3.03u Static equilibrium: on rough surfaces

3 \includegraphics[max width=\textwidth, alt={}, center]{67af8d98-85af-42b1-9e7f-c6380a1f8a3f-2_586_1435_1537_354} A uniform \(\operatorname { rod } P Q\) has weight 72 N . A non-uniform \(\operatorname { rod } Q R\) has weight 54 N and its centre of mass is at \(C\), where \(Q C = 2 C R\). The rods are freely jointed to each other at \(Q\). The rod \(P Q\) is freely jointed to a fixed point of a vertical wall at \(P\) and the rod \(Q R\) rests on horizontal ground at \(R\). The rod \(P Q\) is 2.8 m long and is horizontal. The point \(R\) is 1.44 m below the level of \(P Q\) and 4 m from the wall (see diagram).

1. Find the vertical component of the force exerted by the wall on \(P Q\).
2. Hence show that the normal component of the force exerted by the ground on \(Q R\) is 90 N .
3. Given that the friction at \(R\) is limiting, find the coefficient of friction between the rod \(Q R\) and the ground.

6 A uniform \(\operatorname { rod } A B\), of weight \(W\) and length \(2 l\) is in equilibrium at \(60 ^ { \circ }\) to the horizontal with \(A\) resting against a smooth vertical plane and \(B\) resting on a rough section of a horizontal plane. Another uniform rod \(C D\), of length \(\sqrt { 3 } l\) and weight \(W\), is freely jointed to the mid-point of \(A B\) at \(C\); its other end \(D\) rests on a smooth section of the horizontal plane. \(C D\) is inclined at \(30 ^ { \circ }\) to the horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{dfe477d4-eae6-40e1-b704-1a97485f4c7e-4_508_1075_438_495}

1. Show that the force exerted by the horizontal plane on \(C D\) is \(\frac { 1 } { 2 } W\). Find the normal component of the force exerted by the horizontal plane on \(A B\).
2. Find the magnitude and direction of the force exerted by \(C D\) on \(A B\).
3. Given that \(A B\) is in limiting equilibrium, find the coefficient of friction between \(A B\) and the horizontal plane.

6 Two uniform rods \(A B\) and \(B C\), each of length \(2 l\), are freely jointed at \(B\). The weight of \(A B\) is \(W\) and the weight of \(B C\) is \(2 W\). The rods are in a vertical plane with \(A\) freely pivoted at a fixed point and \(C\) resting in equilibrium on a rough horizontal plane. The normal and frictional components of the force acting on \(B C\) at \(C\) are \(R\) and \(F\) respectively. The rod \(A B\) makes an angle \(30 ^ { \circ }\) to the horizontal and the rod \(B C\) makes an angle \(60 ^ { \circ }\) to the horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{3e8248ca-74f1-443f-a5db-d7da532d2815-4_682_901_479_587}

1. By considering the equilibrium of \(\operatorname { rod } B C\), show that \(W + \sqrt { 3 } F = R\).
2. By taking moments about \(A\) for the equilibrium of the whole system, find another equation involving \(W , F\) and \(R\).
3. Given that the friction at \(C\) is limiting, calculate the value of the coefficient of friction at \(C\).

1

1. 1. Write down the dimensions of force and the dimensions of density. When a wire, with natural length \(l _ { 0 }\) and cross-sectional area \(A\), is stretched to a length \(l\), the tension \(F\) in the wire is given by $$F = \frac { E A \left( l - l _ { 0 } \right) } { l _ { 0 } }$$ where \(E\) is Young's modulus for the material from which the wire is made.
   2. Find the dimensions of Young's modulus \(E\). A uniform sphere of radius \(r\) is made from material with density \(\rho\) and Young's modulus \(E\). When the sphere is struck, it vibrates with periodic time \(t\) given by $$t = k r ^ { \alpha } \rho ^ { \beta } E ^ { \gamma }$$ where \(k\) is a dimensionless constant.
   3. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).
2. Fig. 1 shows a fixed point A that is 1.5 m vertically above a point B on a rough horizontal surface. A particle P of mass 5 kg is at rest on the surface at a distance 0.8 m from B , and is connected to A by a light elastic string with natural length 1.5 m . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c470e80e-b346-4335-9c08-beb5a46cc506-2_405_538_1338_845} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} The coefficient of friction between P and the surface is 0.4 , and P is on the point of sliding. Find the stiffness of the string.

1 The fixed point A is at a height \(4 b\) above a smooth horizontal surface, and C is the point on the surface which is vertically below A. A light elastic string, of natural length \(3 b\) and modulus of elasticity \(\lambda\), has one end attached to A and the other end attached to a block of mass \(m\). The block is released from rest at a point B on the surface where \(\mathrm { BC } = 3 b\), as shown in Fig. 1. You are given that the block remains on the surface and moves along the line BC . \begin{figure}[h] \end{figure}

1. Show that immediately after release the acceleration of the block is \(\frac { 2 \lambda } { 5 m }\).
2. Show that, when the block reaches C , its speed \(v\) is given by \(v ^ { 2 } = \frac { \lambda b } { m }\).
3. Show that the equation \(v ^ { 2 } = \frac { \lambda b } { m }\) is dimensionally consistent. The time taken for the block to move from B to C is given by \(k m ^ { \alpha } b ^ { \beta } \lambda ^ { \gamma }\), where \(k\) is a dimensionless constant.
4. Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\). When the string has natural length 1.2 m and modulus of elasticity 125 N , and the block has mass 8 kg , the time taken for the block to move from B to C is 0.718 s .
5. Find the time taken for the block to move from B to C when the string has natural length 9 m and modulus of elasticity 20 N , and the block has mass 75 kg .

3 The fixed points A and B lie on a line of greatest slope of a smooth inclined plane, with B higher than A . The horizontal distance from A to B is 2.4 m and the vertical distance is 0.7 m . The fixed point C is 2.5 m vertically above B . A light elastic string of natural length 2.2 m has one end attached to C and the other end attached to a small block of mass 9 kg which is in contact with the plane. The block is in equilibrium when it is at A, as shown in Fig. 3. \begin{figure}[h] \end{figure}

1. Show that the modulus of elasticity of the string is 37.73 N . The block starts at A and is at rest. A constant force of 18 N , acting in the direction AB , is then applied to the block so that it slides along the line AB .
2. Find the magnitude and direction of the acceleration of the block
   (A) when it leaves the point A ,
   (B) when it reaches the point B .
3. Find the speed of the block when it reaches the point B .

1. \begin{figure}[h] \end{figure} A particle of mass 0.6 kg is attached to one end of a light elastic spring of natural length 1 m and modulus of elasticity 30 N . The other end of the spring is fixed to a point \(O\) which lies on a smooth plane inclined at an angle \(\alpha\) to the horizontal where \(\tan \alpha = \frac { 3 } { 4 }\) as shown in Figure 1. The particle is held at rest on the slope at a point 1.2 m from \(O\) down the line of greatest slope of the plane.

1. Find the tension in the spring.
2. Find the initial acceleration of the particle.

7. A particle of mass 2 kg is attached to one end of a light elastic string of natural length 1 m and modulus of elasticity 50 N . The other end of the string is attached to a fixed point \(O\) on a rough horizontal plane and the coefficient of friction between the particle and the plane is \(\frac { 10 } { 49 }\). The particle is projected from \(O\) along the plane with an initial speed of \(5 \mathrm {~ms} ^ { - 1 }\).

1. Show that the greatest distance from \(O\) which the particle reaches is 1.84 m .
2. Find, correct to 2 significant figures, the speed at which the particle returns to \(O\).

3 A box weighing 130 N is on a rough plane inclined at \(12 ^ { \circ }\) to the horizontal.
The box is held at rest on the plane by the action of a force of magnitude 70 N acting up the plane in a direction parallel to a line of greatest slope of the plane.
The box is on the point of slipping up the plane.

1. Find the coefficient of friction between the box and the plane. The force of magnitude 70 N is removed.
2. Determine whether or not the box remains in equilibrium.

6 Three particles, A, B and C are in a straight line on a smooth horizontal surface.
The particles have masses \(5 \mathrm {~kg} , 3 \mathrm {~kg}\) and 1 kg respectively. Particles B and C are at rest. Particle A is projected towards B with a speed of \(u \mathrm {~ms} ^ { - 1 }\) and collides with B . The coefficient of restitution between A and B is \(\frac { 1 } { 3 }\). Particle B subsequently collides with C. The coefficient of restitution between B and C is \(\frac { 1 } { 3 }\).

1. Determine whether any further collisions occur.
2. Given that the loss of kinetic energy during the initial collision between A and B is 4.8 J , find the value of \(u\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6b27d322-417e-4cea-85cc-65d3728173c8-5_607_501_294_301} \captionsetup{labelformat=empty} \caption{Fig. 7}
   \end{figure} Fig. 7 shows a uniform rod AB of length \(4 a\) and mass \(m\).
   The end A rests against a rough vertical wall. A light inextensible string is attached to the rod at B and to a point C on the wall vertically above A , where \(\mathrm { AC } = 4 a\). The plane ABC is perpendicular to the wall and the angle ABC is \(30 ^ { \circ }\). The system is in limiting equilibrium. Find the coefficient of friction between the wall and the rod. \section*{END OF QUESTION PAPER}

6 A block B of mass \(m \mathrm {~kg}\) rests on a rough slope inclined at angle \(\alpha\) to the horizontal. The coefficient of friction between \(B\) and the slope is \(\frac { 5 } { 9 }\).

1. When B is in limiting equilibrium, show that \(\tan \alpha = \frac { 5 } { 9 }\).
2. If \(\alpha = 40 ^ { \circ }\), determine the acceleration of B down the slope. A horizontal force of magnitude \(P \mathrm {~N}\) is now applied to B , as shown in the diagram below. At first B is at rest. \includegraphics[max width=\textwidth, alt={}, center]{d1ec7861-dc8b-450b-8e05-c70479ab0dc2-7_381_410_689_242} \(P\) is gradually increased.
3. Show that, for B to slide on the slope, $$\mathrm { P } \left( \cos \alpha - \frac { 5 } { 9 } \sin \alpha \right) > \mathrm { mg } \left( \frac { 5 } { 9 } \cos \alpha + \sin \alpha \right) .$$
4. Determine, in degrees, the least value of \(\alpha\) for which B will not slide no matter how large \(P\) becomes.

5 Fig. 5.1 shows the uniform cross-section of a solid S which is formed from a cylinder by boring two cylindrical tunnels the entire way through the cylinder. The radius of S is 50 cm , and the two tunnels have radii 10 cm and 30 cm . The material making up \(S\) has uniform density.
Coordinates refer to the axes shown in Fig. 5.1 and the units are centimetres. \begin{figure}[h] \end{figure} The centre of mass of \(S\) is ( \(\mathrm { x } , \mathrm { y }\) ).

1. Show that \(\bar { x } = 12\) and find the value of \(\bar { y }\). Solid \(S\) is placed onto two rails, \(A\) and \(B\), whose point of contacts with \(S\) are at ( \(- 30 , - 40\) ) and \(( 30 , - 40 )\) as shown in Fig. 5.2. Two points, \(\mathrm { P } ( 0,50 )\) and \(\mathrm { Q } ( 0 , - 50 )\), are marked on Fig. 5.2. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 5.2} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-6_654_640_1875_251}
   \end{figure} At first, you should assume that the contact between S and the two rails is smooth.
2. Determine the angle PQ makes with the vertical, after S settles into equilibrium. For the remainder of the question, you should assume that the contact between S and A is rough, that the contact between \(S\) and \(B\) is smooth, and that \(S\) does not move when placed on the rails. Fig. 5.3 shows only the forces exerted on S by the rails. The normal contact forces exerted by A and B on S have magnitude \(R _ { \mathrm { A } } \mathrm { N }\) and \(R _ { \mathrm { B } } \mathrm { N }\) respectively. The frictional force exerted by A on S has magnitude \(F\) N. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 5.3} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-7_652_641_593_248}
   \end{figure} The weight of S is \(W \mathrm {~N}\).
3. By taking moments about the origin, express \(F\) in the form \(\lambda W\), where \(\lambda\) is a constant to be determined.
4. Given that S is in limiting equilibrium, find the coefficient of friction between A and S .

6 A uniform beam of length 6 m and mass 10 kg rests horizontally on two supports A and B , which are 3.8 m apart. A particle \(P\) of mass 4 kg is attached 1.95 m from one end of the beam (see Fig. 6.1). \begin{figure}[h] \end{figure} When A is \(x \mathrm {~m}\) from the end of the beam, the supports exert forces of equal magnitude on the beam.

1. Determine the value of \(x\). P is now removed. The same beam is placed on the supports so that B is 0.7 m from the end of the beam. The supports remain 3.8 m apart (see Fig. 6.2). \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Fig. 6.2} \includegraphics[alt={},max width=\textwidth]{a87d62b8-406d-44cd-9ffa-384005329566-8_296_1082_1162_246}
   \end{figure} The contact between A and the beam is smooth. The contact between B and the beam is rough, with coefficient of friction 0.4. A small force of magnitude \(T \mathrm {~N}\) is applied to one end of the beam. The force acts in the same vertical plane as the beam and the angle the force makes with the beam is \(60 ^ { \circ }\). As \(T\) is increased, forces \(\mathrm { T } _ { \mathrm { L } }\) and \(\mathrm { T } _ { \mathrm { S } }\) are defined in the following way.
   \section*{END OF QUESTION PAPER}

3 The diagram shows a uniform beam AB , of weight 80 N and length 7 m , resting in equilibrium in a vertical plane. The end A is in contact with a rough vertical wall, and the angle between the beam and the upward vertical is \(60 ^ { \circ }\). The beam is supported by a smooth peg at a point C , where \(\mathrm { AC } = 2 \mathrm {~m}\). \includegraphics[max width=\textwidth, alt={}, center]{a96a0ebe-8f4f-4d79-9d11-9d348ef72314-4_474_709_445_244}

1. Complete the diagram in the Printed Answer Booklet to show all the forces acting on the beam.
2. 1. Show that the magnitude of the frictional force exerted on the beam by the wall is 25 N .
   2. Hence determine the magnitude of the total contact force exerted on the beam by the wall.
3. Determine the direction of the total contact force exerted on the beam by the wall. The coefficient of friction between the beam and the wall is \(\mu\).
4. Find the range of possible values for \(\mu\).
5. Explain how your answer to part (b)(ii) would change if the peg were situated closer to A but the angle between the beam and the upward vertical remained at \(60 ^ { \circ }\).

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.

5 A uniform rod AB , of mass \(3 m\) and length \(2 a\), rests with the end A on a rough horizontal surface. A small object of mass \(m\) is attached to the rod at B . The rod is maintained in equilibrium at an angle of \(60 ^ { \circ }\) to the horizontal by a force acting at an angle of \(\theta\) to the vertical at a point C , where the distance \(\mathrm { AC } = \frac { 6 } { 5 } a\). The force acting at C is in the same vertical plane as the rod (see Fig. 5). \begin{figure}[h] \end{figure}

1. On the copy of Fig. 5 in the Printed Answer Booklet, mark all the forces acting on the rod. [2]
2. Show that the magnitude of the force acting at C can be expressed as \(\frac { 25 m g } { 6 ( \cos \theta + \sqrt { 3 } \sin \theta ) }\).
3. Given that the rod is in limiting equilibrium and the coefficient of friction between the rod and the surface is \(\frac { 3 } { 4 }\), determine the value of \(\theta\).

2 The diagram shows a uniform beam AB that rests with its end A on rough horizontal ground and its end B against a smooth vertical wall. The beam makes an angle of \(\theta ^ { \circ }\) with the ground. \includegraphics[max width=\textwidth, alt={}, center]{b3e369f4-13f7-457b-9a43-04ed2e2a2bba-3_812_588_347_246} The weight of the beam is \(W N\). The beam is in limiting equilibrium and the coefficient of friction between the beam and the ground is \(\mu\). It is given that the magnitude of the contact force at A is 70 N and the magnitude of the contact force at B is 20 N . Determine, in any order,

• the value of \(W\),
• the value of \(\mu\),
• the value of \(\theta\).

4 Fig. 4 shows a non-uniform rigid plank AB of weight 900 N and length 2.5 m . The centre of mass of the plank is at G which is 2 m from A . The end A rests on rough horizontal ground and does not slip. The plank is held in equilibrium at \(20 ^ { \circ }\) above the horizontal by a force of \(T \mathrm {~N}\) applied at B at an angle of \(55 ^ { \circ }\) above the horizontal as shown in Fig. 4. \begin{figure}[h] \end{figure}

1. Show that \(T = 700\) (correct to 3 significant figures).
2. Determine the possible values of the coefficient of friction between the plank and the ground.

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. The diagram shows a uniform rod \(A B\), of length 8 m and mass 23 kg , in limiting equilibrium with its end \(A\) on rough horizontal ground and point \(C\) resting against a smooth fixed cylinder. The rod is inclined at an angle of \(30 ^ { \circ }\) to the ground. \includegraphics[max width=\textwidth, alt={}, center]{d7f600c5-af4a-4708-bfd9-92b37a95c634-3_240_869_603_598}

The coefficient of friction between the ground and the rod is \(\frac { 2 } { 3 }\).

1. Calculate the magnitude of the normal reaction at \(C\) and the magnitude of the normal reaction to the ground at \(A\).
2. Find the length \(A C\).
3. Suppose instead that the rod is non-uniform with its centre of mass closer to \(A\) than to \(B\). Without carrying out any further calculations, state whether or not this will affect your answers in part (a). Give a reason for your answer.

6 A uniform solid is formed by rotating the region enclosed by the positive \(x\)-axis, the line \(x = 2\) and the curve \(y = \frac { 1 } { 2 } x ^ { 2 }\) through \(360 ^ { \circ }\) around the \(x\)-axis. 6

1. Find the centre of mass of this solid.
   6
2. The solid is placed with its plane face on a rough inclined plane and does not slide. The angle between the inclined plane and the horizontal is gradually increased. When the angle between the inclined plane and the horizontal is \(\alpha\), the solid is on the point of toppling. Find \(\alpha\), giving your answer to the nearest \(0.1 ^ { \circ }\)

9 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
A light elastic string has one end attached to a fixed point, \(A\), on a rough plane inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle, \(P\), of mass 2 kg .
The elastic string has natural length 1.3 metres and modulus of elasticity 65 N .
The particle is pulled down the plane in the direction of the line of greatest slope through \(A\).
The particle is released from rest when it is 2 metres from \(A\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{4fdb2637-6368-422c-99da-85b80efe31c5-14_549_744_861_785} The coefficient of friction between the particle and the plane is 0.6
After the particle is released it moves up the plane.
The particle comes to rest at a point \(B\), which is a distance, \(d\) metres, from \(A\). 9

1. Show that the value of \(d\) is 1.4.
   [0pt] [7 marks] 9
2. Determine what happens after \(P\) reaches the point \(B\). Fully justify your answer.
   [0pt] [3 marks]

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.