3.03s Contact force components: normal and frictional

8. \begin{figure}[h] \end{figure} A particle \(P\) of mass 4 kg is moving up a fixed rough plane at a constant speed of \(16 \mathrm {~ms} ^ { - 1 }\) under the action of a force of magnitude 36 N . The plane is inclined at \(30 ^ { \circ }\) to the horizontal. The force acts in the vertical plane containing the line of greatest slope of the plane through \(P\), and acts at \(30 ^ { \circ }\) to the inclined plane, as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\mu\). Find

1. the magnitude of the normal reaction between \(P\) and the plane,
2. the value of \(\mu\). The force of magnitude 36 N is removed.
3. Find the distance that \(P\) travels between the instant when the force is removed and the instant when it comes to rest.

6 A block of weight 14.7 N is at rest on a horizontal floor. A force of magnitude 4.9 N is applied to the block.

1. The block is in limiting equilibrium when the 4.9 N force is applied horizontally. Show that the coefficient of friction is \(\frac { 1 } { 3 }\).
2. 
   
   [diagram]
   When the force of 4.9 N is applied at an angle of \(30 ^ { \circ }\) above the horizontal, as shown in the diagram, the block moves across the floor. Calculate
   1. the vertical component of the contact force between the floor and the block, and the magnitude of the frictional force,
   2. the acceleration of the block.
   3. Calculate the magnitude of the frictional force acting on the block when the 4.9 N force acts at an angle of \(30 ^ { \circ }\) to the upward vertical, justifying your answer fully.

2 A block, of mass 10 kg , is at rest on a rough horizontal surface, when a horizontal force, of magnitude \(P\) newtons, is applied to the block, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-04_108_962_461_539} The coefficient of friction between the block and the surface is 0.5 .

1. Draw and label a diagram to show all the forces acting on the block.
2. 1. Calculate the magnitude of the normal reaction force acting on the block.
   2. Find the maximum possible magnitude of the friction force between the block and the surface.
   3. Given that \(P = 30\), state the magnitude of the friction force acting on the block.
3. Given that \(P = 80\), find the acceleration of the block.
   \includegraphics[max width=\textwidth, alt={}]{5d474771-fe32-47c6-8bf3-60ff7a25dd12-05_2484_1709_223_153}

2 A wooden block, of mass 4 kg , is placed on a rough horizontal surface. The coefficient of friction between the block and the surface is 0.3 . A horizontal force, of magnitude 30 newtons, acts on the block and causes it to accelerate. \includegraphics[max width=\textwidth, alt={}, center]{7ac7dfd0-4c3e-4eb7-920f-ce5b24ad1281-2_111_771_1146_639}

1. Draw a diagram to show all the forces acting on the block.
2. Calculate the magnitude of the normal reaction force acting on the block.
3. Find the magnitude of the friction force acting on the block.
4. Find the acceleration of the block.

9 A uniform rod, \(P Q\), of length \(2 a\), rests with one end, \(P\), on rough horizontal ground and a point \(T\) resting on a rough fixed prism of semicircular cross-section of radius \(a\), as shown in the diagram. The rod is in a vertical plane which is parallel to the prism's cross-section. The coefficient of friction at both \(P\) and \(T\) is \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-20_451_1093_477_475} The rod is on the point of slipping when it is inclined at an angle of \(30 ^ { \circ }\) to the horizontal. Find the value of \(\mu\).
[0pt] [8 marks] \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-24_2488_1728_219_141}

7. \begin{figure}[h] \end{figure} A wooden crate of mass 20 kg is pulled in a straight line along a rough horizontal floor using a handle attached to the crate.
The handle is inclined at an angle \(\alpha\) to the floor, as shown in Figure 1, where \(\tan \alpha = \frac { 3 } { 4 }\) The tension in the handle is 40 N .
The coefficient of friction between the crate and the floor is 0.14
The crate is modelled as a particle and the handle is modelled as a light rod.
Using the model,

1. find the acceleration of the crate. The crate is now pushed along the same floor using the handle. The handle is again inclined at the same angle \(\alpha\) to the floor, and the thrust in the handle is 40 N as shown in Figure 2 below. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{65e4b254-fb7b-45c2-9702-32f034018193-20_220_923_1457_571} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure}
2. Explain briefly why the acceleration of the crate would now be less than the acceleration of the crate found in part (a).

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.

\includegraphics{figure_3} 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° 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]
2. the value of \(X\). [4]

The force of magnitude \(X\) newtons is now removed.

1. Show that \(P\) remains in equilibrium on the plane. [4]

\includegraphics{figure_2} A parcel of weight \(10\) N lies on a rough plane inclined at an angle of \(30°\) to the horizontal. A horizontal force of magnitude \(P\) newtons acts on the parcel, as shown in Figure 2. The parcel is in equilibrium and on the point of slipping up the plane. The normal reaction of the plane on the parcel is \(18\) N. The coefficient of friction between the parcel and the plane is \(\mu\). Find

1. the value of \(P\), [4]
2. the value of \(\mu\). [5]

The horizontal force is removed.

1. Determine whether or not the parcel moves. [5]

\includegraphics{figure_3} A box of mass 30 kg is being pulled along rough horizontal ground at a constant speed using a rope. The rope makes an angle of 20° with the ground, as shown in Figure 3. The coefficient of friction between the box and the ground is 0.4. The box is modelled as a particle and the rope as a light, inextensible string. The tension in the rope is \(P\) newtons.

1. Find the value of \(P\). [8]

The tension in the rope is now increased to 150 N.

1. Find the acceleration of the box. [6]

\includegraphics{figure_2} A small packet of mass 0.3 kg rests on a rough horizontal surface. The coefficient of friction between the packet and the surface is \(\frac{1}{4}\). Two strings are attached to the packet, making angles of 45° and 30° with the horizontal, and when forces of magnitude 2 N and \(F\) N are exerted through the strings as shown, the packet is on the point of moving in the direction \(\overrightarrow{AB}\). Find the value of \(F\). \hfill [7 marks]

Two heavy boxes, \(M\) and \(N\), are connected securely by a length of rope. The mass of \(M\) is 50 kilograms. The mass of \(N\) is 80 kilograms. \(M\) is placed near the bottom of a rough slope. The slope is inclined at 60° above the horizontal. The rope is passed over a smooth pulley at the top end of the slope so that \(N\) hangs with the rope vertical. The boxes are initially held in this position, with the rope taut and running parallel to the line of greatest slope, as shown in the diagram below. \includegraphics{figure_21} When the boxes are released, \(M\) moves up the slope as \(N\) descends, with acceleration \(a\) m s\(^{-2}\) The tension in the rope is \(T\) newtons.

1. Explain why the equation of motion for \(N\) is $$80g - T = 80a$$ [1 mark]
2. Show that the normal reaction force between \(M\) and the slope is \(25g\) newtons. [1 mark]
3. The coefficient of friction, \(\mu\), between the slope and \(M\) is such that \(0 \leq \mu \leq 1\) Show that $$a \geq \frac{(11 - 5\sqrt{3})g}{26}$$ [6 marks]
4. State one modelling assumption you have made throughout this question. [1 mark]

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]