Friction

1 A block of mass 400 kg rests in limiting equilibrium on horizontal ground. A force of magnitude 2000 N acts on the block at an angle of \(15 ^ { \circ }\) to the upwards vertical. Find the coefficient of friction between the block and the ground, correct to 2 significant figures.

3 A string is attached to a block of mass 4 kg which rests in limiting equilibrium on a rough horizontal table. The string makes an angle of \(24 ^ { \circ }\) above the horizontal and the tension in the string is 30 N .

1. Draw a diagram showing all the forces acting on the block.
2. Find the coefficient of friction between the block and the table.

A crate of mass 300 kg is at rest on rough horizontal ground. The coefficient of friction between the crate and the ground is 0.5. A force of magnitude \(X\) N, acting at an angle \(\alpha\) above the horizontal, is applied to the crate, where \(\sin \alpha = 0.28\). Find the greatest value of \(X\) for which the crate remains at rest. [5]

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]

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.

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 \(\mu\).

3 A ring of mass 0.3 kg is threaded on a horizontal rough rod. The coefficient of friction between the ring and the rod is 0.8 . A force of magnitude 8 N acts on the ring. This force acts at an angle of \(10 ^ { \circ }\) above the horizontal in the vertical plane containing the rod. Find the time taken for the ring to move, from rest, 0.6 m along the rod.

A block of mass \(2\) kg is placed on a rough horizontal surface. The coefficient of friction between the block and the surface is \(0.3\).

1. Calculate the maximum frictional force that can act on the block. [2]
2. A horizontal force of \(5\) N is applied to the block. Calculate the acceleration of the block. [3]

\includegraphics{figure_1} A boy is pulling a sledge of mass 8 kg in a straight line at a constant speed across rough horizontal ground by means of a rope. The rope is inclined at 30° to the ground, as shown in Figure 1. The coefficient of friction between the sledge and the ground is \(\frac{1}{5}\). By modelling the sledge as a particle and the rope as a light inextensible string, find the tension in the rope. [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]

1. Show that \(\mu \geqslant \frac { 6 } { 17 }\). When the applied force acts upwards as in Fig. 2 the block slides along the floor.
2. Find another inequality for \(\mu\).

\includegraphics{figure_1} A small ring \(P\) of mass \(0.03\) kg is threaded on a rough vertical rod. A light inextensible string is attached to the ring and is pulled upwards at an angle of \(15°\) to the horizontal. The tension in the string is \(2.5\) N (see diagram). The ring is in limiting equilibrium and on the point of sliding up the rod. Find the coefficient of friction between the ring and the rod. [4]

5 \includegraphics[max width=\textwidth, alt={}, center]{f1f33ef0-0d4d-4a4a-aadb-28de8dc0ea8d-08_483_840_258_649} The diagram shows a particle \(A\), of mass 1.2 kg , which lies on a plane inclined at an angle of \(40 ^ { \circ }\) to the horizontal and a particle \(B\), of mass 1.6 kg , which lies on a plane inclined at an angle of \(50 ^ { \circ }\) to the horizontal. The particles are connected by a light inextensible string which passes over a small smooth pulley \(P\) fixed at the top of the planes. The parts \(A P\) and \(B P\) of the string are taut and parallel to lines of greatest slope of the respective planes. The two planes are rough, with the same coefficient of friction, \(\mu\), between the particles and the planes. Find the value of \(\mu\) for which the system is in limiting equilibrium.

5 A block of mass \(m \mathrm {~kg}\) is at rest on a horizontal plane. The coefficient of friction between the block and the plane is 0.2 .

1. When a horizontal force of magnitude 5 N acts on the block, the block is on the point of slipping. Find the value of \(m\).
2. \includegraphics[max width=\textwidth, alt={}, center]{8ee41313-b516-48cb-bc87-cd8e54245d28-3_312_711_1244_758} When a force of magnitude \(P \mathrm {~N}\) acts downwards on the block at an angle \(\alpha\) to the horizontal, as shown in the diagram, the frictional force on the block has magnitude 6 N and the block is again on the point of slipping. Find
   1. the value of \(\alpha\) in degrees,
   2. the value of \(P\).

1 A crate of mass 500 kg is being pulled along rough horizontal ground by a horizontal rope attached to a winch. The winch produces a constant pulling force of 2500 N and the crate is moving at constant speed. Find the coefficient of friction between the crate and the ground.

A stone that weighs 15 kg is propelled across the ice in an ice rink with an initial speed of \(4 \text{ m s}^{-1}\). The coefficient of friction between the stone and the ice is \(0.017\). How far does the stone slide before it comes to rest? [5]

The diagram shows a particle of mass \(0.7\) kg resting on a rough horizontal table. The coefficient of friction between the particle and the table is \(0.25\). A light elastic string, of natural length \(50\) cm and modulus of elasticity \(6.86\) N, is attached to the particle. The string is kept at an angle of \(60°\) to the horizontal and is gradually extended by pulling on it until the particle moves. Show that the particle starts to move when the extension in the string is \(17\) cm. \includegraphics{figure_2} [8 marks]

1. Find the value of \(\theta\). At time 4.8 s after leaving \(A\), the particle comes to rest at \(C\).
2. Find the coefficient of friction between \(P\) and the rough part of the plane.

A wooden crate rests on a rough horizontal surface. The coefficient of friction between the crate and the surface is 0.6 A forward force acts on the crate, parallel to the surface. When this force is 600 N, the crate is on the point of moving. Find the weight of the crate. Circle your answer. [1 mark] 1000 N \quad 100 kg \quad 360 N \quad 36 kg

\includegraphics{figure_2} A rough plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\) A particle \(P\) of mass 2 kg is held in equilibrium on the plane by a horizontal force of magnitude \(X\) newtons, as shown in Figure 2. The force acts in a vertical plane which contains a line of greatest slope of the inclined plane.

1. Show that when \(X = 14.7\) there is no frictional force acting on \(P\) [3] The coefficient of friction between \(P\) and the plane is 0.5
2. Find the smallest possible value of \(X\). [8]

4 \includegraphics[max width=\textwidth, alt={}, center]{2bd9f770-65b1-48c2-bf58-24e732bb6988-2_608_723_1247_712} A particle \(P\) has weight 10 N and is in limiting equilibrium on a rough horizontal table. The forces shown in the diagram represent the weight of \(P\), an applied force of magnitude 4 N acting on \(P\) in a direction at \(30 ^ { \circ }\) above the horizontal, and the contact force exerted on \(P\) by the table (the resultant of the frictional and normal components) of magnitude \(C \mathrm {~N}\).

1. Find the value of \(C\).
2. Find the coefficient of friction between \(P\) and the table.

4 Two particles \(P\) and \(Q\), of masses 6 kg and 2 kg respectively, lie at rest 12.5 m apart on a rough horizontal plane. The coefficient of friction between each particle and the plane is 0.4 . Particle \(P\) is projected towards \(Q\) with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that the speed of \(P\) immediately before the collision with \(Q\) is \(10 \sqrt { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   In the collision \(P\) and \(Q\) coalesce to form particle \(R\).
2. Find the loss of kinetic energy due to the collision.
   The coefficient of friction between \(R\) and the plane is 0.4 .
3. Find the distance travelled by particle \(R\) before coming to rest.

9 A particle of mass \(m \mathrm {~kg}\) rests in equilibrium on a rough horizontal table. There is a string attached to the particle. The tension in the string is \(T \mathrm {~N}\) at an angle of \(\theta\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{2e3f056c-58a2-4466-94ea-3fb873e54752-4_205_547_1027_799}

1. Copy and complete the diagram to show all the forces acting on the particle.
2. The coefficient of friction between the particle and the table is \(\mu\) and the particle is on the point of slipping. Show that \(T = \frac { \mu m g } { \cos \theta + \mu \sin \theta }\).
3. Given that \(\mu = 0.75\), find the value of \(\theta\) for which \(T\) is a minimum.

7 \begin{figure}[h] \end{figure} A rectangular block \(B\) of weight 12 N lies in limiting equilibrium on a horizontal surface. A horizontal force of 4 N and a coplanar force of 5 N inclined at \(60 ^ { \circ }\) to the vertical act on \(B\) (see Fig. 1).

1. Find the coefficient of friction between \(B\) and the surface. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4b703cf9-b3d3-4210-b57b-89136595f8a5-04_307_751_1000_696} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} \(B\) is now cut horizontally into two smaller blocks. The upper block has weight 9 N and the lower block has weight 3 N . The 5 N force now acts on the upper block and the 4 N force now acts on the lower block (see Fig. 2). The coefficient of friction between the two blocks is \(\mu\).
2. Given that the upper block is in limiting equilibrium, find \(\mu\).
3. Given instead that \(\mu = 0.1\), find the accelerations of the two blocks.

7 Two forces, which act in a vertical plane, are applied to a crate. The crate has mass 50 kg , and is initially at rest on a rough horizontal surface. One force has magnitude 80 N and acts at an angle of \(30 ^ { \circ }\) to the horizontal and the other has magnitude 40 N and acts at an angle of \(20 ^ { \circ }\) to the horizontal. The forces are shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{01338c87-302c-420f-a473-39b0796ccaed-16_241_999_493_523} The coefficient of friction between the crate and the surface is 0.6 . Model the crate as a particle.

1. Draw a diagram to show the forces acting on the crate.
2. Find the magnitude of the normal reaction force acting on the crate.
3. Does the crate start to move when the two forces are applied to the crate? Show all your working.
4. State one aspect of the possible motion of the crate that is ignored by modelling it as a particle.
   [0pt] [1 mark]