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.
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\caption{Fig. 3}
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- Find the potential energy, \(V\), of the system and hence show that
$$\frac { \mathrm { d } V } { \mathrm {~d} \theta } = a \cos \theta \left( m g - \frac { 4 \lambda } { 5 } ( 1 - \sin \theta ) \right) .$$
- 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 \(0 < \theta < \pi\).
- Find the magnitude of the tension in the string and the vertical force of the hinge on the rod in these positions.
- Show that \(\lambda > \frac { 5 m g } { 4 }\).
- Show that these positions of equilibrium are stable.