3.04b Equilibrium: zero resultant moment and force

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.

5 A uniform rod of mass 4 kg and length 2.4 m can rotate in a vertical plane about a fixed horizontal axis through one end of the rod. The rod is released from rest in a horizontal position and a frictional couple of constant moment 20 Nm opposes the motion.

1. Find the angular acceleration of the rod immediately after it is released.
2. Find the angle that the rod makes with the horizontal when its angular acceleration is zero.
3. Find the maximum angular speed of the rod.
4. The rod first comes to instantaneous rest after rotating through an angle \(\theta\) radians from its initial position. Find an equation for \(\theta\), and verify that \(2.0 < \theta < 2.1\).

6 \includegraphics[max width=\textwidth, alt={}, center]{ab760a4b-e0ec-4256-838f-ed6c762ff18b-3_716_483_890_790} Two small smooth pegs \(P\) and \(Q\) are fixed at a distance \(2 a\) apart on the same horizontal level, and \(A\) is the mid-point of \(P Q\). A light rod \(A B\) of length \(4 a\) is freely pivoted at \(A\) and can rotate in the vertical plane containing \(P Q\), with \(B\) below the level of \(P Q\). A particle of mass \(m\) is attached to the rod at \(B\). A light elastic string, of natural length \(2 a\) and modulus of elasticity \(\lambda\), passes round the pegs \(P\) and \(Q\) and its two ends are attached to the rod at the point \(X\), where \(A X = a\). The angle between the rod and the downward vertical is \(\theta\), where \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\) (see diagram). You are given that the elastic energy stored in the string is \(\lambda a ( 1 + \cos \theta )\).

1. Show that \(\theta = 0\) is a position of equilibrium, and show that the equilibrium is stable if \(\lambda < 4 m g\).
2. Given that \(\lambda = 3 m g\), show that \(\ddot { \theta } = - k \frac { g } { a } \sin \theta\), stating the value of the constant \(k\). Hence find the approximate period of small oscillations of the system about the equilibrium position \(\theta = 0\).

7 \includegraphics[max width=\textwidth, alt={}, center]{ab760a4b-e0ec-4256-838f-ed6c762ff18b-4_783_783_255_641} A uniform circular disc with centre \(C\) has mass \(m\) and radius \(a\). The disc is free to rotate in a vertical plane about a fixed horizontal axis passing through a point \(A\) on the disc, where \(A C = \frac { 1 } { 2 } a\). The disc is slightly disturbed from rest in the position with \(C\) vertically above \(A\). When \(A C\) makes an angle \(\theta\) with the upward vertical the force exerted by the axis on the disc has components \(R\) parallel to \(A C\) and \(S\) perpendicular to \(A C\) (see diagram).

1. Show that the angular speed of the disc is \(\sqrt { \frac { 4 g ( 1 - \cos \theta ) } { 3 a } }\).
2. Find the angular acceleration of the disc, in terms of \(a , g\) and \(\theta\).
3. Find \(R\) and \(S\), in terms of \(m , g\) and \(\theta\).
4. Find the magnitude of the force exerted by the axis on the disc at an instant when \(R = 0\).

6 \includegraphics[max width=\textwidth, alt={}, center]{6e3d5f5e-7ffa-4111-903d-468fb4d20192-4_640_608_267_715} A smooth wire forms a circle with centre \(O\) and radius \(a\), and is fixed in a vertical plane. The highest point on the wire is \(A\). A small ring \(R\) of mass \(m\) moves along the wire. A light elastic string, with natural length \(\frac { 1 } { 2 } a\) and modulus of elasticity \(2 m g\), has one end attached to \(A\) and the other end attached to \(R\). The string \(A R\) makes an angle \(\theta\) (measured anticlockwise) with the downward vertical (see diagram), and you may assume that the string does not become slack.

1. Taking \(A\) as the reference level for gravitational potential energy, show that the total potential energy of the system is \(m g a \left( 6 \cos ^ { 2 } \theta - 4 \cos \theta + \frac { 1 } { 2 } \right)\).
2. Show that there are two positions of equilibrium for which \(0 \leqslant \theta < \frac { 1 } { 2 } \pi\).
3. For each of these positions of equilibrium, determine whether it is stable or unstable.

7 \includegraphics[max width=\textwidth, alt={}, center]{6e3d5f5e-7ffa-4111-903d-468fb4d20192-5_584_686_264_678} \(A B C D\) is a uniform rectangular lamina with mass \(m\) and sides \(A B = 6 a\) and \(A D = 8 a\). The lamina rotates freely in a vertical plane about a fixed horizontal axis passing through \(A\), and it is released from rest in the position with \(D\) vertically above \(A\). When the diagonal \(A C\) makes an angle \(\theta\) below the horizontal, the force acting on the lamina at \(A\) has components \(R\) parallel to \(C A\) and \(S\) perpendicular to \(C A\) (see diagram).

1. Find the moment of inertia of the lamina about the axis through \(A\), in terms of \(m\) and \(a\).
2. Show that the angular speed of the lamina is \(\sqrt { \frac { 3 g ( 4 + 5 \sin \theta ) } { 50 a } }\).
3. Find the angular acceleration of the lamina, in terms of \(a , g\) and \(\theta\).
4. Find \(R\) and \(S\), in terms of \(m , g\) and \(\theta\).

2 A uniform solid circular cone has mass \(M\) and base radius \(R\).

1. Show by integration that the moment of inertia of the cone about its axis of symmetry is \(\frac { 3 } { 10 } M R ^ { 2 }\). (You may assume the standard formula \(\frac { 1 } { 2 } m r ^ { 2 }\) for the moment of inertia of a uniform disc about its axis and that the volume of a cone is \(\frac { 1 } { 3 } \pi r ^ { 2 } h\).) The axis of symmetry of the cone is fixed vertically and the cone is rotating about its axis at an angular speed of \(6 \mathrm { rad } \mathrm { s } ^ { - 1 }\). A frictional couple of constant moment 0.027 Nm is applied to the cone bringing it to rest. Given that the mass of the cone is 2 kg and its base radius is 0.3 m , find
2. the constant angular deceleration of the cone,
3. the time taken for the cone to come to rest from the instant that the couple is applied.

4 A uniform square lamina has mass \(m\) and sides of length \(2 a\).

1. Calculate the moment of inertia of the lamina about an axis through one of its corners perpendicular to its plane. \includegraphics[max width=\textwidth, alt={}, center]{639c658e-0aca-4161-9e77-0f4c494b0b55-3_693_640_434_715} The uniform square lamina has centre \(C\) and is free to rotate in a vertical plane about a fixed horizontal axis passing through one of its corners \(A\). The lamina is initially held such that \(A C\) is vertical with \(C\) above \(A\). The lamina is slightly disturbed from rest from this initial position. When \(A C\) makes an angle \(\theta\) with the upward vertical, the force exerted by the axis on the lamina has components \(X\) parallel to \(A C\) and \(Y\) perpendicular to \(A C\) (see diagram).
2. Show that the angular speed, \(\omega\), of the lamina satisfies \(a \omega ^ { 2 } = \frac { 3 } { 4 } g \sqrt { 2 } ( 1 - \cos \theta )\).
3. Find \(X\) and \(Y\) in terms of \(m , g\) and \(\theta\). \section*{Question 5 begins on page 4.}
   \includegraphics[max width=\textwidth, alt={}]{639c658e-0aca-4161-9e77-0f4c494b0b55-4_767_337_248_863}
   A pendulum consists of a uniform rod \(A B\) of length \(4 a\) and mass \(4 m\) and a spherical shell of radius \(a\), mass \(m\) and centre \(C\). The end \(B\) of the rod is rigidly attached to a point on the surface of the shell in such a way that \(A B C\) is a straight line. The pendulum is initially at rest with \(B\) vertically below \(A\) and it is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through \(A\) (see diagram).

1. Show that the moment of inertia of the pendulum about the axis of rotation is \(47 m a ^ { 2 }\). A particle of mass \(m\) is moving horizontally in the plane in which the pendulum is free to rotate. The particle has speed \(\sqrt { k g a }\), where \(k\) is a positive constant, and strikes the rod at a distance \(3 a\) from \(A\). In the subsequent motion the particle adheres to the rod and the combined rigid body \(P\) starts to rotate.
2. Show that the initial angular speed of \(P\) is \(\frac { 3 } { 56 } \sqrt { \frac { k g } { a } }\).
3. For the case \(k = 4\), find the angle that \(P\) has turned through when \(P\) first comes to instantaneous rest.
4. Find the least value of \(k\) such that the rod reaches the horizontal. \includegraphics[max width=\textwidth, alt={}, center]{639c658e-0aca-4161-9e77-0f4c494b0b55-5_437_903_269_573} A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The rod can rotate in a vertical plane about a smooth fixed horizontal axis passing through \(A\). One end of a light elastic string of natural length \(a\) and modulus of elasticity \(\sqrt { 3 } m g\) is attached to \(A\). The string passes over a small smooth fixed pulley \(C\), where \(A C\) is horizontal and \(A C = a\). The other end of the string is attached to the rod at its mid-point \(D\). The rod makes an angle \(\theta\) below the horizontal (see diagram).

1. Taking \(A\) as the reference level for gravitational potential energy, show that the total potential energy \(V\) of the system is given by $$V = m g a ( \sqrt { 3 } - \sin \theta - \sqrt { 3 } \cos \theta ) .$$
2. Show that \(\theta = \frac { 1 } { 6 } \pi\) is a position of stable equilibrium for the system. The system is making small oscillations about the equilibrium position.
3. By differentiating the energy equation with respect to time, show that $$\frac { 4 } { 3 } a \ddot { \theta } = g ( \cos \theta - \sqrt { 3 } \sin \theta ) .$$
4. Using the substitution \(\theta = \phi + \frac { 1 } { 6 } \pi\), show that the motion is approximately simple harmonic, and find the approximate period of the oscillations. \section*{END OF QUESTION PAPER}

6 A pendulum consists of a uniform rod \(A B\) of length \(2 a\) and mass \(2 m\) and a particle of mass \(m\) that is attached to the end \(B\). The pendulum can rotate in a vertical plane about a smooth fixed horizontal axis passing through \(A\).

1. Show that the moment of inertia of this pendulum about the axis of rotation is \(\frac { 20 } { 3 } m a ^ { 2 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4b50b084-081f-48d2-ad5b-95b2c9e55dfc-4_572_86_852_575} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4b50b084-081f-48d2-ad5b-95b2c9e55dfc-4_582_456_842_1050} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} The pendulum is initially held with \(B\) vertically above \(A\) (see Fig.1) and it is slightly disturbed from this position. When the angle between the pendulum and the upward vertical is \(\theta\) radians the pendulum has angular speed \(\omega \mathrm { rads } ^ { - 1 }\) (see Fig. 2).
2. Show that $$\omega ^ { 2 } = \frac { 6 g } { 5 a } ( 1 - \cos \theta ) .$$
3. Find the angular acceleration of the pendulum in terms of \(g , a\) and \(\theta\). At an instant when \(\theta = \frac { 1 } { 3 } \pi\), the force acting on the pendulum at \(A\) has magnitude \(F\).
4. Find \(F\) in terms of \(m\) and \(g\). It is given that \(a = 0.735 \mathrm {~m}\).
5. Show that the time taken for the pendulum to move from the position \(\theta = \frac { 1 } { 6 } \pi\) to the position \(\theta = \frac { 1 } { 3 } \pi\) is given by $$k \int _ { \frac { 1 } { 6 } \pi } ^ { \frac { 1 } { 3 } \pi } \operatorname { cosec } \left( \frac { 1 } { 2 } \theta \right) \mathrm { d } \theta ,$$ stating the value of the constant \(k\). Hence find the time taken for the pendulum to rotate between these two points. (You may quote an appropriate result given in the List of Formulae (MF1).) \section*{END OF QUESTION PAPER}

1 A uniform rod with centre \(C\) has mass \(2 M\) and length 4a. The rod is free to rotate in a vertical plane about a smooth fixed horizontal axis passing through a point \(A\) on the rod, where \(A C = k a\) and \(0 < k < 2\). The rod is making small oscillations about the equilibrium position with period \(T\).

1. Show that \(T = 2 \pi \sqrt { \frac { a } { 3 g } \left( \frac { 4 + 3 k ^ { 2 } } { k } \right) }\). (You may assume the standard formula \(T = 2 \pi \sqrt { \frac { I } { m g h } }\) for the period of small oscillations of a compound pendulum.)
2. Hence find the value of \(k ^ { 2 }\) for which the period of oscillations is least.

3 \includegraphics[max width=\textwidth, alt={}, center]{57323af2-8cf3-4721-b2c8-a968264be343-2_439_444_1318_822} A uniform rod \(A B\) has mass \(m\) and length \(4 a\). The rod can rotate in a vertical plane about a smooth fixed horizontal axis passing through \(A\). One end of a light elastic string of natural length \(a\) and modulus of elasticity \(\lambda m g\) is attached to \(B\). The other end of the string is attached to a small light ring which slides on a fixed smooth horizontal rail which is in the same vertical plane as the rod. The rail is a vertical distance \(3 a\) above \(A\). The string is always vertical and the rod makes an angle \(\theta\) radians with the horizontal, where \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\) (see diagram).

1. Taking \(A\) as the reference level for gravitational potential energy, find an expression for the total potential energy \(V\) of the system, and show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 2 m g a \cos \theta ( 4 \lambda ( 1 + 2 \sin \theta ) - 1 ) .$$ Determine the positions of equilibrium and the nature of their stability in the cases
2. \(\lambda > \frac { 1 } { 12 }\),
3. \(\lambda < \frac { 1 } { 12 }\). \includegraphics[max width=\textwidth, alt={}, center]{57323af2-8cf3-4721-b2c8-a968264be343-3_392_689_269_671} The diagram shows the curve with equation \(y = \frac { 1 } { 2 } \ln x\). The region \(R\), shaded in the diagram, is bounded by the curve, the \(x\)-axis and the line \(x = 4\). A uniform solid of revolution is formed by rotating \(R\) completely about the \(y\)-axis to form a solid of volume \(V\).

1. Show that \(V = \frac { 1 } { 4 } \pi ( 64 \ln 2 - 15 )\).
2. Find the exact \(y\)-coordinate of the centre of mass of the solid. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{57323af2-8cf3-4721-b2c8-a968264be343-4_385_741_269_646} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} Fig. 1 shows part of the line \(y = \frac { a } { h } x\), where \(a\) and \(h\) are constants. The shaded region bounded by the line, the \(x\)-axis and the line \(x = h\) is rotated about the \(x\)-axis to form a uniform solid cone of base radius \(a\), height \(h\) and volume \(\frac { 1 } { 3 } \pi a ^ { 2 } h\). The mass of the cone is \(M\).

1. Show by integration that the moment of inertia of the cone about the \(y\)-axis is \(\frac { 3 } { 20 } M \left( a ^ { 2 } + 4 h ^ { 2 } \right)\). (You may assume the standard formula \(\frac { 1 } { 4 } m r ^ { 2 }\) for the moment of inertia of a uniform disc about a diameter.) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{57323af2-8cf3-4721-b2c8-a968264be343-4_501_556_1238_726} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A uniform solid cone has mass 3 kg , base radius 0.4 m and height 1.2 m . The cone can rotate about a fixed vertical axis passing through its centre of mass with the axis of the cone moving in a horizontal plane. The cone is rotating about this vertical axis at an angular speed of \(9.6 \mathrm { rad } \mathrm { s } ^ { - 1 }\). A stationary particle of mass \(m \mathrm {~kg}\) becomes attached to the vertex of the cone (see Fig. 2). The particle being attached to the cone causes the angular speed to change instantaneously from \(9.6 \mathrm { rad } \mathrm { s } ^ { - 1 }\) to \(7.8 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
2. Find the value of \(m\). \includegraphics[max width=\textwidth, alt={}, center]{57323af2-8cf3-4721-b2c8-a968264be343-5_534_501_255_767} A triangular frame \(A B C\) consists of three uniform rods \(A B , B C\) and \(C A\), rigidly joined at \(A , B\) and \(C\). Each rod has mass \(m\) and length \(2 a\). The frame is free to rotate in a vertical plane about a fixed horizontal axis passing through \(A\). The frame is initially held such that the axis of symmetry through \(A\) is vertical and \(B C\) is below the level of \(A\). The frame starts to rotate with an initial angular speed of \(\omega\) and at time \(t\) the angle between the axis of symmetry through \(A\) and the vertical is \(\theta\) (see diagram).

1. Show that the moment of inertia of the frame about the axis through \(A\) is \(6 m a ^ { 2 }\).
2. Show that the angular speed \(\dot { \theta }\) of the frame when it has turned through an angle \(\theta\) satisfies $$a \dot { \theta } ^ { 2 } = a \omega ^ { 2 } - k g \sqrt { 3 } ( 1 - \cos \theta ) ,$$ stating the exact value of the constant \(k\).
   Hence find, in terms of \(a\) and \(g\), the set of values of \(\omega ^ { 2 }\) for which the frame makes complete revolutions. At an instant when \(\theta = \frac { 1 } { 6 } \pi\), the force acting on the frame at \(A\) has magnitude \(F\).
3. Given that \(\omega ^ { 2 } = \frac { 2 g } { a \sqrt { 3 } }\), find \(F\) in terms of \(m\) and \(g\). \section*{END OF QUESTION PAPER}

2 A rigid circular hoop of radius \(a\) is fixed in a vertical plane. At the highest point of the hoop there is a small smooth pulley, P. A light inextensible string AB of length \(\frac { 5 } { 2 } a\) is passed over the pulley. A particle of mass \(m\) is attached to the string at \(\mathrm { B } . \mathrm { PB }\) is vertical and angle \(\mathrm { APB } = \theta\). A small smooth ring of mass \(m\) is threaded onto the hoop and attached to the string at A . This situation is shown in Fig. 2. \begin{figure}[h] \end{figure}

1. Show that \(\mathrm { PB } = \frac { 5 } { 2 } a - 2 a \cos \theta\) and hence show that the potential energy of the system relative to P is \(V = - m g a \left( 2 \cos ^ { 2 } \theta - 2 \cos \theta + \frac { 5 } { 2 } \right)\).
2. Hence find the positions of equilibrium and investigate their stability.

2 A rigid circular hoop of radius \(a\) is fixed in a vertical plane. At the highest point of the hoop there is a small smooth pulley, P. A light inextensible string AB of length \(\frac { 5 } { 2 } a\) is passed over the pulley. A particle of mass \(m\) is attached to the string at \(\mathrm { B } . \mathrm { PB }\) is vertical and angle \(\mathrm { APB } = \theta\). A small smooth ring of mass \(m\) is threaded onto the hoop and attached to the string at A . This situation is shown in Fig. 2. \begin{figure}[h] \end{figure}

1. Show that \(\mathrm { PB } = \frac { 5 } { 2 } a - 2 a \cos \theta\) and hence show that the potential energy of the system relative to P is \(V = - m g a \left( 2 \cos ^ { 2 } \theta - 2 \cos \theta + \frac { 5 } { 2 } \right)\).
2. Hence find the positions of equilibrium and investigate their stability.

2 A uniform rigid rod AB of mass \(m\) and length \(4 a\) is freely hinged at the end A to a horizontal rail. The end B is attached to a light elastic string BC of modulus \(\frac { 1 } { 2 } m g\) and natural length \(a\). The end C of the string is attached to a ring which is small, light and smooth. The ring can slide along the rail and is always vertically above B . The angle that AB makes below the rail is \(\theta\). The system is shown in Fig. 2. \begin{figure}[h] \end{figure}

1. Find the potential energy, \(V\), of the system when the string is stretched and show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 4 m g a \cos \theta ( 2 \sin \theta - 1 )$$
2. Hence find any positions of equilibrium of the system and investigate their stability.

3 A uniform rod AB of mass \(m\) and length \(4 a\) is hinged at a fixed point C , where \(\mathrm { AC } = a\), and can rotate freely in a vertical plane. A light elastic string of natural length \(2 a\) and modulus \(\lambda\) is attached at one end to B and at the other end to a small light ring which slides on a fixed smooth horizontal rail which is in the same vertical plane as the rod. The rail is a vertical distance \(2 a\) above C . The string is always vertical. This system is shown in Fig. 3 with the rod inclined at \(\theta\) to the horizontal. \begin{figure}[h] \end{figure}

1. Find an expression for \(V\), the potential energy of the system relative to C , and show that \(\frac { \mathrm { d } V } { \mathrm {~d} \theta } = a \cos \theta \left( \frac { 9 } { 2 } \lambda \sin \theta - m g \right)\).
2. Determine the positions of equilibrium and the nature of their stability in the cases
   (A) \(\lambda > \frac { 2 } { 9 } m g\),
   (B) \(\lambda < \frac { 2 } { 9 } m g\),
   (C) \(\lambda = \frac { 2 } { 9 } m g\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cb86219c-e0b1-4f75-b8b2-50b5a233aa54-3_522_755_342_696} \captionsetup{labelformat=empty} \caption{Fig. 4.1}
   \end{figure}
3. Show, by integration, that the moment of inertia of the cone about its axis of symmetry is \(\frac { 3 } { 10 } M a ^ { 2 }\). [You may assume the standard formula for the moment of inertia of a uniform circular disc about its axis of symmetry and the formula \(V = \frac { 1 } { 3 } \pi r ^ { 2 } h\) for the volume of a cone.] A frustum is made by taking a uniform cone of base radius 0.1 m and height 0.2 m and removing a cone of height 0.1 m and base radius 0.05 m as shown in Fig. 4.2. The mass of the frustum is 2.8 kg . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{cb86219c-e0b1-4f75-b8b2-50b5a233aa54-3_391_517_1352_813} \captionsetup{labelformat=empty} \caption{Fig. 4.2}
   \end{figure} The frustum can rotate freely about its axis of symmetry which is fixed and vertical.
4. Show that the moment of inertia of the frustum about its axis of symmetry is \(0.0093 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). The frustum is accelerated from rest for \(t\) seconds by a couple of magnitude 0.05 N m about its axis of symmetry, until it is rotating at \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
5. Calculate \(t\).
6. Find the position of G , the centre of mass of the frustum. The frustum (rotating at \(10 \mathrm { rad } \mathrm { s } ^ { - 1 }\) ) then receives an impulse tangential to the circumference of the larger circular face. This reduces its angular speed to \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\).
7. To reduce its angular speed further, a parallel impulse of the same magnitude is now applied tangentially in the horizontal plane through G at the curved surface of the frustum. Calculate the resulting angular speed.

5. A uniform circular disc has mass \(m\) and radius \(a\). The disc can rotate freely about an axis that is in the same plane as the disc and tangential to the disc at a point \(A\) on its circumference. The disc hangs at rest in equilibrium with its centre \(O\) vertically below \(A\). A particle \(P\) of mass \(m\) is moving horizontally and perpendicular to the disc with speed \(\sqrt { } ( k g a )\), where \(k\) is a constant. The particle then strikes the disc at \(O\) and adheres to it at \(O\). Given that the disc rotates through an angle of \(90 ^ { \circ }\) before first coming to instantaneous rest, find the value of \(k\).
(Total 10 marks)

8. Four uniform rods, each of mass \(m\) and length \(2 a\), are joined together at their ends to form a plane rigid square framework \(A B C D\) of side \(2 a\). The framework is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\). The axis is perpendicular to the plane of the framework.

1. Show that the moment of inertia of the framework about the axis is \(\frac { 40 m a ^ { 2 } } { 3 }\). The framework is slightly disturbed from rest when \(C\) is vertically above \(A\). Find
2. the angular acceleration of the framework when \(A C\) is horizontal,
3. the angular speed of the framework when \(A C\) is horizontal,
4. the magnitude of the force acting on the framework at \(A\) when \(A C\) is horizontal.

2. Three forces, \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body. \(\mathbf { F } _ { 1 } = ( 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } ) \mathrm { N } , \mathbf { F } _ { 2 } = ( \mathbf { i } + \mathbf { j } - 4 \mathbf { k } )\) N and \(\mathbf { F } _ { 3 } = ( p \mathbf { i } + q \mathbf { j } + r \mathbf { k } ) \mathrm { N }\), where \(p , q\) and \(r\) are constants. All three forces act through the point with position vector ( \(3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\) ) m, relative to a fixed origin. The three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) are equivalent to a single force ( \(5 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k }\) ) N, acting at the origin, together with a couple \(\mathbf { G }\).

1. Find the values of \(p , q\) and \(r\).
2. Find \(\mathbf { G }\).

3. \section*{Figure 1} Figure 1 shows a box in the shape of a cuboid \(P Q R S T U V W\) where \(\overrightarrow { P Q } = 3 \mathbf { i }\) metres, \(\overrightarrow { P S } = 4 \mathbf { j }\) metres and \(\overrightarrow { P T } = 3 \mathbf { k }\) metres. A force \(( 4 \mathbf { i } - 2 \mathbf { j } ) \mathrm { N }\) acts at \(Q\), a force \(( 4 \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) acts at \(R\), a force \(( - 2 \mathbf { j } + \mathbf { k } ) \mathrm { N }\) acts at \(T\), and a force \(( 2 \mathbf { j } + \mathbf { k } ) \mathrm { N }\) acts at \(W\). Given that these are the only forces acting on the box, find

1. the resultant force acting on the box,
2. the resultant vector moment about \(P\) of the four forces acting on the box. When an additional force \(\mathbf { F }\) acts on the box at a point \(X\) on the edge \(P S\), the box is in equilibrium.
3. Find \(\mathbf { F }\).
4. Find the length of \(P X\).

1. Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body. \(\mathbf { F } _ { 1 } = ( 12 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { m } , \mathbf { F } _ { 2 } = ( - 3 \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\) and acts at the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\). The force \(\mathbf { F } _ { 3 }\) acts at the point with position vector \(( 2 \mathbf { i } - \mathbf { k } ) \mathrm { m }\).

Given that this set of forces is equivalent to a couple, find

1. \(\mathbf { F } _ { 3 }\),
2. the magnitude of the couple.
   

Two forces \(\mathbf { F } _ { 1 } = ( 2 \mathbf { i } + \mathbf { j } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( - 2 \mathbf { j } - \mathbf { k } ) \mathrm { N }\) act on a rigid body. The force \(\mathbf { F } _ { 1 }\) acts at the point with position vector \(\mathbf { r } _ { 1 } = ( 3 \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) and the force \(\mathbf { F } _ { 2 }\) acts at the point with position vector \(\mathbf { r } _ { 2 } = ( \mathbf { i } - 2 \mathbf { j } ) \mathrm { m }\). A third force \(\mathbf { F } _ { 3 }\) acts on the body such that \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) are in equilibrium.

1. Find the magnitude of \(\mathbf { F } _ { 3 }\).
2. Find a vector equation of the line of action of \(\mathbf { F } _ { 3 }\). The force \(\mathbf { F } _ { 3 }\) is replaced by a fourth force \(\mathbf { F } _ { 4 }\), acting through the origin \(O\), such that \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 4 }\) are equivalent to a couple.
3. Find the magnitude of this couple.

1. Two forces \(\mathbf { F } _ { 1 } = ( \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { N }\) and \(\mathbf { F } _ { 2 } = ( 3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } ) \mathrm { N }\) act on a rigid body.

The force \(\mathbf { F } _ { 1 }\) acts through the point with position vector ( \(2 \mathbf { i } + \mathbf { k }\) ) m and the force \(\mathbf { F } _ { 2 }\) acts through the point with position vector \(( \mathbf { j } + 2 \mathbf { k } ) \mathrm { m }\).

1. If the two forces are equivalent to a single force \(\mathbf { R }\), find
   1. \(\mathbf { R }\),
   2. a vector equation of the line of action of \(\mathbf { R }\), in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\).
2. If the two forces are equivalent to a single force acting through the point with position vector \(( \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) \mathrm { m }\) together with a couple of moment \(\mathbf { G }\), find the magnitude of \(\mathbf { G }\).

Three forces \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) act on a rigid body. The forces \(\mathbf { F } _ { 1 }\) and \(\mathbf { F } _ { 2 }\) act through the points with position vectors \(\mathbf { r } _ { 1 }\) and \(\mathbf { r } _ { 2 }\) respectively. \(\mathbf { r } _ { 1 } = ( - 2 \mathbf { i } + 3 \mathbf { j } ) \mathrm { m }\), \(\mathbf { F } _ { 1 } = ( 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } ) \mathrm { N }\) \(\mathbf { r } _ { 2 } = ( 3 \mathbf { i } + 2 \mathbf { k } ) \mathrm { m }\), \(\mathbf { F } _ { 2 } = ( - 2 \mathbf { i } + \mathbf { j } - \mathbf { k } ) \mathrm { N }\) Given that the system \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 3 }\) is in equilibrium,

1. find \(\mathbf { F } _ { 3 }\),
2. find a vector equation of the line of action of \(\mathbf { F } _ { 3 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\). The force \(\mathbf { F } _ { 3 }\) is replaced by a force \(\mathbf { F } _ { 4 }\) acting through the point with position vector \(( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }\). The system \(\mathbf { F } _ { 1 } , \mathbf { F } _ { 2 }\) and \(\mathbf { F } _ { 4 }\) is equivalent to a single force ( \(3 \mathbf { i } + \mathbf { j } + \mathbf { k }\) ) N acting through the point with position vector \(( \mathbf { i } + \mathbf { j } + \mathbf { k } ) \mathrm { m }\) together with a couple.
3. Find the magnitude of this couple.