Potential energy with inextensible strings or gravity only

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} 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.

7. \begin{figure}[h] \end{figure} A smooth wire \(A B\), in the shape of a circle of radius \(r\), is fixed in a vertical plane with \(A B\) vertical. A small smooth ring \(R\) of mass \(m\) is threaded on the wire and is connected by a light inextensible string to a particle \(P\) of mass \(m\). The length of the string is greater than the diameter of the circle. The string passes over a small smooth pulley which is fixed at the highest point \(A\) of the wire and angle \(R \hat { A } P = \theta\), as shown in Fig. 2.

1. Show that the potential energy of the system is given by $$2 m g r \left( \cos \theta - \cos ^ { 2 } \theta \right) + \text { constant. }$$
2. Hence determine the values of \(\theta , \theta \geq 0\), for which the system is in equilibrium. (6 marks)
3. Determine the stability of each position of equilibrium. END

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 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 \(\lambda = \frac { 2 } { 3 }\).

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.

\includegraphics{figure_2} Two uniform rods \(AB\) and \(AC\), each of mass \(2m\) and length \(2L\), are freely jointed at \(A\). The mid-points of the rods are \(D\) and \(E\) respectively. A light inextensible string of length \(s\) is fixed to \(E\) and passes round small, smooth light pulleys at \(D\) and \(A\). A particle \(P\) of mass \(m\) is attached to the other end of the string and hangs vertically. The points \(A\), \(B\) and \(C\) lie in the same vertical plane with \(B\) and \(C\) on a smooth horizontal surface. The angles \(PAB\) and \(PAC\) are each equal to \(\theta\) (\(\theta > 0\)), as shown in Fig. 2.

1. Find the length of \(AP\) in terms of \(s\), \(L\) and \(\theta\). [2]
2. Show that the potential energy \(V\) of the system is given by $$V = 2mgL(3\cos\theta + \sin\theta) + \text{constant}.$$ [4]
3. Hence find the value of \(\theta\) for which the system is in equilibrium. [4]
4. Determine whether this position of equilibrium is stable or unstable. [4]

\includegraphics{figure_1} A smooth wire with ends \(A\) and \(B\) is in the shape of a semi-circle of radius \(a\). The mid-point of \(AB\) is \(O\) and is fixed in a vertical plane and hangs below \(AB\) which is horizontal. A small ring \(R\), of mass \(m\sqrt{2}\), is threaded on the wire and is attached to two light inextensible strings. The other end of each string is attached to a particle of mass \(\frac{3m}{2}\). The particles hang vertically under gravity, as shown in Figure 1.

1. Show that, when the radius \(OR\) makes an angle \(2\theta\) with the vertical, the potential energy, \(V\), of the system is given by $$V = \sqrt{2}mga(3 \cos \theta - \cos 2\theta) + \text{constant}.$$ [7]
2. Find the values of \(\theta\) for which the system is in equilibrium. [6]
3. Determine the stability of the position of equilibrium for which \(\theta > 0\). [4]

\includegraphics{figure_1} A framework consists of two uniform rods \(AB\) and \(BC\), each of mass \(m\) and length \(2a\), joined at \(B\). The mid-points of the rods are joined by a light rod of length \(a\sqrt{2}\), so that angle \(ABC\) is a right angle. The framework is free to rotate in a vertical plane about a fixed smooth horizontal axis. This axis passes through the point \(A\) and is perpendicular to the plane of the framework. The angle between the rod \(AB\) and the downward vertical is denoted by \(\theta\), as shown in Fig. 1.

1. Show that the potential energy of the framework is $$-mga(3 \cos \theta + \sin \theta) + \text{constant}.$$ [4]
2. Find the value of \(\theta\) when the framework is in equilibrium, with \(B\) below the level of \(A\). [4]
3. Determine the stability of this position of equilibrium. [4]

\includegraphics{figure_3} A small smooth peg \(P\) is fixed at a distance \(d\) from a fixed smooth vertical wire. A particle of mass \(3m\) is attached to one end of a light inextensible string which passes over \(P\). The particle hangs vertically below \(P\). The other end of the string is attached to a small ring \(R\) of mass \(m\), which is threaded on the wire, as shown in Figure 3.

1. Show that when \(R\) is at a distance \(x\) below the level of \(P\) the potential energy of the system is $$3mg \sqrt{(x^2 + d^2)} - mgx + \text{constant}$$ [4]
2. Hence find \(x\), in terms of \(d\), when the system is in equilibrium. [3]
3. Determine the stability of the position of equilibrium. [3]

\includegraphics{figure_1} A uniform rod \(AB\), of length \(2l\) and mass \(12m\), has its end \(A\) smoothly hinged to a fixed point. One end of a light inextensible string is attached to the other end \(B\) of the rod. The string passes over a small smooth pulley which is fixed at the point \(C\), where \(AC\) is horizontal and \(AC = 2l\). A particle of mass \(m\) is attached to the other end of the string and the particle hangs vertically below \(C\). The angle \(BAC\) is \(\theta\), where \(0 < \theta < \frac{\pi}{2}\), as shown in Figure 1.

1. Show that the potential energy of the system is $$4mgl\left(\sin\frac{\theta}{2} - 3\sin\theta\right) + \text{constant}$$ (4)

1. Find the value of \(\theta\) when the system is in equilibrium and determine the stability of this equilibrium position. (10)