3.04b Equilibrium: zero resultant moment and force

The end \(A\) of a uniform rod \(A B\), of length \(2 a\) and weight \(W\), is freely hinged to a vertical wall. The end \(B\) of the rod is attached to a light elastic string of natural length \(\frac { 3 } { 2 } a\) and modulus of elasticity \(3 W\). The other end of the string is attached to the point \(C\) on the wall, where \(C\) is vertically above \(A\) and \(A C = 2 a\). A particle of weight \(2 W\) is attached to the rod at the point \(D\), where \(D B = \frac { 1 } { 2 } a\). The angle \(A B C\) is equal to \(\theta\) (see diagram). Show that \(\cos \theta = \frac { 3 } { 4 }\) and find the tension in the string in terms of \(W\). Find the magnitude of the reaction force at the hinge.

4 A uniform \(\operatorname { rod } A B\) has length \(2 a\) and weight \(W\). The end \(A\) rests on rough horizontal ground and the end \(B\) rests against a smooth vertical wall. The rod is in a vertical plane that is perpendicular to the wall. The angle between the rod and the horizontal is \(\theta\). A particle of weight \(5 W\) hangs from the rod at the point \(C\), with \(A C = x a\), where \(0 < x < 1\).

1. By taking moments about \(A\), show that the magnitude of the normal reaction at \(B\) is \(\frac { W ( 5 x + 1 ) } { 2 \tan \theta }\).
   [0pt] [3]
   The particle of weight \(5 W\) is now moved a distance \(a\) up the rod, so that \(A C = ( x + 1 ) a\). This results in the magnitude of the normal reaction at \(B\) being double its previous value. The system remains in equilibrium with the rod at angle \(\theta\) with the horizontal.
2. Show that \(x = \frac { 4 } { 5 }\).
   The coefficient of friction between the rod and the ground is \(\frac { 2 } { 3 }\).
3. Given that the rod is about to slip when the particle of weight \(5 W\) is in its second position, find the value of \(\tan \theta\).
   \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Axis \(l\)} \includegraphics[alt={},max width=\textwidth]{c6c8e0fd-6af2-40c9-9513-6581e26e2aec-08_462_693_301_731}
   \end{figure} Three thin uniform rings \(A , B\) and \(C\) are joined together, so that each ring is in contact with each of the other two rings. Ring \(A\) has radius \(2 a\) and mass \(3 M\); rings \(B\) and \(C\) each have radius \(3 a\) and mass \(2 M\). The rings lie in the same plane and the centres of the rings are at the vertices of an isosceles triangle. The object consisting of the three rings is free to rotate about the horizontal axis \(l\) which is tangential to ring \(A\), in the plane of the object and perpendicular to the line of symmetry of the object (see diagram).

4 A uniform \(\operatorname { rod } A B\) has length \(2 a\) and weight \(W\). The end \(A\) rests on rough horizontal ground and the end \(B\) rests against a smooth vertical wall. The angle between the rod and the horizontal is \(\theta\), where \(\tan \theta = \frac { 4 } { 3 }\). One end of a light inextensible rope is attached to a point \(C\) on the rod. The other end is attached to a point where the vertical wall and the horizontal ground meet. The rope is taut and perpendicular to the rod. The rope and rod are in a vertical plane perpendicular to the wall.

1. Show that \(A C = \frac { 18 } { 25 } a\).
   The magnitude of the frictional force at \(A\) is equal to one quarter of the magnitude of the normal reaction force at \(A\).
2. Show that the tension in the rope is \(\frac { 1 } { 4 } W\).
3. Find expressions, in terms of \(W\), for the magnitudes of the normal reaction forces at \(A\) and \(B\).

4 \includegraphics[max width=\textwidth, alt={}, center]{2aaf3493-6509-4668-91a2-9f4708bbbb58-08_677_812_258_664} A uniform rod \(A B\) of length \(4 a\) and weight \(W\) rests with the end \(A\) in contact with a rough vertical wall. A light inextensible string of length \(\frac { 5 } { 2 } a\) has one end attached to the point \(C\) on the rod, where \(A C = \frac { 5 } { 2 } a\). The other end of the string is attached to a point \(D\) on the wall, vertically above \(A\). The vertical plane containing the \(\operatorname { rod } A B\) is perpendicular to the wall. The angle between the rod and the wall is \(\theta\), where \(\tan \theta = 2\) (see diagram). The end \(A\) of the rod is on the point of slipping down the wall and the coefficient of friction between the rod and the wall is \(\mu\). Find, in either order, the tension in the string and the value of \(\mu\).

5 \includegraphics[max width=\textwidth, alt={}, center]{34dd6523-7c0c-4842-bbda-56ad8d3f9766-10_456_684_264_731} A uniform \(\operatorname { rod } A B\) of length \(2 x\) and weight \(W\) rests on the smooth rim of a fixed hemispherical bowl of radius \(a\). The end \(B\) of the rod is in contact with the rough inner surface of the bowl. The coefficient of friction between the rod and the bowl at \(B\) is \(\frac { 1 } { 3 }\). A particle of weight \(\frac { 1 } { 4 } W\) is attached to the end \(A\) of the rod. The end \(B\) is about to slip upwards when \(A B\) is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 3 } { 4 }\) (see diagram).

1. By resolving parallel to the rod, show that the normal component of the reaction of the bowl on the rod at \(B\) is \(\frac { 3 } { 4 } W\).
2. Find, in terms of \(W\), the reaction between the rod and the smooth rim of the bowl.
3. Find \(x\) in terms of \(a\).

A uniform rod \(A B\), of weight \(W\) and length \(2 a\), rests with the end \(A\) on a rough horizontal plane. A light inextensible string \(B C\) is attached to the rod at \(B\) and passes over a small smooth fixed peg \(P\), which is at a distance \(h\) vertically above \(A\). A particle is attached at \(C\) and hangs vertically. The points \(A , B\) and \(C\) are all in the same vertical plane. In equilibrium the rod is inclined at an angle \(\theta\) to the horizontal (see diagram). The coefficient of friction between the rod and the plane is \(\mu\). Show that $$\mu \geqslant \frac { 2 a \cos \theta } { h + 2 a \sin \theta }$$ Given that the particle attached at \(C\) has weight \(k W\), angle \(A B P = 90 ^ { \circ }\) and \(h = 3 a\), find

1. the value of \(k\),
2. the horizontal component of the force on \(P\), in terms of \(W\).

A uniform rod \(A B\), of weight \(W\) and length \(2 a\), rests with the end \(A\) on a rough horizontal plane. A light inextensible string \(B C\) is attached to the rod at \(B\) and passes over a small smooth fixed peg \(P\), which is at a distance \(h\) vertically above \(A\). A particle is attached at \(C\) and hangs vertically. The points \(A , B\) and \(C\) are all in the same vertical plane. In equilibrium the rod is inclined at an angle \(\theta\) to the horizontal (see diagram). The coefficient of friction between the rod and the plane is \(\mu\). Show that $$\mu \geqslant \frac { 2 a \cos \theta } { h + 2 a \sin \theta }$$ Given that the particle attached at \(C\) has weight \(k W\), angle \(A B P = 90 ^ { \circ }\) and \(h = 3 a\), find

1. the value of \(k\),
2. the horizontal component of the force on \(P\), in terms of \(W\).

A uniform rod \(A B\), of weight \(W\) and length \(2 a\), rests with the end \(A\) on a rough horizontal plane. A light inextensible string \(B C\) is attached to the rod at \(B\) and passes over a small smooth fixed peg \(P\), which is at a distance \(h\) vertically above \(A\). A particle is attached at \(C\) and hangs vertically. The points \(A , B\) and \(C\) are all in the same vertical plane. In equilibrium the rod is inclined at an angle \(\theta\) to the horizontal (see diagram). The coefficient of friction between the rod and the plane is \(\mu\). Show that $$\mu \geqslant \frac { 2 a \cos \theta } { h + 2 a \sin \theta }$$ Given that the particle attached at \(C\) has weight \(k W\), angle \(A B P = 90 ^ { \circ }\) and \(h = 3 a\), find

1. the value of \(k\),
2. the horizontal component of the force on \(P\), in terms of \(W\).

Two identical uniform rough spheres \(A\) and \(B\), each of weight \(W\) and radius \(a\), are at rest on a rough horizontal plane, and are not in contact with each other. A third identical sphere \(C\) rests on \(A\) and \(B\) with its centre in the same vertical plane as the centres of \(A\) and \(B\). The line joining the centres of \(A\) and \(C\) and the line joining the centres of \(B\) and \(C\) are each inclined at an angle \(\theta\) to the vertical (see diagram). The coefficient of friction between each sphere and the plane is \(\mu\). The coefficient of friction between \(C\) and \(A\), and between \(C\) and \(B\), is \(\mu ^ { \prime }\). The system remains in equilibrium. Show that $$\mu \geqslant \frac { \sin \theta } { 3 ( 1 + \cos \theta ) } \quad \text { and } \quad \mu ^ { \prime } \geqslant \frac { \sin \theta } { 1 + \cos \theta } .$$

Two identical uniform rough spheres \(A\) and \(B\), each of weight \(W\) and radius \(a\), are at rest on a rough horizontal plane, and are not in contact with each other. A third identical sphere \(C\) rests on \(A\) and \(B\) with its centre in the same vertical plane as the centres of \(A\) and \(B\). The line joining the centres of \(A\) and \(C\) and the line joining the centres of \(B\) and \(C\) are each inclined at an angle \(\theta\) to the vertical (see diagram). The coefficient of friction between each sphere and the plane is \(\mu\). The coefficient of friction between \(C\) and \(A\), and between \(C\) and \(B\), is \(\mu ^ { \prime }\). The system remains in equilibrium. Show that $$\mu \geqslant \frac { \sin \theta } { 3 ( 1 + \cos \theta ) } \quad \text { and } \quad \mu ^ { \prime } \geqslant \frac { \sin \theta } { 1 + \cos \theta } .$$

3 \includegraphics[max width=\textwidth, alt={}, center]{bcd7ee99-e382-4cb6-aa39-d8b385b01319-2_506_623_977_760} Two uniform rods \(A B\) and \(B C\), each of length \(2 a\) and mass \(m\), are smoothly hinged at \(B\). They rest in equilibrium with \(C\) in contact with a smooth vertical wall and \(A\) in contact with a rough horizontal floor. The rods are in a vertical plane perpendicular to the wall. The rods \(A B\) and \(B C\) make angles \(\alpha\) and \(\beta\) respectively with the horizontal (see diagram). Show that

1. the reaction at \(C\) has magnitude \(\frac { 1 } { 2 } m g \cot \beta\),
2. \(\tan \alpha = 3 \tan \beta\). The coefficient of friction at \(A\) is \(\mu\). Given that \(\alpha = 60 ^ { \circ }\), find the least possible value of \(\mu\).

4 \includegraphics[max width=\textwidth, alt={}, center]{b486decd-75b8-44bd-889f-2472f1163871-3_567_575_258_785} A uniform circular disc, with centre \(O\) and weight \(W\), rests in equilibrium on a horizontal floor and against a vertical wall. The plane of the disc is vertical and perpendicular to the wall. The disc is in contact with the floor at \(A\) and with the wall at \(B\). A force of magnitude \(P\) acts tangentially on the disc at the point \(C\) on the edge of the disc, where the radius \(O C\) makes an angle \(\theta\) with the upward vertical, and \(\tan \theta = \frac { 4 } { 3 }\) (see diagram). The coefficient of friction between the disc and the floor and between the disc and the wall is \(\frac { 1 } { 2 }\). Show that the sum of the magnitudes of the frictional forces at \(A\) and \(B\) is equal to \(P\). Given that the equilibrium is limiting at both \(A\) and \(B\),

1. show that \(P = \frac { 15 } { 34 } \mathrm {~W}\),
2. find the ratio of the magnitude of the normal reaction at \(A\) to the magnitude of the normal reaction at \(B\).

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 \(k = 5\).

3 \includegraphics[max width=\textwidth, alt={}, center]{62d0d8cb-8f8c-4298-9705-71a735a9a4e7-3_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 \(k = 5\).

3 \includegraphics[max width=\textwidth, alt={}, center]{184020e1-7ff2-4172-8d33-baff963afa76-3_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 \(k = 5\).

4 \includegraphics[max width=\textwidth, alt={}, center]{9b520e69-a14e-47e5-97d7-998f5145844b-06_465_663_262_742} A small ring \(P\) of weight \(W\) is free to slide on a rough horizontal wire, one end of which is attached to a vertical wall at \(Q\). The end \(A\) of a thin uniform \(\operatorname { rod } A B\) of length \(2 a\) and weight \(\frac { 5 } { 2 } W\) is freely hinged to the wall at the point \(A\) which is a distance \(a\) vertically below \(Q\). A light elastic string of natural length \(2 a\) has one end attached to the ring \(P\) and the other end attached to the rod at \(B\). The string is at right angles to the rod and \(A , B , P\) and \(Q\) lie in a vertical plane. The system is in limiting equilibrium with \(A B\) making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\) (see diagram).

1. Find the tension in the string in terms of \(W\).
2. Find the coefficient of friction between the ring and the wire.
3. Find the magnitude of the resultant force on the rod at the hinge in terms of \(W\).
4. Find the modulus of elasticity of the string in terms of \(W\). \includegraphics[max width=\textwidth, alt={}, center]{9b520e69-a14e-47e5-97d7-998f5145844b-08_862_698_260_721} A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F = 4 a\) and \(F G = 2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B = 6 a\) and \(B C = 4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac { 11 } { 12 } m\) is fixed to the mid-point of \(A B\) (see diagram).

4 \includegraphics[max width=\textwidth, alt={}, center]{1651d08b-b20f-4f2e-9f47-0a1a5d0a839a-06_465_663_262_742} A small ring \(P\) of weight \(W\) is free to slide on a rough horizontal wire, one end of which is attached to a vertical wall at \(Q\). The end \(A\) of a thin uniform \(\operatorname { rod } A B\) of length \(2 a\) and weight \(\frac { 5 } { 2 } W\) is freely hinged to the wall at the point \(A\) which is a distance \(a\) vertically below \(Q\). A light elastic string of natural length \(2 a\) has one end attached to the ring \(P\) and the other end attached to the rod at \(B\). The string is at right angles to the rod and \(A , B , P\) and \(Q\) lie in a vertical plane. The system is in limiting equilibrium with \(A B\) making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\) (see diagram).

1. Find the tension in the string in terms of \(W\).
2. Find the coefficient of friction between the ring and the wire.
3. Find the magnitude of the resultant force on the rod at the hinge in terms of \(W\).
4. Find the modulus of elasticity of the string in terms of \(W\). \includegraphics[max width=\textwidth, alt={}, center]{1651d08b-b20f-4f2e-9f47-0a1a5d0a839a-08_862_698_260_721} A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F = 4 a\) and \(F G = 2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B = 6 a\) and \(B C = 4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac { 11 } { 12 } m\) is fixed to the mid-point of \(A B\) (see diagram).

4 \includegraphics[max width=\textwidth, alt={}, center]{2ab1a594-6c37-4c78-b53c-33c13bf6eb21-06_465_663_262_742} A small ring \(P\) of weight \(W\) is free to slide on a rough horizontal wire, one end of which is attached to a vertical wall at \(Q\). The end \(A\) of a thin uniform \(\operatorname { rod } A B\) of length \(2 a\) and weight \(\frac { 5 } { 2 } W\) is freely hinged to the wall at the point \(A\) which is a distance \(a\) vertically below \(Q\). A light elastic string of natural length \(2 a\) has one end attached to the ring \(P\) and the other end attached to the rod at \(B\). The string is at right angles to the rod and \(A , B , P\) and \(Q\) lie in a vertical plane. The system is in limiting equilibrium with \(A B\) making an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\) (see diagram).

1. Find the tension in the string in terms of \(W\).
2. Find the coefficient of friction between the ring and the wire.
3. Find the magnitude of the resultant force on the rod at the hinge in terms of \(W\).
4. Find the modulus of elasticity of the string in terms of \(W\). \includegraphics[max width=\textwidth, alt={}, center]{2ab1a594-6c37-4c78-b53c-33c13bf6eb21-08_862_698_260_721} A uniform picture frame of mass \(m\) is made by removing a rectangular lamina \(E F G H\) in which \(E F = 4 a\) and \(F G = 2 a\) from a larger rectangular lamina \(A B C D\) in which \(A B = 6 a\) and \(B C = 4 a\). The side \(E F\) is parallel to the side \(A B\). The point of intersection of the diagonals \(A C\) and \(B D\) coincides with the point of intersection of the diagonals \(E G\) and \(F H\). One end of a light inextensible string of length \(10 a\) is attached to \(A\) and the other end is attached to \(B\). The frame is suspended from the mid-point \(O\) of the string. A small object of mass \(\frac { 11 } { 12 } m\) is fixed to the mid-point of \(A B\) (see diagram).

5. A non-uniform rod \(A B\), of mass 5 kg and length 4 m , rests with one end \(A\) on rough horizontal ground. The centre of mass of the rod is \(d\) metres from \(A\). The rod is held in limiting equilibrium at an angle \(\theta\) to the horizontal by a force \(\mathbf { P }\), which acts in a direction perpendicular to the rod at \(B\), as shown in Figure 2. The line of action of \(\mathbf { P }\) lies in the same vertical plane as the rod.

1. Find, in terms of \(d , g\) and \(\theta\),
   1. the magnitude of the vertical component of the force exerted on the rod by the ground,
   2. the magnitude of the friction force acting on the rod at \(A\). Given that \(\tan \theta = \frac { 5 } { 12 }\) and that the coefficient of friction between the rod and the ground is \(\frac { 1 } { 2 }\),
2. find the value of \(d\).
   

2. \begin{figure}[h] \end{figure} A uniform \(\operatorname { rod } A B\) has weight 70 N and length 3 m . It rests in a horizontal position on two smooth supports placed at \(P\) and \(Q\), where \(A P = 0.5 \mathrm {~m}\), as shown in Fig. 1 . The reaction on the rod at \(P\) has magnitude 20 N . Find

1. the magnitude of the reaction on the rod at \(Q\),
2. the distance \(A Q\).
   . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a9f91ceb-662a-40cd-956b-815052b8f1a0-01_190_476_964_1905} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A particle \(P\) of mass 2 kg is held in equilibrium under gravity by two light inextensible strings. One string is horizontal and the other is inclined at an angle \(\alpha\) to the horizontal, as shown in Fig. 2. The tension in the horizontal string is 15 N . The tension in the other string is \(T\) newtons.
   1. Find the size of the angle \(\alpha\).
      (6 marks)
   2. Find the value of \(T\). You must ensure that your answers to parts of questions are clearly labelled.
      You must show sufficient working to make your methods clear to the Examiner. Answers without working may gain no credit.

5 \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-3_319_650_1219_749} A uniform \(\operatorname { rod } A B\) of length 60 cm and weight 15 N is freely suspended from its end \(A\). The end \(B\) of the rod is attached to a light inextensible string of length 80 cm whose other end is fixed to a point \(C\) which is at the same horizontal level as \(A\). The rod is in equilibrium with the string at right angles to the rod (see diagram).

1. Show that the tension in the string is 4.5 N .
2. Find the magnitude and direction of the force acting on the rod at \(A\).

7 \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-4_76_243_269_365} \includegraphics[max width=\textwidth, alt={}, center]{35477eb8-59e0-4de6-889c-1f5841f65eec-4_332_1427_322_360} A barrier is modelled as a uniform rectangular plank of wood, \(A B C D\), rigidly joined to a uniform square metal plate, \(D E F G\). The plank of wood has mass 50 kg and dimensions 4.0 m by 0.25 m . The metal plate has mass 80 kg and side 0.5 m . The plank and plate are joined in such a way that \(C D E\) is a straight line (see diagram). The barrier is smoothly pivoted at the point \(D\). In the closed position, the barrier rests on a thin post at \(H\). The distance \(C H\) is 0.25 m .

1. Calculate the contact force at \(H\) when the barrier is in the closed position. In the open position, the centre of mass of the barrier is vertically above \(D\).
2. Calculate the angle between \(A B\) and the horizontal when the barrier is in the open position.

3 \includegraphics[max width=\textwidth, alt={}, center]{d6d87705-be4b-407d-b699-69fb441d88a7-2_710_572_721_788} A uniform solid hemisphere of weight 12 N and radius 6 cm is suspended by two vertical strings. One string is attached to the point \(O\), the centre of the plane face, and the other string is attached to the point \(A\) on the rim of the plane face. The hemisphere hangs in equilibrium and \(O A\) makes an angle of \(60 ^ { \circ }\) with the vertical (see diagram).

1. Find the horizontal distance from the centre of mass of the hemisphere to the vertical through \(O\).
2. Calculate the tensions in the strings.

9 A uniform plank \(A B\) has weight 100 N and length 4 m . The plank rests horizontally in equilibrium on two smooth supports \(C\) and \(D\), where \(A C = x \mathrm {~m}\) and \(C D = 0.5 \mathrm {~m}\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{d5ab20c8-afd5-473e-8238-96762bd3786d-6_181_1271_1101_395} The magnitude of the reaction of the support on the plank at \(C\) is 75 N . Modelling the plank as a rigid rod, find

1. the magnitude of the reaction of the support on the plank at \(D\),
2. the value of \(x\). A stone block, which is modelled as a particle, is now placed at the end of the plank at \(B\) and the plank is on the point of tilting about \(D\).
3. Find the weight of the stone block.
4. Explain the limitation of modelling
   1. the stone block as a particle,
   2. the plank as a rigid rod.