Movement of a material point in the gravity field of a homogeneous ring

A ring-like mass exerts a well computable gravitational attraction on a material point located along a straight line, being perpendicular to the plane of the ring in its centre. In the state of weightlessness, an oscillatory movement will develop owing to this effect. The period, T, of oscillation depends on the gravity constant, on the density and dimensions of the ring, as well as on the amplitude of the oscillation. Its exact computation can be based on the determination of the gravity potential function of the ring. The oscillation has the following form:

$\text{T=}\text{1}\text{√f√}\textcolor{red}{}\text{.I(r,R,z)}$

where f is the gravitational constant.

is the density of the ring, r and R are the radii of the ring, z is the distance of the turning point of the oscillation from the centre of the ring, while I/r,R,z/ is an improper integral which can be computed with any desired accuracy owing to the favourable function-theoretical character of the potential. We computed the oscillation period for various possible values of the parameters and obtained time data of an order of magnitude which falls into a well observable interval.The outlined conceptual experiment for the improved determination of the gravitational constant may present, of course, many technical difficulties and error sources /e.g. the path of the oscillating point is quite unstable owing to the extremely small acting forces, electric charges and also radiation pressure might be present, the gravity field would show a gradient on the spot of the experiment, etc./. Nevertheless, it seems to be worthwhile to consider carrying out such an experiment, using the possibilities offered by modern techniques in observing distances and time. For the path distortions caused by errors, we give a few estimates, but in case of realization of the experiment, a more detailed error analysis must be made.