Write down the dimensions of velocity, acceleration and force.
The force \(F\) of gravitational attraction between two objects with masses \(m _ { 1 }\) and \(m _ { 2 }\), at a distance \(r\) apart, is given by
$$F = \frac { G m _ { 1 } m _ { 2 } } { r ^ { 2 } }$$
where \(G\) is the universal constant of gravitation.
Show that the dimensions of \(G\) are \(\mathrm { M } ^ { - 1 } \mathrm {~L} ^ { 3 } \mathrm {~T} ^ { - 2 }\).
In SI units (based on the kilogram, metre and second) the value of \(G\) is \(6.67 \times 10 ^ { - 11 }\).
Find the value of \(G\) in imperial units based on the pound \(( 0.4536 \mathrm {~kg} )\), foot \(( 0.3048 \mathrm {~m} )\) and second.
For a planet of mass \(m\) and radius \(r\), the escape velocity \(v\) from the planet's surface is given by
$$v = \sqrt { \frac { 2 G m } { r } }$$
Show that this formula is dimensionally consistent.
For a planet in circular orbit of radius \(R\) round a star of mass \(M\), the time \(t\) taken to complete one orbit is given by
$$t = k G ^ { \alpha } M ^ { \beta } R ^ { \gamma }$$
where \(k\) is a dimensionless constant.
Use dimensional analysis to find \(\alpha , \beta\) and \(\gamma\).