If the gravitational force between two objects were proportional to 1/R (and not as 1/R²), where R is the distance between them, then a particle in a circular path (under such a force) would have its orbital speed v, proportional to

If the gravitational force between two objects were proportional to 1/R instead of 1/R², the dynamics of objects in circular orbits would be very different. In such a case, the force acting on a particle in a circular orbit would decrease less rapidly with increasing distance R. This would change the dependence of the orbital speed of the particle on its distance from the center of attraction.

For an object in stable circular orbit, the necessary centripetal force for maintaining its orbit has to be supplied by the gravitational force. Gravitational force that depends on 1/R, the usual dependency of necessary orbital speed on R is broken. For this case, orbital speed in terms of given condition leads to a value of v not depending on R. This means that the orbital speed is independent of the distance from the center of attraction.

This behavior is in contrast to the 1/R² dependence of the actual gravitational force, where the orbital speed decreases with an increase in distance. The hypothetical 1/R force would result in strange orbital dynamics, as particles would have the same speed at all distances, which would fundamentally change the structure and stability of orbits in such a system.