3.03m Equilibrium: sum of resolved forces = 0

2. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(3 m g\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held in equilibrium by a horizontal force of magnitude \(\frac { 4 } { 3 } m g\) applied to \(P\). This force acts in the vertical plane containing the string, as shown in Figure 1. Find (a) the tension in the string,
(b) the elastic energy stored in the string.

7. \begin{figure}[h] \end{figure} The fixed points \(A\) and \(B\) are 4 m apart on a smooth horizontal floor. One end of a light elastic string, of natural length 1.8 m and modulus of elasticity 45 N , is attached to a particle \(P\) and the other end is attached to \(A\). One end of another light elastic string, of natural length 1.2 m and modulus of elasticity 20 N , is attached to \(P\) and the other end is attached to \(B\). The particle \(P\) rests in equilibrium at the point \(O\), where \(A O B\) is a straight line, as shown in Figure 6.

1. Show that \(A O = 2.2 \mathrm {~m}\). The point \(C\) lies on the straight line \(A O B\) with \(A C = 2.7 \mathrm {~m}\). The mass of \(P\) is 0.6 kg . The particle \(P\) is held at \(C\) and then released from rest.
2. Show that, while both strings are taut, \(P\) moves with simple harmonic motion with centre \(O\). The point \(D\) lies on the straight line \(A O B\) with \(A D = 1.8 \mathrm {~m}\). When \(P\) reaches \(D\) the string \(P B\) breaks.
3. Find the time taken by \(P\) to move directly from \(C\) to \(A\).

2. \begin{figure}[h] \end{figure} Two elastic ropes each have natural length 30 cm and modulus of elasticity \(\lambda \mathrm { N }\). One end of each rope is attached to a lead weight \(P\) of mass 2 kg and the other ends are attached to two points \(A\) and \(B\) on a horizontal ceiling, where \(A B = 72 \mathrm {~cm}\). The weight hangs in equilibrium 15 cm below the ceiling, as shown in Fig. 2. By modelling \(P\) as a particle and the ropes as light elastic strings,

1. find, to one decimal place, the value of \(\lambda\).
2. State how you have used the fact that \(P\) is modelled as a particle.

4. \begin{figure}[h] \end{figure} A particle of weight \(W\) lies on a rough plane.The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\) .The coefficient of friction between the particle and the plane is \(\frac { 1 } { 2 }\) The particle is held in equilibrium by a force of magnitude 1.2 W .The force makes an angle \(\theta\) with the plane,where \(0 < \theta < \pi\) ,and acts in a vertical plane containing a line of greatest slope of the plane,as shown in Figure 2.

1. Find the value of \(\theta\) for which there is no frictional force acting on the particle. The minimum value of \(\theta\) for the particle to remain in equilibrium is \(\beta\)
2. Show that $$\beta = \arccos \left( \frac { \sqrt { 5 } } { 3 } \right) - \arctan \left( \frac { 1 } { 2 } \right)$$
3. Find the range of values of \(\theta\) for which the particle remains in equilibrium with the frictional force acting up the plane.

4 \includegraphics[max width=\textwidth, alt={}, center]{c9e725ad-561b-4e98-9b8f-7c9d3c8e67e6-3_494_255_258_945} Particles \(P\) and \(Q\), of masses 0.4 kg and 0.3 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley and the sections of the string not in contact with the pulley are vertical. \(P\) rests in limiting equilibrium on a plane inclined at \(60 ^ { \circ }\) to the horizontal (see diagram).

1. (a) Calculate the components, parallel and perpendicular to the plane, of the contact force exerted by the plane on \(P\).
   (b) Find the coefficient of friction between \(P\) and the plane. \(P\) is held stationary and a particle of mass 0.2 kg is attached to \(Q\). With the string taut, \(P\) is released from rest.
2. Calculate the tension in the string and the acceleration of the particles. \includegraphics[max width=\textwidth, alt={}, center]{c9e725ad-561b-4e98-9b8f-7c9d3c8e67e6-3_579_1195_1553_475} The \(( t , v )\) diagram represents the motion of two cyclists \(A\) and \(B\) who are travelling along a horizontal straight road. At time \(t = 0 , A\), who cycles with constant speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), overtakes \(B\) who has initial speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). From time \(t = 0 B\) cycles with constant acceleration for 20 s . When \(t = 20\) her speed is \(11 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), which she subsequently maintains.

2 \includegraphics[max width=\textwidth, alt={}, center]{4c6c9323-8238-4ec2-94a1-6e8188a34521-02_597_885_676_630} Three horizontal forces of magnitudes \(F \mathrm {~N} , 8 \mathrm {~N}\) and 10 N act at a point and are in equilibrium. The \(F \mathrm {~N}\) and 8 N forces are perpendicular to each other, and the 10 N force acts at an obtuse angle \(( 90 + \alpha ) ^ { \circ }\) to the \(F \mathrm {~N}\) force (see diagram). Calculate

1. \(\alpha\),
2. \(F\).

6 \includegraphics[max width=\textwidth, alt={}, center]{ce4c43e6-da4f-4c02-ab0f-01a21717949c-3_348_1109_1345_516} A small smooth ring \(R\) of weight 7 N is threaded on a light inextensible string. The ends of the string are attached to fixed points \(A\) and \(B\) at the same horizontal level. A horizontal force of magnitude 5 N is applied to \(R\). The string is taut. In the equilibrium position the angle \(A R B\) is a right angle, and the portion of the string attached to \(B\) makes an angle \(\theta\) with the horizontal (see diagram).

1. Explain why the tension \(T \mathrm {~N}\) is the same in each part of the string.
2. By resolving horizontally and vertically for the forces acting on \(R\), form two simultaneous equations in \(T \cos \theta\) and \(T \sin \theta\).
3. Hence find \(T\) and \(\theta\).

1 \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-2_305_295_264_868} Two perpendicular forces of magnitudes \(F \mathrm {~N}\) and 8 N act at a point \(O\) (see diagram). Their resultant has magnitude 17 N .

1. Calculate \(F\) and find the angle which the resultant makes with the 8 N force. A third force of magnitude \(E \mathrm {~N}\), acting in the same plane as the two original forces, is now applied at the point \(O\). The three forces of magnitudes \(E N , F N\) and \(8 N\) are in equilibrium.
2. State the value of \(E\) and the angle between the directions of the \(E \mathrm {~N}\) and 8 N forces.

6 \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-4_328_698_255_657} A particle \(P\) lies on a slope inclined at \(30 ^ { \circ }\) to the horizontal. \(P\) is attached to one end of a taut light inextensible string which passes through a small smooth ring \(Q\) of mass \(m \mathrm {~kg}\). The portion \(P Q\) of the string is horizontal and the other portion of the string is inclined at \(40 ^ { \circ }\) to the vertical. A horizontal force of magnitude \(H \mathrm {~N}\), acting away from \(P\), is applied to \(Q\) (see diagram). The tension in the string is 6.4 N , and the string is in the vertical plane containing the line of greatest slope on which \(P\) lies. Both \(P\) and \(Q\) are in equilibrium.

1. Calculate \(m\).
2. Calculate \(H\).
3. Given that the weight of \(P\) is 32 N , and that \(P\) is in limiting equilibrium, show that the coefficient of friction between \(P\) and the slope is 0.879 , correct to 3 significant figures. \(Q\) and the string are now removed.
4. Determine whether \(P\) remains in equilibrium.

4 \includegraphics[max width=\textwidth, alt={}, center]{8b79facc-e37f-45c3-95c0-9f2a30ca8fe4-3_394_963_276_552} Two forces of magnitudes 6 N and 10 N separated by an angle of \(110 ^ { \circ }\) act on a particle \(P\), which rests on a horizontal surface (see diagram).

1. Find the magnitude of the resultant of the 6 N and 10 N forces, and the angle between the resultant and the 10 N force. The two forces act in the same vertical plane. The particle \(P\) has weight 20 N and rests in equilibrium on the surface. Given that the surface is smooth, find
2. the magnitude of the force exerted on \(P\) by the surface,
3. the angle between the surface and the 10 N force.

7 An explorer is trying to pull a loaded sledge of total mass 100 kg along horizontal ground using a light rope. The only resistance to motion of the sledge is from friction between it and the ground. \begin{figure}[h] \end{figure} Initially she pulls with a force of 121 N on the rope inclined at \(34 ^ { \circ }\) to the horizontal, as shown in Fig. 7, but the sledge does not move.

1. Draw a diagram showing all the forces acting on the sledge. Show that the frictional force between the ground and the sledge is 100 N , correct to 3 significant figures. Calculate the normal reaction of the ground on the sledge. The sledge is given a small push to set it moving at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The explorer continues to pull on the rope with the same force and the same angle as before. The frictional force is also unchanged.
2. Describe the subsequent motion of the sledge. The explorer now pulls the rope, still at an angle of \(34 ^ { \circ }\) to the horizontal, so that the tension in it is 155 N . The frictional force is now 95 N .
3. Calculate the acceleration of the sledge. In a new situation, there is no rope and the sledge slides down a uniformly rough slope inclined at \(26 ^ { \circ }\) to the horizontal. The sledge starts from rest and reaches a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 2 seconds.
4. Calculate the frictional force between the slope and the sledge.

2 Fig. 2 shows a sack of rice of weight 250 N hanging in equilibrium supported by a light rope AB . End A of the rope is attached to the sack. The rope passes over a small smooth fixed pulley. \begin{figure}[h] \end{figure} Initially, end B of the rope is attached to a vertical wall as shown in Fig. 2.

1. Calculate the horizontal and the vertical forces acting on the wall due to the rope. End B of the rope is now detached from the wall and attached instead to the top of the sack. The sack is in equilibrium with both sections of the rope vertical.
2. Calculate the tension in the rope.

8 A cylindrical tub of mass 250 kg is on a horizontal floor. Resistance to its motion other than that due to friction is negligible. The first attempt to move the tub is by pulling it with a force of 150 N in the \(\mathbf { i }\) direction, as shown in Fig. 8.1. \begin{figure}[h] \end{figure}

1. Calculate the acceleration of the tub if friction is ignored. In fact, there is friction and the tub does not move.
2. Write down the magnitude and direction of the frictional force opposing the pull. Two more forces are now added to the 150 N force in a second attempt to move the tub, as shown in Fig. 8.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{6cca1e5e-82b0-487d-8048-b9db7745dea6-5_502_935_1411_607} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure} Angle \(\theta\) is acute and chosen so that the resultant of the three forces is in the \(\mathbf { i }\) direction.
3. Determine the value of \(\theta\) and the resultant of the three forces. With this resultant force, the tub moves with constant acceleration and travels 1 metre from rest in 2 seconds.
4. Show that the magnitude of the friction acting on the tub is 661 N , correct to 3 significant figures. When the speed of the tub is \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it comes to a part of the floor where the friction on the tub is 200 N greater. The pulling forces stay the same.
5. Find the velocity of the tub when it has moved a further 1.65 m .

   4	□ box P □
   \multirow[t]{10}{*}{4 }
   4

4 Fig. 4 shows a block of mass 15 kg on a smooth plane inclined at \(20 ^ { \circ }\) to the horizontal. The block is held in equilibrium by a horizontal force of magnitude \(P \mathrm {~N}\). \begin{figure}[h] \end{figure}

1. Show all the forces acting on the block.
2. Calculate \(P\).

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.

3 \includegraphics[max width=\textwidth, alt={}, center]{020ebd88-b920-40ce-84cf-5c26d45e2935-2_355_695_1073_726} A uniform solid hemisphere, of radius \(a\) and mass \(M\), is placed with its curved surface in contact with a rough plane that is inclined at an angle \(\alpha\) to the horizontal. A particle \(P\) of mass \(m\) is attached to the rim of the hemisphere. The system rests in equilibrium with the rim of the hemisphere horizontal and \(P\) at the point on the rim that is closest to the inclined plane (see diagram). Given that the coefficient of friction between the plane and the hemisphere is \(\frac { 1 } { 2 }\), show that

1. \(\tan \alpha \leqslant \frac { 1 } { 2 }\),
2. \(m \leqslant \frac { M ( 1 + \sqrt { } 5 ) } { 4 }\).

3 \includegraphics[max width=\textwidth, alt={}, center]{3daca234-9b7f-41d4-bbaa-d35615a120fc-2_419_1102_1859_520} The diagram shows two uniform rods \(B A\) and \(A C\), smoothly hinged at \(A\). The rod \(B A\) has length \(8 a\) and weight \(W\); the rod \(A C\) has length \(6 a\) and weight \(2 W\). The rods are in equilibrium in a vertical plane with \(B\) and \(C\) resting on a rough horizontal floor and angle \(C A B\) equal to \(90 ^ { \circ }\). Show that the normal contact force at \(B\) is \(\frac { 26 } { 25 } W\). The coefficient of friction between each rod and the floor is \(\mu\). Find the least possible value of \(\mu\).

The points \(C\) and \(D\) are at a distance \(( 2 \sqrt { } 3 ) a\) apart on a horizontal surface. A rough peg \(A\) is fixed at a vertical distance \(6 a\) above \(C\) and a smooth peg \(B\) is fixed at a vertical distance \(4 a\) above \(D\). A uniform rectangular frame \(P Q R S\), with \(P Q = 3 a\) and \(Q R = 6 a\), is made of rigid thin wire and has weight \(W\). It rests in equilibrium in a vertical plane with \(P S\) on \(A\) and \(S R\) on \(B\), and with angle \(S A C = 30 ^ { \circ }\) (see diagram).

1. Show that \(A B = 4 a\) and that angle \(S A B = 30 ^ { \circ }\).
2. Show that the normal reaction at \(A\) is \(\frac { 1 } { 2 } W\).
3. Find the frictional force at \(A\).

The points \(C\) and \(D\) are at a distance \(( 2 \sqrt { } 3 ) a\) apart on a horizontal surface. A rough peg \(A\) is fixed at a vertical distance \(6 a\) above \(C\) and a smooth peg \(B\) is fixed at a vertical distance \(4 a\) above \(D\). A uniform rectangular frame \(P Q R S\), with \(P Q = 3 a\) and \(Q R = 6 a\), is made of rigid thin wire and has weight \(W\). It rests in equilibrium in a vertical plane with \(P S\) on \(A\) and \(S R\) on \(B\), and with angle \(S A C = 30 ^ { \circ }\) (see diagram).

1. Show that \(A B = 4 a\) and that angle \(S A B = 30 ^ { \circ }\).
2. Show that the normal reaction at \(A\) is \(\frac { 1 } { 2 } W\).
3. Find the frictional force at \(A\).

4 \includegraphics[max width=\textwidth, alt={}, center]{833c338f-53c1-436e-a772-0cdaf17fa72d-3_559_1303_255_422} The diagram shows a central cross-section CDEF of a uniform solid cube of weight \(W\) and with edges of length \(2 a\). The cube rests on a rough horizontal plane. A thin uniform \(\operatorname { rod } A B\), of weight \(W\) and length \(6 a\), is hinged to the plane at \(A\). The rod rests in smooth contact with the cube at \(C\), with angle \(C A D\) equal to \(30 ^ { \circ }\). The rod is in the same vertical plane as \(C D E F\). The coefficient of friction between the plane and the cube is \(\mu\). Given that the system is in equilibrium, show that \(\mu \geqslant \frac { 3 } { 25 } \sqrt { } 3\). [6] Find the magnitude of the force acting on the \(\operatorname { rod }\) at \(A\).

1 \includegraphics[max width=\textwidth, alt={}, center]{a8e37fb1-14c7-4004-b186-d607878e200d-2_744_504_255_824} A uniform ladder \(A B\), of length \(3 a\) and weight \(W\), rests with the end \(A\) in contact with smooth horizontal ground and the end \(B\) against a smooth vertical wall. One end of a light inextensible rope is attached to the ladder at the point \(C\), where \(A C = a\). The other end of the rope is fixed to the point \(D\) at the base of the wall and the rope \(D C\) is in the same vertical plane as the ladder \(A B\). The ladder rests in equilibrium in a vertical plane perpendicular to the wall, with the ladder making an angle \(\theta\) with the horizontal and the rope making an angle \(\alpha\) with the horizontal (see diagram). It is given that \(\tan \theta = 2 \tan \alpha\). Find, in terms of \(W\) and \(\alpha\), the tension in the rope and the magnitudes of the forces acting on the ladder at \(A\) and at \(B\).

3 \includegraphics[max width=\textwidth, alt={}, center]{58728f93-bfdb-4f76-a9b9-3a1d1592bfc9-3_898_1116_258_518} The end \(P\) of a uniform rod \(P Q\), of weight \(k W\) and length \(8 a\), is rigidly attached to a point on the surface of a uniform sphere with centre \(C\), weight \(W\) and radius \(a\). The end \(Q\) is rigidly attached to a point on the surface of an identical sphere with centre \(D\). The points \(C , P , Q\) and \(D\) are in a straight line. The object consisting of the rod and two spheres rests with one sphere in contact with a rough horizontal surface, at the point \(A\), and the other sphere in contact with a smooth vertical wall, at the point \(B\). The angle between \(C D\) and the horizontal is \(\theta\). The point \(B\) is at a height of \(7 a\) above the base of the wall (see diagram). The points \(A , B , C , D , P\) and \(Q\) are all in the same vertical plane.

1. Show that \(\sin \theta = \frac { 3 } { 5 }\). The object is in limiting equilibrium and the coefficient of friction at \(A\) is \(\mu\).
2. Find the numerical value of \(\mu\).
3. Given that the resultant force on the object at \(A\) is \(W \sqrt { } ( 65 )\), show that \(k = 5\).