Find unknown parameter from period

4
\includegraphics[max width=\textwidth, alt={}, center]{137d2806-f45c-4121-8ee9-bf89580e1cca-3_906_1538_248_301} The end \(A\) of a uniform \(\operatorname { rod } A B\), of mass \(4 m\) and length \(3 a\), is rigidly attached to a point on a uniform spherical shell, of mass \(\lambda m\) and radius \(3 a\). The end \(B\) of the rod is rigidly attached to a point on a uniform ring. The ring has centre \(O\), mass \(4 m\) and radius \(\frac { 1 } { 2 } a\). The ring and the rod are in the same vertical plane. The line \(O B A\), extended, passes through the centre of the spherical shell. \(B C\) is a diameter of the ring (see diagram). Show that the moment of inertia of this system, about a fixed horizontal axis through \(C\) perpendicular to the plane of the ring, is \(( 30 + 55 \lambda ) m a ^ { 2 }\). Given that the system performs small oscillations of period \(2 \pi \sqrt { } \left( \frac { 5 a } { g } \right)\) about this axis, find the value of \(\lambda\).

4
\includegraphics[max width=\textwidth, alt={}, center]{a473cbb8-877f-48df-8751-c76d96396734-3_906_1538_248_301} The end \(A\) of a uniform \(\operatorname { rod } A B\), of mass \(4 m\) and length \(3 a\), is rigidly attached to a point on a uniform spherical shell, of mass \(\lambda m\) and radius \(3 a\). The end \(B\) of the rod is rigidly attached to a point on a uniform ring. The ring has centre \(O\), mass \(4 m\) and radius \(\frac { 1 } { 2 } a\). The ring and the rod are in the same vertical plane. The line \(O B A\), extended, passes through the centre of the spherical shell. \(B C\) is a diameter of the ring (see diagram). Show that the moment of inertia of this system, about a fixed horizontal axis through \(C\) perpendicular to the plane of the ring, is \(( 30 + 55 \lambda ) m a ^ { 2 }\). Given that the system performs small oscillations of period \(2 \pi \sqrt { } \left( \frac { 5 a } { g } \right)\) about this axis, find the value of \(\lambda\).

5
\includegraphics[max width=\textwidth, alt={}, center]{833c338f-53c1-436e-a772-0cdaf17fa72d-3_316_949_1320_598} The end \(B\) of a uniform rod \(A B\), of mass \(3 M\) and length \(4 a\), is rigidly attached to a point on the circumference of a uniform disc. The disc has centre \(O\), mass \(2 M\) and radius \(a\), and \(A B O\) is a straight line. The disc and the rod are in the same vertical plane. A particle \(P\), of mass \(M\), is attached to the rod at a distance \(k a\) from \(A\), where \(k\) is a positive constant (see diagram). Show that the moment of inertia of this system, about a fixed horizontal axis \(l\) through \(A\) perpendicular to the plane of the disc, is \(\left( 67 + k ^ { 2 } \right) M a ^ { 2 }\). The system is free to rotate about \(l\) and performs small oscillations of period \(4 \pi \sqrt { } \left( \frac { a } { g } \right)\). Find the possible values of \(k\).

An object is formed from a square frame \(A B C D\) with a square lamina attached in one corner of the frame. The frame consists of four identical thin rods, each of mass \(M\) and length \(2 a\). The lamina has mass \(k M\) and edges of length \(a\). It has one vertex at \(C\) and adjacent sides in contact with \(C B\) and \(C D\) (see diagram).

1. Show that the moment of inertia of the object about an axis \(l\) through \(A\) perpendicular to the plane of the object is \(\frac { 2 } { 3 } M a ^ { 2 } ( 7 k + 20 )\).
   The object is released from rest with the edge \(A B\) horizontal and \(D\) vertically above \(A\). The object rotates freely about the fixed axis \(l\). The angular speed of the object is \(\frac { 1 } { 2 } \sqrt { } \left( \frac { 5 g } { a } \right)\) when \(D\) is first vertically below \(A\).
2. Find the value of \(k\).

5 An object is formed from a uniform circular disc, of radius \(2 a\) and mass \(3 M\), and a uniform \(\operatorname { rod } A B\), of length \(3 a\) and mass \(k M\), where \(k\) is a constant. The centre of the disc is \(O\). The end \(B\) of the rod is rigidly joined to a point on the circumference of the disc so that \(O B A\) is a straight line. The fixed horizontal axis \(l\) is in the plane of the object, passes through \(A\) and is perpendicular to \(A B\).

1. Show that the moment of inertia of the object about the axis \(l\) is \(3 M a ^ { 2 } ( 26 + k )\).
   The object is free to rotate about \(l\).
2. Show that small oscillations of the object about \(l\) are approximately simple harmonic. Given that the period of these oscillations is \(4 \pi \sqrt { } \left( \frac { a } { g } \right)\), find the value of \(k\).