3.03r Friction: concept and vector form

4 A rigid thin uniform rod AB with length 2.4 m and weight 30 N is used in different situations.

1. In the first situation, the rod rests on a small support 0.6 m from B and is held horizontally in equilibrium by a vertical string attached to A, as shown in Fig. 4.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{42b6ee17-f0ae-4687-8392-281ba724a607-5_196_707_456_680} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure} Calculate the tension in the string and the force of the support on the rod.
2. In the second situation, the rod rests in equilibrium on the point of slipping with end A on a horizontal floor and the rod resting at P on a fixed block of height 0.9 m , as shown in Fig. 4.2. The rod is perpendicular to the edge of the block on which it rests and is inclined at \(\theta\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{42b6ee17-f0ae-4687-8392-281ba724a607-5_208_746_1101_657} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure} (A) Suppose that the contact between the block and the rod is rough with coefficient of friction 0.6 and contact between the end A and the floor is smooth. Show that \(\tan \theta = 0.6\).
   (B) Suppose instead that the contact between the block and the rod is smooth and the contact between the end A and the floor is rough. The rod is now in limiting equilibrium at a different angle \(\theta\) such that the distance AP is 1.5 m . Calculate the normal reaction of the block on the rod. Calculate the coefficient of friction between the rod and the floor.

3

1. A car of mass 900 kg is travelling at a steady speed of \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a hill inclined at arcsin 0.1 to the horizontal. The power required to do this is 20 kW . Calculate the resistance to the motion of the car.
2. A small box of mass 11 kg is placed on a uniform rough slope inclined at arc \(\cos \frac { 12 } { 13 }\) to the horizontal. The coefficient of friction between the box and the slope is \(\mu\).
   1. Show that if the box stays at rest then \(\mu \geqslant \frac { 5 } { 12 }\). For the remainder of this question, the box moves on a part of the slope where \(\mu = 0.2\).
      The box is projected up the slope from a point P with an initial speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It travels a distance of 1.5 m along the slope before coming instantaneously to rest. During this motion, the work done against air resistance is 6 joules per metre.
   2. Calculate the value of \(v\). As the box slides back down the slope, it passes through its point of projection P and later reaches its initial speed at a point Q . During this motion, once again the work done against air resistance is 6 joules per metre.
   3. Calculate the distance PQ.

3 A uniform plank is 2.8 m long and has weight 200 N . The centre of mass is G.

1. Fig. 3.1 shows the plank horizontal and in equilibrium, resting on supports at A and B . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-5_229_1125_434_459} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
   \end{figure} Calculate the reactions of the supports on the plank at A and at B .
2. Fig. 3.2 shows the plank horizontal and in equilibrium between a support at C and a peg at D . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-5_236_1141_993_461} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure} Calculate the reactions of the support and the peg on the plank at C and at D , showing the directions of these forces on a diagram. Fig. 3.3 shows the plank in equilibrium between a support at P and a peg at Q . The plank is inclined at \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8d4aeab2-332a-442f-b1e7-0bbf8a945f0f-5_424_1099_1692_475} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
   \end{figure}
3. Calculate the normal reactions at P and at Q .
4. Just one of the contacts is rough. Determine which one it is if the value of the coefficient of friction is as small as possible. Find this value of the coefficient of friction.

4 Jack and Jill are raising a pail of water vertically using a light inextensible rope. The pail and water have total mass 20 kg . In parts (i) and (ii), all non-gravitational resistances to motion may be neglected.

1. How much work is done to raise the pail from rest so that it is travelling upwards at \(0.5 \mathrm {~ms} ^ { - 1 }\) when at a distance of 4 m above its starting position?
2. What power is required to raise the pail at a steady speed of \(0.5 \mathrm {~ms} ^ { - 1 }\) ? Jack falls over and hurts himself. He then slides down a hill.
   His mass is 35 kg and his speed increases from \(1 \mathrm {~ms} ^ { - 1 }\) to \(3 \mathrm {~ms} ^ { - 1 }\) while descending through a vertical height of 3 m .
3. How much work is done against friction? In Jack's further motion, he slides down a slope at an angle \(\alpha\) to the horizontal where \(\sin \alpha = 0.1\). The frictional force on him is now constant at 150 N . For this part of the motion, Jack's initial speed is \(3 \mathrm {~ms} ^ { - 1 }\).
4. How much further does he slide before coming to rest?

3 \begin{enumerate}[label=(\alph*)] \item Fig. 3.1 shows a framework in a vertical plane constructed of light, rigid rods \(\mathrm { AB } , \mathrm { BC } , \mathrm { AD }\) and BD . The rods are freely pin-jointed to each other at \(\mathrm { A } , \mathrm { B }\) and D and to a vertical wall at C and D. There are vertical loads of \(L \mathrm {~N}\) at A and \(3 L \mathrm {~N}\) at B . Angle DAB is \(30 ^ { \circ }\), angle DBC is \(60 ^ { \circ }\) and ABC is a straight, horizontal line. \begin{figure}[h] \end{figure}

1. Draw a diagram showing the loads and the internal forces in the four rods.
2. Find the internal forces in the rods in terms of \(L\), stating whether each rod is in tension or in thrust (compression). [You may leave answers in surd form. Note that you are not required to find the external forces acting at C and at D.]

\item Fig. 3.2 shows uniform beams PQ and QR , each of length 2 lm and of weight \(W \mathrm {~N}\). The beams are freely hinged at Q and are in equilibrium on a rough horizontal surface when inclined at \(60 ^ { \circ }\) to the horizontal. You are given that the total force acting at Q on QR due to the hinge is horizontal. This force, \(U \mathrm {~N}\), is shown in Fig. 3.3. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Show that the frictional force between the floor and each beam is \(\frac { \sqrt { 3 } } { 6 } W \mathrm {~N}\).

1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{130d0f63-83ac-484f-9c0b-a633e0d87743-5_641_885_269_671} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} A small sphere of mass 0.15 kg is attached to one end, B, of a light, inextensible piece of fishing line of length 2 m . The other end of the line, A , is fixed and the line can swing freely. The sphere swings with the line taut from a point where the line is at an angle of \(40 ^ { \circ }\) with the vertical, as shown in Fig. 4.
   1. Explain why no work is done on the sphere by the tension in the line.
   2. Show that the sphere has dropped a vertical distance of about 0.4679 m when it is at the lowest point of its swing and calculate the amount of gravitational potential energy lost when it is at this point.
   3. Assuming that there is no air resistance and that the sphere swings from rest from the position shown in Fig. 4, calculate the speed of the sphere at the lowest point of its swing.
   4. Now consider the case where
      Calculate the speed of the sphere at the lowest point of its swing.
   5. A block of mass 3 kg slides down a uniform, rough slope that is at an angle of \(30 ^ { \circ }\) to the horizontal. The acceleration of the block is \(\frac { 1 } { 8 } g\). Show that the coefficient of friction between the block and the slope is \(\frac { 1 } { 4 } \sqrt { 3 }\).

3 A non-uniform beam AB has weight 85 N . The length of the beam is 5 m and its centre of mass is 3 m from A . In this question all the forces act in the same vertical plane. Fig. 3.1 shows the beam in horizontal equilibrium, supported at its ends. \begin{figure}[h] \end{figure}

1. Calculate the reactions of the supports on the beam. Using a smooth hinge, the beam is now attached at A to a vertical wall. The beam is held in equilibrium at an angle \(\alpha\) to the horizontal by means of a horizontal force of magnitude 27.2 N acting at B . This situation is shown in Fig. 3.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{81efb50d-c89d-4ce1-94d7-592c946f6176-4_725_675_1153_347} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{81efb50d-c89d-4ce1-94d7-592c946f6176-4_732_565_1153_1231} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
   \end{figure}
2. Show that \(\tan \alpha = \frac { 15 } { 8 }\). The hinge and 27.2 N force are removed. The beam now rests with B on a rough horizontal floor and A on a smooth vertical wall, as shown in Fig. 3.3. It is at the same angle \(\alpha\) to the horizontal. There is now a force of 34 N acting at right angles to the beam at its centre in the direction shown. The beam is in equilibrium and on the point of slipping.
3. Draw a diagram showing the forces acting on the beam. Show that the frictional force acting on the beam is 7.4 N .
   Calculate the value of the coefficient of friction between the beam and the floor.

4 A box of mass 16 kg is on a uniformly rough horizontal floor with an applied force of fixed direction but varying magnitude \(P\) N acting as shown in Fig. 4. You may assume that the box does not tip for any value of \(P\). The coefficient of friction between the box and the floor is \(\mu\). \begin{figure}[h] \end{figure} Initially the box is at rest and on the point of slipping with \(P = 58\).

1. Show that \(\mu\) is about 0.25 . In the rest of this question take \(\mu\) to be exactly 0.25 .
   The applied force on the box is suddenly increased so that \(P = 70\) and the box moves against friction with the floor and another horizontal retarding force, \(S\). The box reaches a speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from rest after 5 seconds; during this time the box slides 3 m .
2. Calculate the work done by the applied force of 70 N and also the average power developed by this force over the 5 seconds.
3. By considering the values of time, distance and speed, show that an object starting from rest that travels 3 m while reaching a speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 5 seconds cannot be moving with constant acceleration. The reason that the acceleration is not constant is that the retarding force \(S\) is not constant.
4. Calculate the total work done by the retarding force \(S\).

1

1. A stone of mass 0.6 kg falls vertically 1.5 m from A to B against resistance. Its downward speeds at A and \(B\) are \(5.5 \mathrm {~ms} ^ { - 1 }\) and \(7.5 \mathrm {~ms} ^ { - 1 }\) respectively.
   1. Calculate the change in kinetic energy and the change in gravitational potential energy of the stone as it falls from A to B .
   2. Calculate the work done against resistance to the motion of the stone as it falls from A to B .
   3. Assuming the resistive force is constant, calculate the power with which the resistive force is retarding the stone when it is at A .
2. A uniform plank is inclined at \(40 ^ { \circ }\) to the horizontal. A box of mass 0.8 kg is on the point of sliding down it. The coefficient of friction between the box and the plank is \(\mu\).
   1. Show that \(\mu = \tan 40 ^ { \circ }\). The plank is now inclined at \(20 ^ { \circ }\) to the horizontal.
   2. Calculate the work done when the box is pushed 3 m up the plank, starting and finishing at rest.

5 \includegraphics[max width=\textwidth, alt={}, center]{af1f9f1b-f6c0-4044-9864-5b9ce309d3fa-03_462_1109_283_569} Two uniform rods \(A B\) and \(B C\) have weights 64 N and 40 N respectively. The rods are freely jointed to each other at \(B\). The rod \(A B\) is freely jointed to a fixed point on horizontal ground at \(A\) and the rod \(B C\) rests against a vertical wall at \(C\). The rod \(B C\) is 1.8 m long and is horizontal. A particle of weight 9 N is attached to the rod \(B C\) at the point 0.4 m from \(C\). The point \(A\) is 1.2 m below the level of \(B C\) and 3.8 m from the wall (see diagram). The system is in equilibrium.

1. Show that the magnitude of the frictional force at \(C\) is 27 N .
2. Calculate the horizontal and vertical components of the force exerted on \(A B\) at \(B\).
3. Given that friction is limiting at \(C\), find the coefficient of friction between the \(\operatorname { rod } B C\) and the wall.

6 Fig. 6.1 shows a cross-section through a block of mass 5 kg which is on top of a trolley of mass 11 kg . The trolley is on top of a smooth horizontal surface. The coefficient of friction between the block and the trolley is 0.3 . Throughout this question you may assume that there are no other resistances to motion on either the block or the trolley. \begin{figure}[h] \end{figure} Initially, both the block and trolley are at rest. A constant force of magnitude 50 N is now applied horizontally to the trolley, as shown in Fig. 6.1.

1. Show that in the subsequent motion the block will slide.
2. Find the acceleration of
   1. the block,
   2. the trolley. The same block and trolley are again at rest. An obstruction, in the form of a fixed horizontal pole, is placed in front of the block, the pole is 91 cm above the trolley and the width of the block is 56 cm as shown in Fig. 6.2, as well as the forward direction of motion. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-6_426_1324_1793_269} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
      \end{figure} It is given that the block is uniform and that the contact between the pole and the block is smooth. A small horizontal force is now applied to the trolley in the forward direction of motion and gradually increased.
3. Determine whether the block will topple or slide.

4 Fig. 4 shows a thin rigid non-uniform rod PQ of length 0.5 m . End P rests on a rough circular peg. A force of \(T \mathrm {~N}\) acts at the end Q at \(60 ^ { \circ }\) to QP . The weight of the rod is 40 N and its centre of mass is 0.3 m from P . \begin{figure}[h] \end{figure} The rod does not slip on the peg and is in equilibrium with PQ horizontal.

1. Show that the vertical component of \(T\) is 24 N .
2. \(F\) is the contact force at P between the rod and the peg. Find
   • the vertical component of \(F\),
   • the horizontal component of \(F\).
   • Given that the rod is about to slip on the peg, find the coefficient of friction between the rod and the peg.

3 The diagram shows two blocks P and Q of masses 0.5 kg and 2 kg respectively, on a horizontal surface. The points \(\mathrm { A } , \mathrm { B }\) and C lie on the surface in a straight line. There is a wall at C . The surface between B and C is smooth, and the surface between A and B is rough, such that the coefficient of friction between P and AB is \(\frac { 2 } { 3 }\). \includegraphics[max width=\textwidth, alt={}, center]{cad8805d-59f6-4ed2-81f4-9e8c749461f5-3_229_1271_1601_278} P is projected with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) directly towards Q , which is at rest. As a result of the collision between P and Q, P changes direction and subsequently comes to rest at A. You may assume that P only collides with Q once.

1. Determine the coefficient of restitution between P and Q .
2. Calculate the impulse exerted on P by Q during their collision. After colliding with P , Q strikes the wall, which is perpendicular to the direction of the motion of Q , and comes to rest exactly halfway between A and B . The collision between Q and the wall is perfectly elastic.
3. Determine the coefficient of friction between Q and AB .

3 The diagram shows the three points A, B and C that lie along a line of greatest slope on a rough plane which is inclined at an angle of \(25 ^ { \circ }\) to the horizontal. \includegraphics[max width=\textwidth, alt={}, center]{0a790ad0-7eda-40f1-9894-f156766ae46f-3_392_1136_383_242} A block of mass 6 kg is placed at B and is projected up the plane towards C with an initial speed of \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The block travels 3.5 m before coming instantaneously to rest at C , before sliding back down the plane. When the block is sliding back down the plane it attains its initial speed at A , which lies \(x \mathrm {~m}\) down the plane from B . It is given that the work done against resistance throughout the motion is 4 joules per metre.

1. Use an energy method to determine the following.
   1. The value of \(u\)
   2. The value of \(x\) A student claims that half of the energy lost due to resistances is accounted for by friction between the block and the plane, and the other half by air resistance.
2. Assuming that the student's claim is correct, determine the coefficient of friction between the block and the plane.

8 A capsule consists of a uniform hollow right circular cylinder of radius \(r\) and length \(2 h\) attached to two uniform hollow hemispheres of radius \(r\).
The centres of the plane faces of the hemispheres coincide with the centres, A and B , of the ends of the cylinder. \begin{figure}[h] \end{figure} Fig. 8 represents a vertical cross-section through a plane of symmetry of the capsule as it rests in limiting equilibrium with a point C of one hemisphere on a rough horizontal floor and a point D of the other hemisphere against a rough vertical wall. The total weight of the capsule is \(W\) and acts at a point midway between A and B . The plane containing \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D is vertical, with AB making an acute angle \(\theta\) with the downward vertical.

1. Complete the copy of Fig. 8 in the Printed Answer Booklet to show all the remaining forces acting on the capsule. The coefficient of friction at each point of contact is \(\frac { 1 } { 3 }\).
2. By resolving vertically and horizontally, determine the magnitude of the normal contact force between the floor and the capsule in terms of \(W\).
3. By determining an expression for \(r\) in terms of \(h\) and \(\theta\), show that \(\tan \theta > \frac { 3 } { 4 }\).

6. \includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-7_606_506_365_781} A uniform ladder \(A B\), of mass 10 kg and length 5 m , rests with one end \(A\) against a smooth vertical wall and the other end \(B\) on rough horizontal ground. The ladder is inclined at an angle \(\theta\) to the horizontal. A woman of mass 75 kg stands on the ladder so that her weight acts at a distance \(x \mathrm {~m}\) from \(B\).

1. Show that the frictional force, \(F \mathrm {~N}\), between the ladder and the horizontal ground is given by $$F = 5 g \cot \theta ( 1 + 3 x ) .$$ For safety reasons, it is recommended that \(\theta\) is chosen such that the ratio \(C B : C A\) is \(1 : 4\).
2. Determine the least value of the coefficient of friction such that the ladder will not slip however high the woman climbs.
3. State one modelling assumption that you have made in your solution.

1. \begin{figure}[h] \end{figure} A small book of mass \(m\) is held on a rough straight desk lid which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The book is released from rest at a distance of 0.5 m from the edge of the desk lid, as shown in Figure 1. The book slides down the desk lid and then hits the floor that is 0.8 m below the edge of the desk lid. The coefficient of friction between the book and the desk lid is 0.4 The book is modelled as a particle which, after leaving the desk lid, is assumed to move freely under gravity.

1. Find, in terms of \(m\) and \(g\), the magnitude of the normal reaction on the book as it slides down the desk lid.
2. Use the work-energy principle to find the speed of the book as it hits the floor.

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

A particle \(P\) is held at rest at a point \(A\) on the plane.
The particle \(P\) is then projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\), up a line of greatest slope of the plane. In an initial model, the plane is modelled as being smooth and air resistance is modelled as being negligible. Using this model and the principle of conservation of mechanical energy,

1. find the speed of \(P\) at the instant when it has travelled a distance \(\frac { 25 } { 6 } \mathrm {~m}\) up the plane from \(A\). In a refined model, the plane is now modelled as being rough, with the coefficient of friction between \(P\) and the plane being \(\frac { 3 } { 5 }\) Air resistance is still modelled as being negligible.
   Using this refined model and the work-energy principle,
2. find the speed of \(P\) at the instant when it has travelled a distance \(\frac { 25 } { 6 } \mathrm {~m}\) up the plane from \(A\).

1. A cyclist is travelling around a circular track which is banked at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\)

The cyclist moves with constant speed in a horizontal circle of radius \(r\).
In an initial model,

• the cyclist and her cycle are modelled as a particle
• the track is modelled as being rough so that there is sideways friction between the tyres of the cycle and the track, with coefficient of friction \(\mu\), where \(\mu < \frac { 4 } { 3 }\) Using this model, the maximum speed that the cyclist can travel around the track in a horizontal circle of radius \(r\), without slipping sideways, is \(V\).
  1. Show that \(V = \sqrt { \frac { ( 3 + 4 \mu ) r g } { 4 - 3 \mu } }\)

In a new simplified model,

• the cyclist and her cycle are modelled as a particle
• the motion is now modelled so that there is no sideways friction between the tyres of the cycle and the track

Using this new model, the speed that the cyclist can travel around the track in a horizontal circle of radius \(r\), without slipping sideways, is \(U\).

Find \(U\) in terms of \(r\) and \(g\).

Show that \(U < V\).

1. A girl is cycling round a circular track.

The girl and her bicycle have a combined mass of 55 kg .
The coefficient of friction between the track surface and the tyres of the bicycle is \(\mu\).
The track is banked at an angle of \(15 ^ { \circ }\) to the horizontal.
The girl and her bicycle are modelled as a particle moving in a horizontal circle of radius 50 m
The minimum speed at which the girl can cycle round this circle without slipping is \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Using the model, find the value of \(\mu\).

4. \begin{figure}[h] \end{figure} A uniform solid cylinder of base radius \(r\) and height \(\frac { 4 } { 3 } r\) has the same density as a uniform solid hemisphere of radius \(r\). The plane face of the hemisphere is joined to a plane face of the cylinder to form the composite solid \(S\) shown in Figure 3. The point \(O\) is the centre of the plane face of \(S\).

1. Show that the distance from \(O\) to the centre of mass of \(S\) is \(\frac { 73 } { 72 } r\) The solid \(S\) is placed with its plane face on a rough horizontal plane. The coefficient of friction between \(S\) and the plane is \(\mu\). A horizontal force \(P\) is applied to the highest point of \(S\). The magnitude of \(P\) is gradually increased.
2. Find the range of values of \(\mu\) for which \(S\) will slide before it starts to tilt.

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.

8 A rough slope is inclined at an angle of \(25 ^ { \circ }\) to the horizontal. A box of weight 80 newtons is on the slope. A rope is attached to the box and is parallel to the slope. The tension in the rope is of magnitude \(T\) newtons. The diagram shows the slope, the box and the rope. \includegraphics[max width=\textwidth, alt={}, center]{c220e6c4-2676-4022-8301-7d720dc082b2-7_307_469_500_840}

1. The box is held in equilibrium by the rope.
   1. Show that the normal reaction force between the box and the slope is 72.5 newtons, correct to three significant figures.
   2. The coefficient of friction between the box and the slope is 0.32 . Find the magnitude of the maximum value of the frictional force which can act on the box.
   3. Find the least possible tension in the rope to prevent the box from moving down the slope.
   4. Find the greatest possible tension in the rope.
   5. Show that the mass of the box is approximately 8.16 kg .
2. The rope is now released and the box slides down the slope. Find the acceleration of the box.

8 A crate, of mass 200 kg , is initially at rest on a rough horizontal surface. A smooth ring is attached to the crate. A light inextensible rope is passed through the ring, and each end of the rope is attached to a tractor. The lower part of the rope is horizontal and the upper part is at an angle of \(20 ^ { \circ }\) to the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{fe8c1ea4-cf4d-4741-8af5-03e8c2c88559-5_344_1186_518_420} When the tractor moves forward, the crate accelerates at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The coefficient of friction between the crate and the surface is 0.4 . Assume that the tension, \(T\) newtons, is the same in both parts of the rope.

1. Draw and label a diagram to show the forces acting on the crate.
2. Express the normal reaction between the surface and the crate in terms of \(T\).
3. Find \(T\).

6 A box, of mass 3 kg , is placed on a slope inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The box slides down the slope. Assume that air resistance can be ignored.

1. A simple model assumes that the slope is smooth.
   1. Draw a diagram to show the forces acting on the box.
   2. Show that the acceleration of the box is \(4.9 \mathrm {~ms} ^ { - 2 }\).
2. A revised model assumes that the slope is rough. The box slides down the slope from rest, travelling 5 metres in 2 seconds.
   1. Show that the acceleration of the box is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   2. Find the magnitude of the friction force acting on the box.
   3. Find the coefficient of friction between the box and the slope.
   4. In reality, air resistance affects the motion of the box. Explain how its acceleration would change if you took this into account.