6. The gravitational attraction \(F \mathrm {~N}\) between two point masses \(m _ { 1 } \mathrm {~kg}\) and \(m _ { 2 } \mathrm {~kg}\) at a distance \(x \mathrm {~m}\) apart is given by \(F = \frac { k m _ { 1 } m _ { 2 } } { x ^ { 2 } }\), where \(k\) is a constant. Given that a small body of mass 1 kg experiences a force of \(g \mathrm {~N}\) at the surface of the Earth, which has radius \(R \mathrm {~m}\) and mass \(M \mathrm {~kg}\),

1. show that \(k = \frac { g R ^ { 2 } } { M }\). A small communications satellite of mass \(m \mathrm {~kg}\) is put into a circular orbit of radius \(r \mathrm {~m}\) around the Earth. Modelling the Earth as a particle of mass \(M \mathrm {~kg}\), and using the value of \(k\) from (a), (b) prove that the period of rotation, \(T \mathrm {~s}\), of the satellite is given by \(T = \frac { 2 \pi } { R } \sqrt { \frac { r ^ { 3 } } { g } }\). (4 marks) To cover transmission to any point on the Earth, three small satellites \(X , Y\) and \(Z\), each of mass \(m \mathrm {~kg}\), are placed in a common circular orbit of radius \(r\) and form an equilateral triangle as shown.
   \includegraphics[max width=\textwidth, alt={}, center]{1bcb4e33-b27c-48f0-9540-9ec553e7fe40-2_223_238_1215_1709}
2. Show that the period of rotation of \(X\) is given by \(T \sqrt { \frac { 3 M } { 3 M + m \sqrt { 3 } } } \mathrm {~s}\), where \(T \mathrm {~s}\) is the period found in (b).