6.04e Rigid body equilibrium: coplanar forces

3 \includegraphics[max width=\textwidth, alt={}, center]{09d3e8ca-0062-4f62-8453-7acaff591db5-2_661_711_918_717} Two uniform rods \(A B\) and \(A C\), of equal lengths, and of weights 200 N and 360 N respectively, are freely jointed at \(A\). The mid-points of the rods are joined by a taut light inextensible string. The rods are in equilibrium in a vertical plane with \(B\) and \(C\) in contact with a smooth horizontal surface. The point \(A\) is 2.1 m above the surface and \(B C = 1.4 \mathrm {~m}\) (see diagram).

1. Show that the force exerted on \(A B\) at \(B\) has magnitude 240 N and find the tension in the string.
2. Find the horizontal and vertical components of the force exerted on \(A B\) at \(A\) and state their directions.

5 \includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-3_549_447_253_849} Two uniform rods \(A B\) and \(B C\), each of length 1.4 m and weight 80 N , are freely jointed to each other at \(B\), and \(A B\) is freely jointed to a fixed point at \(A\). They are held in equilibrium with \(A B\) at an angle \(\alpha\) to the horizontal, and \(B C\) at an angle of \(60 ^ { \circ }\) to the horizontal, by a light string, perpendicular to \(B C\), attached to \(C\) (see diagram).

1. By taking moments about \(B\) for \(B C\), calculate the tension in the string. Hence find the horizontal and vertical components of the force acting on \(B C\) at \(B\).
2. Find \(\alpha\). \includegraphics[max width=\textwidth, alt={}, center]{a04e6d4e-2437-4761-87ee-43e6771fbbd9-3_691_665_1370_740} A circus performer \(P\) of mass 80 kg is suspended from a fixed point \(O\) by an elastic rope of natural length 5.25 m and modulus of elasticity \(2058 \mathrm {~N} . P\) is in equilibrium at a point 5 m above a safety net. A second performer \(Q\), also of mass 80 kg , falls freely under gravity from a point above \(P\). \(P\) catches \(Q\) and together they begin to descend vertically with initial speed \(3.5 \mathrm {~ms} ^ { - 1 }\) (see diagram). The performers are modelled as particles.

5 \includegraphics[max width=\textwidth, alt={}, center]{85402f4a-8d55-47d8-ba48-5b837609b0f4-3_581_903_267_621} Two uniform rods \(X A\) and \(X B\) are freely jointed at \(X\). The lengths of the rods are 1.5 m and 1.3 m respectively, and their weights are 150 N and 130 N respectively. The rods are in equilibrium in a vertical plane with \(A\) and \(B\) in contact with a rough horizontal surface. \(A\) and \(B\) are at distances horizontally from \(X\) of 0.9 m and 0.5 m respectively, and \(X\) is 1.2 m above the surface (see diagram).

1. The normal components of the contact forces acting on the rods at \(A\) and \(B\) are \(R _ { A } \mathrm {~N}\) and \(R _ { B } \mathrm {~N}\) respectively. Show that \(R _ { A } = 125\) and find \(R _ { B }\).
2. Find the frictional components of the contact forces acting on the rods at \(A\) and \(B\).
3. Find the horizontal and vertical components of the force exerted on \(X A\) at \(X\), stating their directions.

5 \includegraphics[max width=\textwidth, alt={}, center]{3243c326-a51c-462f-a57c-a150d0044ea9-3_510_716_662_676} Two uniform rods \(A B\) and \(B C\), each of length \(4 L\), are freely jointed at \(B\), and rest in a vertical plane with \(A\) and \(C\) on a smooth horizontal surface. The weight of \(A B\) is \(W\) and the weight of \(B C\) is \(2 W\). The rods are joined by a horizontal light inextensible string fixed to each rod at a point distance \(L\) from \(B\), so that each rod is inclined at an angle of \(60 ^ { \circ }\) to the horizontal (see diagram).

1. By considering the equilibrium of the whole body, show that the force acting on \(B C\) at \(C\) is \(1.75 W\) and find the force acting on \(A B\) at \(A\).
2. Find the tension in the string in terms of \(W\).
3. Find the horizontal and vertical components of the force acting on \(A B\) at \(B\), and state the direction of the component in each case.

5 \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-3_576_535_258_804} A particle \(P\) of mass 0.3 kg is moving in a vertical circle. It is attached to the fixed point \(O\) at the centre of the circle by a light inextensible string of length 1.5 m . When the string makes an angle of \(40 ^ { \circ }\) with the downward vertical, the speed of \(P\) is \(6.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Air resistance may be neglected.

1. Find the radial and transverse components of the acceleration of \(P\) at this instant. In the subsequent motion, with the string still taut and making an angle \(\theta ^ { \circ }\) with the downward vertical, the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Use conservation of energy to show that \(v ^ { 2 } \approx 19.7 + 29.4 \cos \theta ^ { \circ }\).
3. Find the tension in the string in terms of \(\theta\).
4. Find the value of \(v\) at the instant when the string becomes slack. \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-3_574_842_1640_664} A step-ladder is modelled as two uniform rods \(A B\) and \(A C\), freely jointed at \(A\). The rods are in equilibrium in a vertical plane with \(B\) and \(C\) in contact with a rough horizontal surface. The rods have equal lengths; \(A B\) has weight 150 N and \(A C\) has weight 270 N . The point \(A\) is 2.5 m vertically above the surface, and \(B C = 1.6 \mathrm {~m}\) (see diagram).

1. Find the horizontal and vertical components of the force acting on \(A C\) at \(A\).
2. The coefficient of friction has the same value \(\mu\) at \(B\) and at \(C\), and the step-ladder is on the point of slipping. Giving a reason, state whether the equilibrium is limiting at \(B\) or at \(C\), and find \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-4_648_227_269_982} Two points \(A\) and \(B\) lie on a vertical line with \(A\) at a distance 2.6 m above \(B\). A particle \(P\) of mass 10 kg is joined to \(A\) by an elastic string and to \(B\) by another elastic string (see diagram). Each string has natural length 0.8 m and modulus of elasticity 196 N . The strings are light and air resistance may be neglected.

1. Verify that \(P\) is in equilibrium when \(P\) is vertically below \(A\) and the length of the string \(P A\) is 1.5 m . The particle is set in motion along the line \(A B\) with both strings remaining taut. The displacement of \(P\) below the equilibrium position is denoted by \(x\) metres.
2. Show that the tension in the string \(P A\) is \(245 ( 0.7 + x )\) newtons, and the tension in the string \(P B\) is \(245 ( 0.3 - x )\) newtons.
3. Show that the motion of \(P\) is simple harmonic.
4. Given that the amplitude of the motion is 0.25 m , find the proportion of time for which \(P\) is above the mid-point of \(A B\).

4 The region bounded by the curve \(y = \sqrt { x }\), the \(x\)-axis and the lines \(x = 1\) and \(x = 4\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution.

1. Find the \(x\)-coordinate of the centre of mass of this solid. From this solid, the cylinder with radius 1 and length 3 with its axis along the \(x\)-axis (from \(x = 1\) to \(x = 4\) ) is removed.
2. Show that the centre of mass of the remaining object, Q , has \(x\)-coordinate 3 . This object Q has weight 96 N and it is supported, with its axis of symmetry horizontal, by a string passing through the cylindrical hole and attached to fixed points A and B (see Fig. 4). AB is horizontal and the sections of the string attached to A and B are vertical. There is sufficient friction to prevent slipping. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5bb02383-91c0-4454-aaea-0bd6af6ba325-5_837_819_1034_628} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
3. Find the support forces, \(R\) and \(S\), acting on the string at A and B
   (A) when the string is light,
   (B) when the string is heavy and uniform with a total weight of 6 N .

4

1. The region bounded by the curve \(y = x ^ { 3 }\) for \(0 \leqslant x \leqslant 2\), the \(x\)-axis and the line \(x = 2\), is occupied by a uniform lamina. Find the coordinates of the centre of mass of this lamina. [8]
2. The region bounded by the circular arc \(y = \sqrt { 4 - x ^ { 2 } }\) for \(1 \leqslant x \leqslant 2\), the \(x\)-axis and the line \(x = 1\), is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution, as shown in Fig. 4.1. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{39e14918-5017-43c0-9b74-7c68717ad5f3-5_627_499_593_785} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure}
   1. Show that the \(x\)-coordinate of the centre of mass of this solid of revolution is 1.35 . This solid is placed on a rough horizontal surface, with its flat face in a vertical plane. It is held in equilibrium by a light horizontal string attached to its highest point and perpendicular to its flat face, as shown in Fig. 4.2. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{39e14918-5017-43c0-9b74-7c68717ad5f3-5_573_613_1662_728} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
      \end{figure}
   2. Find the least possible coefficient of friction between the solid and the horizontal surface.

4

1. A uniform solid of revolution is obtained by rotating through \(2 \pi\) radians about the \(y\)-axis the region bounded by the curve \(y = 8 - 2 x ^ { 2 }\) for \(0 \leqslant x \leqslant 2\), the \(x\)-axis and the \(y\)-axis.
   1. Find the \(y\)-coordinate of the centre of mass of this solid. The solid is now placed on a rough plane inclined at an angle \(\theta\) to the horizontal. It rests in equilibrium with its circular face in contact with the plane as shown in Fig. 4. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{2a4afead-e772-4d86-bc8d-86ffa5bca507-4_511_568_616_831} \captionsetup{labelformat=empty} \caption{Fig. 4}
      \end{figure}
   2. Given that the solid is on the point of toppling, find \(\theta\).
2. Find the \(y\)-coordinate of the centre of mass of a uniform lamina in the shape of the region bounded by the curve \(y = 8 - 2 x ^ { 2 }\) for \(- 2 \leqslant x \leqslant 2\), and the \(x\)-axis.

4 The region \(A\) is bounded by the curve \(y = x ^ { 2 } + 5\) for \(0 \leqslant x \leqslant 3\), the \(x\)-axis, the \(y\)-axis and the line \(x = 3\). The region \(B\) is bounded by the curve \(y = x ^ { 2 } + 5\) for \(0 \leqslant x \leqslant 3\), the \(y\)-axis and the line \(y = 14\). These regions are shown in Fig. 4. \begin{figure}[h] \end{figure}

1. Find the coordinates of the centre of mass of a uniform lamina occupying the region \(A\).
2. The region \(B\) is rotated through \(2 \pi\) radians about the \(y\)-axis to form a uniform solid of revolution \(R\). Find the \(y\)-coordinate of the centre of mass of the solid \(R\).
3. The region \(A\) is rotated through \(2 \pi\) radians about the \(y\)-axis to form a uniform solid of revolution \(S\). Using your answer to part (ii), or otherwise, find the \(y\)-coordinate of the centre of mass of the solid \(S\).

4

1. A uniform lamina occupies the region bounded by the \(x\)-axis, the \(y\)-axis and the curve \(y = 3 - \sqrt { x }\) for \(0 \leqslant x \leqslant 9\). Find the coordinates of the centre of mass of this lamina.
2. Fig. 4.1 shows the region bounded by the line \(x = 2\) and the part of the circle \(y ^ { 2 } = 25 - x ^ { 2 }\) for which \(2 \leqslant x \leqslant 5\). This region is rotated about the \(x\)-axis to form a uniform solid of revolution \(S\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-5_675_659_479_705} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure}
   1. Find the \(x\)-coordinate of the centre of mass of \(S\). The solid \(S\) rests in equilibrium with its curved surface in contact with a rough plane inclined at \(25 ^ { \circ }\) to the horizontal. Fig. 4.2 shows a vertical section containing AB , which is a diameter and also a line of greatest slope of the flat surface of \(S\). This section also contains XY, which is a line of greatest slope of the plane. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{86dd0c01-970d-4b67-9a6c-5df276a4a2be-5_494_560_1615_749} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
      \end{figure}
   2. Find the angle \(\theta\) that AB makes with the horizontal.

4

1. A uniform solid of revolution \(S\) is formed by rotating the region enclosed between the \(x\)-axis and the curve \(y = x \sqrt { 4 - x }\) for \(0 \leqslant x \leqslant 4\) through \(2 \pi\) radians about the \(x\)-axis, as shown in Fig. 4.1. O is the origin and the end A of the solid is at the point \(( 4,0 )\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-5_520_625_408_703} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure}
   1. Find the \(x\)-coordinate of the centre of mass of the solid \(S\). The solid \(S\) has weight \(W\), and it is freely hinged to a fixed point at O . A horizontal force, of magnitude \(W\) acting in the vertical plane containing OA , is applied at the point A , as shown in Fig. 4.2. \(S\) is in equilibrium. \begin{figure}[h]
      \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-5_346_512_1361_781} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
      \end{figure}
   2. Find the angle that OA makes with the vertical.
      [0pt] [Question 4(b) is printed overleaf]
2. Fig. 4.3 shows the region bounded by the \(x\)-axis, the \(y\)-axis, the line \(y = 8\) and the curve \(y = ( x - 2 ) ^ { 3 }\) for \(0 \leqslant y \leqslant 8\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{db60e7d9-bec5-47f7-9e27-38b7d112851e-6_631_695_370_683} \captionsetup{labelformat=empty} \caption{Fig. 4.3}
   \end{figure} Find the coordinates of the centre of mass of a uniform lamina occupying this region.

4 The region \(R\) is bounded by the \(x\)-axis, the \(y\)-axis, the curve \(y = \mathrm { e } ^ { - x }\) and the line \(x = k\), where \(k\) is a positive constant.

1. The region \(R\) is rotated through \(2 \pi\) radians about the \(x\)-axis to form a uniform solid of revolution. Find the \(x\)-coordinate of the centre of mass of this solid, and show that it can be written in the form $$\frac { 1 } { 2 } - \frac { k } { \mathrm { e } ^ { 2 k } - 1 } .$$
2. The solid in part (i) is placed with its larger circular face in contact with a rough plane inclined at \(60 ^ { \circ }\) to the horizontal, as shown in Fig. 4, and you are given that no slipping occurs. \begin{figure}[h]
   [diagram]

   \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} Show that, whatever the value of \(k\), the solid will not topple.
3. A uniform lamina occupies the region \(R\). Find, in terms of \(k\), the coordinates of the centre of mass of this lamina. \section*{END OF QUESTION PAPER}

4

1. A uniform lamina occupies the region bounded by the \(x\)-axis and the curve \(y = \frac { x ^ { 2 } ( a - x ) } { a ^ { 2 } }\) for \(0 \leqslant x \leqslant a\). Find the coordinates of the centre of mass of this lamina.
2. The region \(A\) is bounded by the \(x\)-axis, the \(y\)-axis, the curve \(y = \sqrt { x ^ { 2 } + 16 }\) and the line \(x = 3\). The region \(B\) is bounded by the \(y\)-axis, the curve \(y = \sqrt { x ^ { 2 } + 16 }\) and the line \(y = 5\). These regions are shown in Fig. 4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{70a2c3ce-7bdb-4ddd-92fc-f7dcbdfdcfaf-5_604_460_605_792} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure}
   1. Find the \(x\)-coordinate of the centre of mass of the uniform solid of revolution formed when the region \(A\) is rotated through \(2 \pi\) radians about the \(x\)-axis.
   2. Using your answer to part (i), or otherwise, find the \(x\)-coordinate of the centre of mass of the uniform solid of revolution formed when the region \(B\) is rotated through \(2 \pi\) radians about the \(x\)-axis. \section*{END OF QUESTION PAPER}

5. \begin{figure}[h] \end{figure} A firework is modelled as a uniform solid formed by joining the plane surface of a right circular cone of height \(2 r\) and base radius \(r\), to one of the plane surfaces of a cylinder of height \(h\) and base radius \(r\) as shown in Figure 2. Using this model,

1. show that the distance of the centre of mass of the firework from its plane base is $$\frac { 3 h ^ { 2 } + 4 h r + 2 r ^ { 2 } } { 2 ( 3 h + 2 r ) }$$ The firework is to be launched from rough ground inclined at an angle \(\alpha\) to the horizontal. Given that the firework does not slip or topple and that \(h = 4 r\),
2. Find, correct to the nearest degree, the maximum value of \(\alpha\).

4. \begin{figure}[h] \end{figure} A stand used to reach high shelves in a storeroom is in the shape of a frustum of a cone. It is modelled as a uniform solid formed by removing a right circular cone of height \(2 h\) from a similar cone of height \(3 h\) and base radius \(3 r\) as shown in Figure 1.

1. Show that the centre of mass of the stand is a distance of \(\frac { 33 } { 76 } h\) from its larger plane face.
   (7 marks)
   The stand is stored hanging in equilibrium from a point on the circumference of the larger plane face. Given that \(h = 2 r\),
2. find, correct to the nearest degree, the acute angle which the plane faces of the stand make with the vertical.
   (4 marks)

6. \begin{figure}[h] \end{figure} Figure 1 shows a bowl formed by removing from a solid hemisphere of radius \(\frac { 3 } { 2 } r\) a smaller hemisphere of radius \(r\) having the same axis of symmetry and the same plane face.

1. Show that the centre of mass of the bowl is a distance of \(\frac { 195 } { 304 } r\) from its plane face.
   (7 marks)
   The bowl has mass \(M\) and is placed with its curved surface on a smooth horizontal plane. A stud of mass \(\frac { 1 } { 2 } M\) is attached to the outer rim of the bowl. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8b7133ed-3748-46cb-99d2-570ee33c7393-4_517_729_1318_539} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} When the bowl is in equilibrium its plane surface is inclined at an angle \(\alpha\) to the horizontal as shown in Figure 2.
2. Find tan \(\alpha\).
   (6 marks)

4. \begin{figure}[h] \end{figure} Four identical uniform rods, each of mass \(m\) and length \(2 a\), are freely jointed to form a rhombus \(A B C D\). The rhombus is suspended from \(A\) and is prevented from collapsing by an elastic string which joins \(A\) to \(C\), with \(\angle B A D = 2 \theta , 0 \leq \theta \leq \frac { 1 } { 3 } \pi\), as shown in Fig. 2. The natural length of the elastic string is \(2 a\) and its modulus of elasticity is \(4 m g\).

1. Show that the potential energy, \(V\), of the system is given by $$V = 4 m g a \left[ ( 2 \cos \theta - 1 ) ^ { 2 } - 2 \cos \theta \right] + \text { constant } .$$
2. Hence find the non-zero value of \(\theta\) for which the system is in equilibrium.
3. Determine whether this position of equilibrium is stable or unstable.

4. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of length \(2 a\) and mass \(8 m\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\). One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\frac { 4 } { 5 } \mathrm { mg }\), is fixed to \(B\). The other end of the string is attached to a small ring which is free to slide on a smooth straight horizontal wire which is fixed in the same vertical plane as \(A B\) at a height 7a vertically above \(A\). The rod \(A B\) makes an angle \(\theta\) with the upward vertical at \(A\), as shown in Fig. 2.

1. Show that the potential energy \(V\) of the system is given by $$V = \frac { 8 } { 5 } m g a \left( \cos ^ { 2 } \theta - \cos \theta \right) + \text { constant. }$$
2. Hence find the values of \(\theta , 0 \leq \theta \leq \pi\), for which the system is in equilibrium.
3. Determine the nature of these positions of equilibrium.

4. \begin{figure}[h] \end{figure} A uniform rod \(P Q\), of length \(2 a\) and mass \(m\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through the end \(P\). The end \(Q\) is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\frac { m g } { 2 \sqrt { 3 } }\). The other end of the string is attached to a fixed point \(O\), where \(O P\) is horizontal and \(O P = 2 a\), as shown in Fig. 2. \(\angle O P Q\) is denoted by \(2 \theta\).

1. Show that, when the string is taut, the potential energy of the system is $$- \frac { m g a } { \sqrt { 3 } } ( 2 \cos 2 \theta + \sqrt { 3 } \sin 2 \theta + 2 \sin \theta ) + \text { constant } .$$
2. Verify that there is a position of equilibrium at \(\theta = \frac { \pi } { 6 }\).
3. Determine whether this is a position of stable equilibrium.

7. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of length \(2 a\) and mass \(k M\) where \(k\) is a constant, is free to rotate in a vertical plane about the fixed point \(A\). One end of a light inextensible string of length \(6 a\) is attached to the end \(B\) of the rod and passes over a small smooth pulley which is fixed at the point \(P\). The line \(A P\) is horizontal and of length \(2 a\). The other end of the string is attached to a particle of mass \(M\) which hangs vertically below the point \(P\), as shown in Figure 3. The angle \(P A B\) is \(2 \theta\), where \(0 ^ { \circ } \leq \theta \leq 180 ^ { \circ }\).

1. Show that the potential energy of the system is $$M g a ( 4 \sin \theta - k \sin 2 \theta ) + \text { constant. }$$ The system has a position of equilibrium when \(\cos \theta = \frac { 3 } { 4 }\).
2. Find the value of \(k\).
3. Hence find the value of \(\cos \theta\) at the other position of equilibrium.
4. Determine the stability of each of the two positions of equilibrium.

4. \begin{figure}[h] \end{figure} A light inextensible string of length \(2 a\) has one end attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(m\). A second light inextensible string of length \(L\), where \(L > \frac { 12 a } { 5 }\), has one of its ends attached to \(P\) and passes over a small smooth peg fixed at a point \(B\). The line \(A B\) is horizontal and \(A B = 2 a\). The other end of the second string is attached to a particle of mass \(\frac { 7 } { 20 } m\), which hangs vertically below \(B\), as shown in Figure 2.

1. Show that the potential energy of the system, when the angle \(P A B = 2 \theta\), is $$\frac { 1 } { 5 } m g a ( 7 \sin \theta - 10 \sin 2 \theta ) + \text { constant. }$$
2. Show that there is only one value of \(\cos \theta\) for which the system is in equilibrium and find this value.
3. Determine the stability of the position of equilibrium.
   \section*{June 2009}

5. \begin{figure}[h] \end{figure} The end \(A\) of a uniform rod \(A B\), of length \(2 a\) and mass \(4 m\), is smoothly hinged to a fixed point. The end \(B\) is attached to one end of a light inextensible string which passes over a small smooth pulley, fixed at the same level as \(A\). The distance from \(A\) to the pulley is \(4 a\). The other end of the string carries a particle of mass \(m\) which hangs freely, vertically below the pulley, with the string taut. The angle between the rod and the downward vertical is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\), as shown in Figure 1.

1. Show that the potential energy of the system is $$2 m g a ( \sqrt { } ( 5 - 4 \sin \theta ) - 2 \cos \theta ) + \text { constant }$$
2. Hence, or otherwise, show that any value of \(\theta\) which corresponds to a position of equilibrium of the system satisfies the equation $$4 \sin ^ { 3 } \theta - 6 \sin ^ { 2 } \theta + 1 = 0 .$$
3. Given that \(\theta = \frac { \pi } { 6 }\) corresponds to a position of equilibrium, determine its stability. \section*{L \(\_\_\_\_\)}

7. \begin{figure}[h] \end{figure} Figure 3 shows a framework \(A B C\), consisting of two uniform rods rigidly joined together at \(B\) so that \(\angle A B C = 90 ^ { \circ }\). The rod \(A B\) has length \(2 a\) and mass \(4 m\), and the rod \(B C\) has length \(a\) and mass \(2 m\). The framework is smoothly hinged at \(A\) to a fixed point, so that the framework can rotate in a fixed vertical plane. One end of a light elastic string, of natural length \(2 a\) and modulus of elasticity \(3 m g\), is attached to \(A\). The string passes through a small smooth ring \(R\) fixed at a distance \(2 a\) from \(A\), on the same horizontal level as \(A\) and in the same vertical plane as the framework. The other end of the string is attached to \(B\). The angle \(A R B\) is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\).

1. Show that the potential energy \(V\) of the system is given by $$V = 8 a m g \sin 2 \theta + 5 a m g \cos 2 \theta + \text { constant }$$
2. Find the value of \(\theta\) for which the system is in equilibrium.
3. Determine the stability of this position of equilibrium.

5. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of length \(4 a\) and weight \(W\), is free to rotate in a vertical plane about a fixed smooth horizontal axis which passes through the point \(C\) of the rod, where \(A C = 3 a\). One end of a light inextensible string of length \(L\), where \(L > 10 a\), is attached to the end \(A\) of the rod and passes over a small smooth fixed peg at \(P\) and another small smooth fixed peg at \(Q\). The point \(Q\) lies in the same vertical plane as \(P , A\) and \(B\). The point \(P\) is at a distance \(3 a\) vertically above \(C\) and \(P Q\) is horizontal with \(P Q = 4 a\). A particle of weight \(\frac { 1 } { 2 } W\) is attached to the other end of the string and hangs vertically below \(Q\). The rod is inclined at an angle \(2 \theta\) to the vertical, where \(- \pi < 2 \theta < \pi\), as shown in Figure 1.

1. Show that the potential energy of the system is $$W a ( 3 \cos \theta - \cos 2 \theta ) + \text { constant }$$
2. Find the positions of equilibrium and determine their stability.

6. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has mass \(4 m\) and length \(4 l\). The rod can turn freely in a vertical plane about a fixed smooth horizontal axis through \(A\). A particle of mass \(k m\), where \(k < 7\), is attached to the rod at \(B\). One end of a light elastic string, of natural length \(l\) and modulus of elasticity 4 mg , is attached to the point \(D\) of the rod, where \(A D = 3 l\). The other end of the string is attached to a fixed point \(E\) which is vertically above \(A\), where \(A E = 3 l\), as shown in Figure 2. The angle between the rod and the upward vertical is \(2 \theta\), where \(\arcsin \left( \frac { 1 } { 6 } \right) < \theta \leqslant \frac { \pi } { 2 }\).

1. Show that, while the string is stretched, the potential energy of the system is $$8 m g l \left\{ ( 7 - k ) \sin ^ { 2 } \theta - 3 \sin \theta \right\} + \text { constant }$$ There is a position of equilibrium with \(\theta \leqslant \frac { \pi } { 6 }\).
2. Show that \(k \leqslant 4\) Given that \(k = 4\),
3. show that this position of equilibrium is stable.