4. The radius of the Earth is \(R \mathrm {~m}\). The force of attraction towards the centre of the Earth experienced by a body of mass \(m \mathrm {~kg}\) at a distance \(x \mathrm {~m}\) from the centre is of magnitude \(\frac { k m } { x ^ { 2 } } \mathrm {~N}\), where \(k\) is a constant.

1. Show that \(k = g R ^ { 2 }\). Two satellites \(A\) and \(B\), each of mass \(m \mathrm {~kg}\), are moving in circular orbits around the Earth at distances \(3 R \mathrm {~m}\) and \(4 R \mathrm {~m}\) respectively from the centre of the Earth. Given that the satellites move in the same plane and that they lie along the same radial line from the centre at any time,
2. show that the ratio of the speed of \(B\) to that of \(A\) is \(4 : 3\). If, in addition, the satellites are linked with a taut, straight wire in the same plane and along the same radial line,
3. find, in terms of \(m\) and \(g\), the magnitude of the force in the wire.
   (8 marks) \section*{MECHANICS 3 (A) TEST PAPER 5 Page 2}