6 The magnitude of the gravitational force \(F\) between two planets of masses \(m _ { 1 }\) and \(m _ { 2 }\) with centres at a distance \(d\) apart is given by
$$F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }$$
where \(G\) is a constant.
6
- Show that \(G\) must have dimensions \(L ^ { 3 } M ^ { - 1 } T ^ { - 2 }\), where \(L\) represents length, \(M\) represents mass and \(T\) represents time.
6 - The lifetime \(t\) of a planet is thought to depend on its mass \(m\), its radius \(r\), the constant \(G\) and a dimensionless constant \(k\) such that
$$t = k m ^ { a } r ^ { b } G ^ { c }$$
where \(a , b\) and \(c\) are constants.
Determine the values of \(a , b\) and \(c\).