given that the tension in the string can be increased to 200 N before the block begins to move up the slope,

1. find, correct to 3 significant figures, the magnitude of the frictional force and state the direction in which it acts. [4 marks]
2. Without performing any further calculations, state whether the reaction calculated in part (b) will increase, decrease or remain the same in the refined model. Give a reason for your answer. [3 marks]

In this question use \(g = 9.8\) m s\(^{-2}\). The diagram shows a box, of mass 8.0 kg, being pulled by a string so that the box moves at a constant speed along a rough horizontal wooden board. The string is at an angle of 40° to the horizontal. The tension in the string is 50 newtons. \includegraphics{figure_16a} The coefficient of friction between the box and the board is \(\mu\) Model the box as a particle.

1. Show that \(\mu = 0.83\) [4 marks]
2. One end of the board is lifted up so that the board is now inclined at an angle of 5° to the horizontal. The box is pulled up the inclined board. The string remains at an angle of 40° to the board. The tension in the string is increased so that the box accelerates up the board at 3 m s\(^{-2}\) \includegraphics{figure_16b}
   1. Draw a diagram to show the forces acting on the box as it moves. [1 mark]
   2. Find the tension in the string as the box accelerates up the slope at 3 m s\(^{-2}\). [7 marks]