Compound pendulum oscillations

4 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\) and \(P\) is held with the string taut and horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt { } ( 2 a g )\) so that it begins to move along a circular path. The string becomes slack when \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\).

1. Show that \(\cos \theta = \frac { 2 } { 3 }\).
2. Find the greatest height, above the horizontal through \(O\), reached by \(P\) in its subsequent motion. \includegraphics[max width=\textwidth, alt={}, center]{0f39ff02-a4fc-49ce-b87e-f70bef5a58b6-10_1049_744_260_696} A thin uniform \(\operatorname { rod } A B\) has mass \(\lambda M\) and length \(2 a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre \(O\), mass \(3 M\) and radius \(a\). The end \(B\) of the rod is rigidly attached to the surface of a uniform solid sphere with centre \(C\), mass \(5 M\) and radius \(a\). The rod lies along the line joining the centres of the spheres, so that \(C B A O\) is a straight line. The horizontal axis \(L\) is perpendicular to the rod and passes through the point of the rod that is a distance \(\frac { 1 } { 2 } a\) from \(B\) (see diagram). The object consisting of the rod and the two spheres can rotate freely about \(L\).

4 A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\) and \(P\) is held with the string taut and horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt { } ( 2 a g )\) so that it begins to move along a circular path. The string becomes slack when \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\).

1. Show that \(\cos \theta = \frac { 2 } { 3 }\).
2. Find the greatest height, above the horizontal through \(O\), reached by \(P\) in its subsequent motion. \includegraphics[max width=\textwidth, alt={}, center]{4240c99e-10ba-443e-8021-1872e6e64ccf-10_1051_744_258_696} A thin uniform \(\operatorname { rod } A B\) has mass \(\lambda M\) and length \(2 a\). The end \(A\) of the rod is rigidly attached to the surface of a uniform hollow sphere (spherical shell) with centre \(O\), mass \(3 M\) and radius \(a\). The end \(B\) of the rod is rigidly attached to the surface of a uniform solid sphere with centre \(C\), mass \(5 M\) and radius \(a\). The rod lies along the line joining the centres of the spheres, so that \(C B A O\) is a straight line. The horizontal axis \(L\) is perpendicular to the rod and passes through the point of the rod that is a distance \(\frac { 1 } { 2 } a\) from \(B\) (see diagram). The object consisting of the rod and the two spheres can rotate freely about \(L\).

5. \begin{figure}[h] \end{figure} A uniform circular lamina has radius \(2 a\) and centre \(C\). The points \(P , Q , R\) and \(S\) on the lamina are the vertices of a square with centre \(C\) and \(C P = a\). Four circular discs, each of radius \(\frac { a } { 2 }\), with centres \(P , Q , R\) and \(S\), are removed from the lamina. The remaining lamina forms a template \(T\), as shown in Figure 1. The radius of gyration of \(T\) about an axis through \(C\), perpendicular to \(T\), is \(k\).

1. Show that \(k ^ { 2 } = \frac { 55 a ^ { 2 } } { 24 }\) The template \(T\) is free to rotate in a vertical plane about a fixed smooth horizontal axis which is perpendicular to \(T\) and passes through a point on its outer rim.
2. Write down an equation of rotational motion for \(T\) and deduce that the period of small oscillations of \(T\) about its stable equilibrium position is $$2 \pi \sqrt { } \left( \frac { 151 a } { 48 g } \right)$$