3. A small pebble of mass \(m\) is placed in a viscous liquid and sinks vertically from rest through the liquid. When the speed of the pebble is \(v\) the magnitude of the resistance due to the liquid is modelled as \(m k v ^ { 2 }\), where \(k\) is a positive constant. Find the speed of the pebble after it has fallen a distance \(D\) through the liquid.
\includegraphics[max width=\textwidth, alt={}, center]{618fdb9c-cc0b-4a80-a148-4311c908c94e-3_813_699_397_674} Figure 1 shows a uniform rod \(A B\), of mass \(m\) and length \(4 a\), resting on a smooth fixed sphere of radius \(a\). A light elastic string, of natural length \(a\) and modulus of elasticity \(\frac { 3 } { 4 } m g\), has one end attached to the lowest point \(C\) of the sphere and the other end attached to \(A\). The points \(A\), \(B\) and \(C\) lie in a vertical plane with \(\angle B A C = 2 \theta\), where \(\theta < \frac { \pi } { 4 }\). Given that \(A C\) is always horizontal,

1. show that the potential energy of the system is $$\frac { m g a } { 8 } \left( 16 \sin 2 \theta + 3 \cot ^ { 2 } \theta - 6 \cot \theta \right) + \text { constant } ,$$
2. show that there is a value of \(\theta\) for which the system is in equilibrium such that \(0.535 < \theta < 0.545\).
3. Determine whether this position of equilibrium is stable or unstable.