A satellite of mass `m` is in a circular orbit of radius `r` round the Earth. Calculate its angular momentum with respect to the centre of the orbit in terms of the mass `M` of the Earth and `G`.

Explanation

The situation is shown in figure
The angular momentum of the satellite with respect to the centre of orbit is given by
`vec(L) = vec(r ) xx m vec(v)`
Where `vec(r )` is the position vector of satellite with respect to the centre of orbit and `vec(v)` is its velocity vector of satellite.
In case of circular orbit, the angle between `vec(r )` and `vec(v)` is `90^(@)`. Hence

`L = mvr sin 90^(@) = m v r` ...(1)
The direction is perpendicular to the plane of the orbit.
We know orbital speed of satellite is
`v = sqrt((GM)/(r ))` ...(2)
From equation (1) and (2), we get
`L = msqrt((GM)/(r )) rArr L = (GM m^(2) r)^(1//2)`
Now we will understand the concept of double star system through an example.