6.04e Rigid body equilibrium: coplanar forces

6. A smooth wire, with ends \(A\) and \(B\), is in the shape of a semicircle of radius \(r\). The line \(A B\) is horizontal and the midpoint of \(A B\) is \(O\). The wire is fixed in a vertical plane. A small ring \(R\) of mass \(2 m\) is threaded on the wire and is attached to two light inextensible strings. One string passes through a small smooth ring fixed at \(A\) and is attached to a particle of mass \(\sqrt { 6 } m\). The other string passes through a small smooth ring fixed at \(B\) and is attached to a second particle of mass \(\sqrt { 6 } \mathrm {~m}\). The particles hang freely under gravity, as shown in Figure 3. The angle between the radius \(O R\) and the downward vertical is \(2 \theta\), where \(- \frac { \pi } { 4 } < \theta < \frac { \pi } { 4 }\)

1. Show that the potential energy of the system is $$2 m g r ( 2 \sqrt { 3 } \cos \theta - \cos 2 \theta ) + \text { constant }$$
2. Find the values of \(\theta\) for which the system is in equilibrium.
3. Determine the stability of the position of equilibrium for which \(\theta > 0\)

8 \includegraphics[max width=\textwidth, alt={}, center]{98647526-b52a-4316-9a09-48d756b8f599-3_493_748_1393_708} The diagram shows a uniform rod \(A B\), of mass \(m\) and length \(2 a\), free to rotate in a vertical plane about a fixed horizontal axis through \(A\). A light elastic string has natural length \(a\) and modulus of elasticity \(\frac { 1 } { 2 } m g\). The string joins \(B\) to a light ring \(R\) which slides along a smooth horizontal wire fixed at a height \(a\) above \(A\) and in the same vertical plane as \(A B\). The string \(B R\) remains vertical. The angle between \(A B\) and the horizontal is denoted by \(\theta\), where \(0 < \theta < \pi\).

1. Taking the reference level for gravitational potential energy to be the horizontal through \(A\), show that the total potential energy of the system is $$m g a \left( \sin ^ { 2 } \theta - \sin \theta \right) .$$
2. Find the three values of \(\theta\) for which the system is in equilibrium.
3. For each position of equilibrium, determine whether it is stable or unstable.

7 \includegraphics[max width=\textwidth, alt={}, center]{de53978b-aa96-4fa2-a928-81a16450154e-4_557_1036_278_553} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is pivoted to a fixed point at \(A\) and is free to rotate in a vertical plane. Two fixed vertical wires in this plane are a distance \(6 a\) apart and the point \(A\) is half-way between the two wires. Light smooth rings \(R _ { 1 }\) and \(R _ { 2 }\) slide on the wires and are connected to \(B\) by light elastic strings, each of natural length \(a\) and modulus of elasticity \(\frac { 1 } { 4 } m g\). The strings \(B R _ { 1 }\) and \(B R _ { 2 }\) are always horizontal and the angle between \(A B\) and the upward vertical is \(\theta\), where \(- \frac { 1 } { 2 } \pi < \theta < \frac { 1 } { 2 } \pi\) (see diagram).

1. Taking \(A\) as the reference level for gravitational potential energy, show that the total potential energy of the system is $$m g a \left( 1 + \cos \theta + \sin ^ { 2 } \theta \right) .$$
2. Given that \(\theta = 0\) is a position of stable equilibrium, find the approximate period of small oscillations about this position.

6 \includegraphics[max width=\textwidth, alt={}, center]{fb9e4e4a-953b-4e52-858e-438b4009e79c-3_428_595_221_806} A uniform rod \(A B\), of mass \(m\) and length \(2 a\), is free to rotate in a vertical plane about a fixed horizontal axis through \(A\). A light elastic string has natural length \(a\) and modulus of elasticity \(m g\); one end is attached to \(B\) and the other end is attached to a light ring \(R\) which can slide along a smooth vertical wire. The wire is in the same vertical plane as \(A B\), and is at a distance \(a\) from \(A\). The rod \(A B\) makes an angle \(\theta\) with the upward vertical, where \(0 < \theta < \frac { 1 } { 2 } \pi\) (see diagram).

1. Give a reason why the string \(R B\) is always horizontal.
2. By considering potential energy, find the value of \(\theta\) for which the system is in equilibrium.
3. Determine whether this position of equilibrium is stable or unstable.

7 A uniform rod \(A B\) has mass \(m\) and length \(2 a\). The point \(P\) on the rod is such that \(A P = \frac { 2 } { 3 } a\).

1. Prove by integration that the moment of inertia of the rod about an axis through \(P\) perpendicular to \(A B\) is \(\frac { 4 } { 9 } m a ^ { 2 }\). The axis through \(P\) is fixed and horizontal, and the rod can rotate without resistance in a vertical plane about this axis. The rod is released from rest in a horizontal position. Find, in terms of \(m\) and \(g\),
2. the force acting on the rod at \(P\) immediately after the release of the rod,
3. the force acting on the rod at \(P\) at an instant in the subsequent motion when \(B\) is vertically below \(P\).

7 \includegraphics[max width=\textwidth, alt={}, center]{b86c4b97-13a9-4aaf-8c95-20fe043b4532-3_585_801_991_647} A light rod \(A B\) of length \(2 a\) can rotate freely in a vertical plane about a fixed horizontal axis through \(A\). A particle of mass \(m\) is attached to the rod at \(B\). A fixed smooth ring \(R\) lies in the same vertical plane as the rod, where \(A R = a\) and \(A R\) makes an angle \(\frac { 1 } { 4 } \pi\) above the horizontal. A light elastic string, of natural length \(a\) and modulus of elasticity \(m g \sqrt { } 2\), passes through the ring \(R\); one end is fixed to \(A\) and the other end is fixed to \(B\). The rod makes an angle \(\theta\) below the horizontal, where \(- \frac { 1 } { 4 } \pi < \theta < \frac { 3 } { 4 } \pi\) (see diagram).

1. Use the cosine rule to show that \(R B ^ { 2 } = a ^ { 2 } ( 5 - ( 2 \sqrt { } 2 ) \cos \theta + ( 2 \sqrt { } 2 ) \sin \theta )\).
2. Show that \(\theta = 0\) is a position of stable equilibrium.
3. Show that \(\frac { \mathrm { d } ^ { 2 } \theta } { \mathrm {~d} t ^ { 2 } } = - k \sin \theta\), expressing the constant \(k\) in terms of \(a\) and \(g\), and hence write down the approximate period of small oscillations about the equilibrium position \(\theta = 0\).

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 \(R = 0\).

3 The region \(R\) is bounded by the \(x\)-axis, the \(y\)-axis, the curve \(y = a \mathrm { e } ^ { \frac { x } { a } }\) and the line \(x = a \ln 2\) (where \(a\) is a positive constant). A uniform solid of revolution, of mass \(M\), is formed by rotating \(R\) through \(2 \pi\) radians about the \(x\)-axis. Find, in terms of \(M\) and \(a\), the moment of inertia about the \(x\)-axis of this solid of revolution.
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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}

4

1. Write down the moment of inertia of a uniform circular disc of mass \(m\) and radius \(2 a\) about a diameter. A uniform solid cylinder has mass \(M\), radius \(2 r\) and height \(h\).
2. Show by integration, and using the result from part (i), that the moment of inertia of the cylinder about a diameter of an end face is $$M \left( r ^ { 2 } + \frac { 1 } { 3 } h ^ { 2 } \right)$$ and hence find the moment of inertia of the cylinder about a diameter through the centre of the cylinder. \includegraphics[max width=\textwidth, alt={}, center]{4b50b084-081f-48d2-ad5b-95b2c9e55dfc-3_919_897_260_591} A smooth circular wire hoop, with centre \(O\) and radius \(r\), is fixed in a vertical plane. The highest point on the wire is \(H\). A small bead \(B\) of mass \(m\) is free to move along the wire. A light inextensible string of length \(a\), where \(a > 2 r\), has one end attached to the bead. The other end of the string passes over a small smooth pulley at \(H\) and carries at its end a particle \(P\) of mass \(\lambda m\), where \(\lambda\) is a positive constant. The part of the string \(H P\) is vertical and the part of the string \(B H\) makes an angle \(\theta\) radians with the downward vertical where \(0 \leqslant \theta \leqslant \frac { 1 } { 3 } \pi\) (see diagram). You may assume that \(P\) remains above the lowest point of the wire.

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.

3 A uniform rigid rod AB of length \(2 a\) and mass \(m\) is smoothly hinged to a fixed point at A so that it can rotate freely in a vertical plane. A light elastic string of modulus \(\lambda\) and natural length \(a\) connects the midpoint of AB to a fixed point C which is vertically above A with \(\mathrm { AC } = a\). The rod makes an angle \(2 \theta\) with the upward vertical, where \(\frac { 1 } { 3 } \pi \leqslant 2 \theta \leqslant \pi\). This is shown in Fig. 3. \begin{figure}[h] \end{figure}

1. Find the potential energy, \(V\), of the system relative to A in terms of \(m , \lambda , a\) and \(\theta\). Show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 2 a \cos \theta ( 2 \lambda \sin \theta - 2 m g \sin \theta - \lambda ) .$$ Assume now that the system is set up so that the result (*) continues to hold when \(\pi < 2 \theta \leqslant \frac { 5 } { 3 } \pi\).
2. In the case \(\lambda < 2 m g\), show that there is a stable position of equilibrium at \(\theta = \frac { 1 } { 2 } \pi\). Show that there are no other positions of equilibrium in this case.
3. In the case \(\lambda > 2 m g\), find the positions of equilibrium for \(\frac { 1 } { 3 } \pi \leqslant 2 \theta \leqslant \frac { 5 } { 3 } \pi\) and determine for each whether the equilibrium is stable or unstable, justifying your conclusions.

2 A uniform rod AB of length 0.5 m and mass 0.5 kg is freely hinged at A so that it can rotate in a vertical plane. Attached at B are two identical light elastic strings BC and BD each of natural length 0.5 m and stiffness \(2 \mathrm {~N} \mathrm {~m} ^ { - 1 }\). The ends C and D are fixed at the same horizontal level as A and with \(\mathrm { AC } = \mathrm { CD } = 0.5 \mathrm {~m}\). The system is shown in Fig. 2.1 with the angle \(\mathrm { BAC } = \theta\). You may assume that \(\frac { 1 } { 3 } \pi \leqslant \theta \leqslant \frac { 5 } { 3 } \pi\) so that both strings are taut. \begin{figure}[h] \end{figure}

1. Show that the length of BC in metres is \(\sin \frac { 1 } { 2 } \theta\).
2. Find the potential energy, \(V \mathrm {~J}\), of the system relative to AD in terms of \(\theta\). Hence show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 1.5 \sin \theta - 1.225 \cos \theta - \frac { 0.5 \sin \theta } { \sqrt { 1.25 - \cos \theta } } - 0.5 \cos \frac { 1 } { 2 } \theta .$$
3. Fig. 2.2 shows a graph of the function \(\mathrm { f } ( \theta ) = 1.5 \sin \theta - 1.225 \cos \theta - \frac { 0.5 \sin \theta } { \sqrt { 1.25 - \cos \theta } } - 0.5 \cos \frac { 1 } { 2 } \theta\) for \(0 \leqslant \theta \leqslant 2 \pi\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bc637a95-b469-493b-8fd4-d3b12049878b-2_453_1264_2021_397} \captionsetup{labelformat=empty} \caption{Fig. 2.2}
   \end{figure} Use the graph both to estimate, correct to 1 decimal place, the values of \(\theta\) for which the system is in equilibrium and also to determine their stability.

3 A uniform rigid rod AB of mass \(m\) and length \(2 a\) is freely hinged to a horizontal floor at A . The end B is attached to a light elastic string of modulus \(\lambda\) and natural length \(5 a\). The other end of the string is attached to a small, light, smooth ring C which can slide along a horizontal rail. The rail is a distance \(7 a\) above the floor and C is always vertically above B . The angle that AB makes with the floor is \(\theta\). The system is shown in Fig. 3. \begin{figure}[h] \end{figure}

1. Find the potential energy, \(V\), of the system and hence show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = a \cos \theta \left( m g - \frac { 4 \lambda } { 5 } ( 1 - \sin \theta ) \right) .$$
2. Show that there is a position of equilibrium when \(\theta = \frac { 1 } { 2 } \pi\) and determine whether or not it is stable. There are two further positions of equilibrium when \(0 < \theta < \pi\).
3. Find the magnitude of the tension in the string and the vertical force of the hinge on the rod in these positions.
4. Show that \(\lambda > \frac { 5 m g } { 4 }\).
5. Show that these positions of equilibrium are stable.

2 Fig. 2 shows a system in a vertical plane. A uniform rod AB of length \(2 a\) and mass \(m\) is freely hinged at A . The angle that AB makes with the horizontal is \(\theta\), where \(- \frac { 2 } { 3 } \pi < \theta < \frac { 2 } { 3 } \pi\). Attached at B is a light spring BC of natural length \(a\) and stiffness \(\frac { m g } { a }\). The other end of the spring is attached to a small light smooth ring C which can slide freely along a vertical rail. The rail is at a distance of \(a\) from A and the spring is always horizontal. \begin{figure}[h] \end{figure}

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

3 Fig. 3 shows a uniform rigid rod AB of length \(2 a\) and mass \(2 m\). The rod is freely hinged at A so that it can rotate in a vertical plane. One end of a light inextensible string of length \(l\) is attached to B . The string passes over a small smooth fixed pulley at C , where C is vertically above A and \(\mathrm { AC } = 6 a\). A particle of mass \(\lambda m\), where \(\lambda\) is a positive constant, is attached to the other end of the string and hangs freely, vertically below C . The rod makes an angle \(\theta\) with the upward vertical, where \(0 \leqslant \theta \leqslant \pi\). You may assume that the particle does not interfere with the rod AB or the section of the string BC . \begin{figure}[h] \end{figure}

1. Find the potential energy, \(V\), of the system relative to a situation in which the rod AB is horizontal, and hence show that $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = 2 m g a \sin \theta \left( \frac { 3 \lambda } { \sqrt { 10 - 6 \cos \theta } } - 1 \right) .$$
2. Show that \(\theta = 0\) and \(\theta = \pi\) are the only values of \(\theta\) for which the system is in equilibrium whatever the value of \(\lambda\).
3. Show that, if there is a third value of \(\theta\) for which the system is in equilibrium, then \(\frac { 2 } { 3 } < \lambda < \frac { 4 } { 3 }\).
4. Given that there are three positions of equilibrium, establish whether each of these positions is stable or unstable. It is given that, for small values of \(\theta\), $$\frac { \mathrm { d } V } { \mathrm {~d} \theta } \approx 2 m g a \left[ \left( \frac { 3 } { 2 } \lambda - 1 \right) \theta - \left( \frac { 13 } { 16 } \lambda - \frac { 1 } { 6 } \right) \theta ^ { 3 } \right] .$$
5. Investigate the stability of the equilibrium position given by \(\theta = 0\) in the case when \(\lambda = \frac { 2 } { 3 }\).

6. \begin{figure}[h] \end{figure} A rough uniform rod, of mass \(m\) and length \(4 a\), is rod is held on a rough horizontal table. The rod is perpendicular to the edge of the table and a length \(3 a\) projects horizontally over the edge, as shown in Fig. 1.

1. Show that the moment of inertia of the rod about the edge of the table is \(\frac { 7 } { 3 } m a ^ { 2 }\). The rod is released from rest and rotates about the edge of the table. When the rod has turned through an angle \(\theta\), its angular speed is \(\dot { \theta }\). Assuming that the rod has not started to slip,
2. show that \(\dot { \theta } ^ { 2 } = \frac { 6 g \sin \theta } { 7 a }\),
3. find the angular acceleration of the rod,
4. find the normal reaction of the table on the rod. The coefficient of friction between the rod and the edge of the table is \(\mu\).
5. Show that the rod starts to slip when \(\tan \theta = \frac { 4 } { 13 } \mu\)