Q:

Where is the force free point on a radial rod?Imagine a rod which falls radially towards a central mass. The rod is under tension due to tidal forces. There is a force free point FFP (describing a geodesic) on the rod because the forces upwards and downwards cancel each other.Where is the FFP located? In the middle of the rod or somewhere in the lower part because the tidal acceleration increases with decreasing r-coordinate?This question seems a curiosity but of pedagogical value though.

- Timm Deeg (age 75)

Germany

- Timm Deeg (age 75)

Germany

A:

At least in the Newtonian approximation (excellent for most situations) you can just take the integral of (1/r^{2}) divided by the rod length to get the inverse square of the effective average distance.If the distance from the central mass to the middle of the rod is r and the length of the rod is b, I get that the point you're looking for is at distance

r(1-(b^{2}/4r^{2}))^{1/2} from the central mass. If b is much less the r, then the point is shifted down ~b^{2}/8r from the middle of the rod.<r, it's="" shifted="" down by="" ~b2="" 8r="" from="" the="" middle="" of="" rod.=""

Mike W.

*(published on 06/22/2018)*