Equilibrium with friction on horizontal surface

7
\includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-4_414_865_1512_641} Two light strings are attached to a block of mass 20 kg . The block is in equilibrium on a horizontal surface \(A B\) with the strings taut. The strings make angles of \(60 ^ { \circ }\) and \(30 ^ { \circ }\) with the horizontal, on either side of the block, and the tensions in the strings are \(T \mathrm {~N}\) and 75 N respectively (see diagram).

1. Given that the surface is smooth, find the value of \(T\) and the magnitude of the contact force acting on the block.
2. It is given instead that the surface is rough and that the block is on the point of slipping. The frictional force on the block has magnitude 25 N and acts towards \(A\). Find the coefficient of friction between the block and the surface. \footnotetext{Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. University of Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. }

5
\includegraphics[max width=\textwidth, alt={}, center]{8d64372d-0b9a-4b93-8c41-7096c813f714-4_620_623_255_760} A small block of mass 1.25 kg is on a horizontal surface. Three horizontal forces, with magnitudes and directions as shown in the diagram, are applied to the block. The angle \(\theta\) is such that \(\cos \theta = 0.28\) and \(\sin \theta = 0.96\). A horizontal frictional force also acts on the block, and the block is in equilibrium.

1. Show that the magnitude of the frictional force is 7.5 N and state the direction of this force.
2. Given that the block is in limiting equilibrium, find the coefficient of friction between the block and the surface. The force of magnitude 6.1 N is now replaced by a force of magnitude 8.6 N acting in the same direction, and the block begins to move.
3. Find the magnitude and direction of the acceleration of the block.

4
\includegraphics[max width=\textwidth, alt={}, center]{2026cad4-8494-4139-ad21-d8a17ac2b955-3_276_570_264_790} A stone slab of mass 320 kg rests in equilibrium on rough horizontal ground. A force of magnitude \(X \mathrm {~N}\) acts upwards on the slab at an angle of \(\theta\) to the vertical, where \(\tan \theta = \frac { 7 } { 24 }\) (see diagram).

1. Find, in terms of \(X\), the normal component of the force exerted on the slab by the ground.
2. Given that the coefficient of friction between the slab and the ground is \(\frac { 3 } { 8 }\), find the value of \(X\) for which the slab is about to slip.

4
\includegraphics[max width=\textwidth, alt={}, center]{c7133fc4-9a14-43fd-b5ed-788da72291cd-3_383_791_262_678} Forces of magnitude \(X \mathrm {~N}\) and 40 N act on a block \(B\) of mass 15 kg , which is in equilibrium in contact with a horizontal surface between points \(A\) and \(C\) on the surface. The forces act in the same vertical plane and in the directions shown in the diagram.

1. Given that the surface is smooth, find the value of \(X\).
2. It is given instead that the surface is rough and that the block is in limiting equilibrium. The frictional force acting on the block has magnitude 10 N in the direction towards \(A\). Find the coefficient of friction between the block and the surface.

3 A box of mass 5 kg is at rest on a rough horizontal floor.

1. Find the value of the normal reaction of the floor on the box. The box remains at rest on the floor when a force of 10 N is applied to it at an angle of \(40 ^ { \circ }\) to the upward vertical, as shown in Fig. 3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{52d6c914-b204-4587-a82e-fbab6693fcf8-2_293_472_2131_794} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure}
2. Draw a diagram showing all the forces acting on the box.
3. Calculate the new value of the normal reaction of the floor on the box and also the frictional force.

3 \begin{figure}[h] \end{figure} Fig. 3 shows a block of mass 15 kg on a rough, horizontal plane. A light string is fixed to the block at A, passes over a smooth, fixed pulley B and is attached at C to a sphere. The section of the string between the block and the pulley is inclined at \(40 ^ { \circ }\) to the horizontal and the section between the pulley and the sphere is vertical. The system is in equilibrium and the tension in the string is 58.8 N .

1. The sphere has a mass of \(m \mathrm {~kg}\). Calculate the value of \(m\).
2. Calculate the frictional force acting on the block.
3. Calculate the normal reaction of the plane on the block.

3 A box of mass 5 kg is at rest on a rough horizontal floor.

1. Find the value of the normal reaction of the floor on the box. The box remains at rest on the floor when a force of 10 N is applied to it at an angle of \(40 ^ { \circ }\) to the upward vertical, as shown in Fig. 3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{94f23528-931c-47b6-89aa-4b6edd25cc30-2_286_470_1067_803} \captionsetup{labelformat=empty} \caption{Fig. 3}
   \end{figure}
2. Draw a diagram showing all the forces acting on the box.
3. Calculate the new value of the normal reaction of the floor on the box and also the frictional force.

8 A trolley C of mass 8 kg with rusty axle bearings is initially at rest on a horizontal floor.
The trolley stays at rest when it is pulled by a horizontal string with tension 25 N , as shown in Fig. 8.1. \begin{figure}[h] \end{figure}

1. State the magnitude of the horizontal resistance opposing the pull. A second trolley D of mass 10 kg is connected to trolley C by means of a light, horizontal rod.
   The string now has tension 50 N , and is at an angle of \(25 ^ { \circ }\) to the horizontal, as shown in Fig. 8.2. The two trolleys stay at rest. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2efbb554-fe60-42ce-9213-8c66bfdb1d85-5_305_1191_1050_701} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure}
2. Calculate the magnitude of the total horizontal resistance acting on the two trolleys opposing the pull.
3. Calculate the normal reaction of the floor on trolley C . The axle bearings of the trolleys are oiled and the total horizontal resistance to the motion of the two trolleys is now 20 N . The two trolleys are still pulled by the string with tension 50 N , as shown in Fig. 8.2.
4. Calculate the acceleration of the trolleys. In a new situation, the trolleys are on a slope at \(5 ^ { \circ }\) to the horizontal and are initially travelling down the slope at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistances are 15 N to the motion of D and 5 N to the motion of C . There is no string attached. The rod connecting the trolleys is parallel to the slope. This situation is shown in Fig. 8.3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{2efbb554-fe60-42ce-9213-8c66bfdb1d85-5_355_1294_2156_429} \captionsetup{labelformat=empty} \caption{Fig. 8.3}
   \end{figure}
5. Calculate the speed of the trolleys after 2 seconds and also the force in the rod connecting the trolleys, stating whether this rod is in tension or thrust (compression).

5 Fig. 5 shows a block of mass 10 kg at rest on a rough horizontal floor. A light string, at an angle of \(30 ^ { \circ }\) to the vertical, is attached to the block. The tension in the string is 50 N . The block is in equilibrium. \begin{figure}[h] \end{figure}

1. Show all the forces acting on the block.
2. Show that the frictional force acting on the block is 25 N .
3. Calculate the normal reaction of the floor on the block.
4. Calculate the magnitude of the total force the floor is exerting on the block.

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\includegraphics[max width=\textwidth, alt={}, center]{e69f8d73-764e-4f13-a126-faec02c4ad08-07_159_896_488_244} A child attempts to drag a sledge along horizontal ground by means of a rope attached to the sledge. The rope makes an angle of \(15 ^ { \circ }\) with the horizontal (see diagram). Given that the sledge remains at rest and that the frictional force acting on the sledge is 60 N , find the tension in the rope.

3 Fig. 3 shows a particle of weight 8 N on a rough horizontal table.
The particle is being pulled by a horizontal force of 10 N .
It remains at rest in equilibrium. \begin{figure}[h] \end{figure}

1. What information given in the question, tells you that the forces shown in Fig. 3 cannot be the only forces acting on the particle?
2. The only other forces acting on the particle are due to the particle being on the table. State the types of these forces and their magnitudes.

5 A sledge of mass 8 kg is at rest on a rough horizontal surface. A child tries to move the sledge by pushing it with a pole, as shown in the diagram, but the sledge does not move. The pole is at an angle of \(30 ^ { \circ }\) to the horizontal and exerts a force of 40 newtons on the sledge.
\includegraphics[max width=\textwidth, alt={}, center]{8c6f9ac0-c24f-48d0-9fb2-883651e791d7-4_221_922_513_552} Model the sledge as a particle.

1. Draw a diagram to show the four forces acting on the sledge.
2. Show that the normal reaction force between the sledge and the surface has magnitude 98.4 N .
3. Find the magnitude of the friction force that acts on the sledge.
4. Find the least possible value of the coefficient of friction between the sledge and the surface.

2. A monk uses a small brush to clean the stone floor of a monastery by pushing the brush with a force of \(P\) Newtons at an angle of \(60 ^ { \circ }\) to the vertical. He moves the brush at a constant speed. The mass of the brush is 0.5 kg and the coefficient of friction between the brush and the floor is \(\frac { 1 } { \sqrt { 3 } }\). The brush is modelled as a particle and air resistance is ignored.

1. Show that \(P = \frac { g } { 2 }\) Newtons.
2. Explain why it is reasonable to ignore air resistance in this situation.

3 A trolley C of mass 8 kg with rusty axle bearings is initially at rest on a horizontal floor.
The trolley stays at rest when it is pulled by a horizontal string with tension 25 N , as shown in Fig. 8.1. \begin{figure}[h] \end{figure}

1. State the magnitude of the horizontal resistance opposing the pull. A second trolley D of mass 10 kg is connected to trolley C by means of a light, horizontal rod.
   The string now has tension 50 N , and is at an angle of \(25 ^ { \circ }\) to the horizontal, as shown in Fig. 8.2. The two trolleys stay at rest. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f5f9b9b7-6766-4f8e-b011-506051104123-3_297_1180_971_741} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure}
2. Calculate the magnitude of the total horizontal resistance acting on the two trolleys opposing the pull.
3. Calculate the normal reaction of the floor on trolley C . The axle bearings of the trolleys are oiled and the total horizontal resistance to the motion of the two trolleys is now 20 N . The two trolleys are still pulled by the string with tension 50 N , as shown in Fig. 8.2.
4. Calculate the acceleration of the trolleys. In a new situation, the trolleys are on a slope at \(5 ^ { \circ }\) to the horizontal and are initially travelling down the slope at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistances are 15 N to the motion of D and 5 N to the motion of C . There is no string attached. The rod connecting the trolleys is parallel to the slope. This situation is shown in Fig. 8.3. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f5f9b9b7-6766-4f8e-b011-506051104123-3_351_1285_2038_466} \captionsetup{labelformat=empty} \caption{Fig. 8.3}
   \end{figure}
5. Calculate the speed of the trolleys after 2 seconds and also the force in the rod connecting the PhysicsAptMaths, statter \&REther this rod is in tension or thrust (compression).

1 Fig. 5 shows a block of mass 10 kg at rest on a rough horizontal floor. A light string, at an angle of \(30 ^ { \circ }\) to the vertical, is attached to the block. The tension in the string is 50 N . The block is in equilibrium. \begin{figure}[h] \end{figure}

1. Show all the forces acting on the block.
2. Show that the frictional force acting on the block is 25 N .
3. Calculate the normal reaction of the floor on the block.
4. Calculate the magnitude of the total force the floor is exerting on the block.

5 Fig. 5 shows a block of mass 10 kg at rest on a rough horizontal floor. A light string, at an angle of \(30 ^ { \circ }\) to the vertical, is attached to the block. The tension in the string is 50 N . The block is in equilibrium. \begin{figure}[h] \end{figure}

1. Show all the forces acting on the block.
2. Show that the frictional force acting on the block is 25 N .
3. Calculate the normal reaction of the floor on the block.
4. Calculate the magnitude of the total force the floor is exerting on the block.

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\includegraphics[max width=\textwidth, alt={}, center]{699c5e1e-1476-42cb-b3c4-ca08c4d81cb6-07_362_754_1123_242} A block \(D\) of weight 50 N lies at rest in equilibrium on a fixed rough horizontal surface. A force of magnitude 15 N is applied to \(D\) at an angle \(\theta\) to the horizontal (see diagram).

1. Complete the diagram in the Printed Answer Booklet showing all the forces acting on \(D\). It is given that \(D\) remains at rest and the coefficient of friction between \(D\) and the surface is 0.2 .
2. Show that $$15 \cos \theta - 3 \sin \theta \leqslant 10 .$$ \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{11} \includegraphics[alt={},max width=\textwidth]{699c5e1e-1476-42cb-b3c4-ca08c4d81cb6-08_318_1488_251_244}
   \end{figure} A golfer hits a ball from a point \(A\) with a speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(15 ^ { \circ }\) above the horizontal. While the ball is in the air, it is modelled as a particle moving under the influence of gravity. Take the acceleration due to gravity to be \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The ball first lands at a point \(B\) which is 4 m below the level of \(A\) (see diagram).
3. Determine the time taken for the ball to travel from \(A\) to \(B\).
4. Determine the horizontal distance of \(B\) from \(A\).
5. Determine the direction of motion of the ball 1.5 seconds after the golfer hits the ball. The horizontal distance from \(A\) to \(B\) is found to be greater than the answer to part (b).
6. State one factor that could account for this difference.