Center of gravity of an orbitting body not at center of gravity

[Pds3.14]

It just occurred to me that in the right gravitational field, say, in a low orbit 100 km from the core of a neutron star with a mass equal to that of the sun, an orbiting object's center of gravity is somewhat noticeably displaced from its center of mass. Specifically, 1 meter closer to the neutron star, there is about a 1.03 km/s^2 stronger gravity, obviously, this is a ludicrous amount of force that would quite like kill anyone at that altitude (if radiation and heat hadn't done that already). With a field strength variation that high (1030 square hertz) compared to the gravity of 51520 km/s^2. The gravity on a 1-meter dumbbell facing the neutron star will move the center of mass by about 1/100000th of a meter.
Or a more powerful example using low orbit of a black hole generating 1 G 10 meters away.
At 10 meters away, the variation is roughly 2 square hertz, so you would be fairly okay in terms of not being turned into spaghetti, but the previous dumbbell, or roughly speaking, a person, would experience roughly a 10 centimeter displacement in center-of gravity away from center off mass. This is all at non-relativistic speeds of about 10 m/s orbital velocity and non-relativistic gravity of 8-12 m/s^2.

Curiously, if I go out to 100 meters distant, the distance removed from the center of mass that the center of gravity is goes down by a factor of 100, e.g. It is proportional to the inverse square. However, the overall strength of the effect is therefore clearly associated with the inverse fourth power.
Which, taken to its logical extreme, means that at 1 meter, with a gravity of 100 G and a variation of 2 kilohertz/second (YAY for unit cancelation), this force would push the center of G 10/11ths of the way toward the hole. What effects does this have in real life, is it partially responsible for tidal locking?

 

[sophiecentaur]

Even in real life, the microgravity inside an orbiting spacecraft can be detected and will affect what objects will do inside it. The effect when bigger masses are involved will be correspondingly greater.

Tidal locking has to involve some loss mechainsm, I believe. Without friction effects, the mutual steady slowing down and speeding up of the two bodies would become oscillatory, energy passing back and forth between orbital and rotational forms. The tide, of course, happens because of the difference in gravitational attraction from place to place. There is an equivalent to 'spaghettification' with the atmospheres of small planets in orbit near to stars, when the higher velocity molecules happen to 'escape' up/down into the star when they reach the null between the two bodies.

 

[Simon Bridge]

If you had a fluid on the object, it would slosh around.
If you stood on it, it would feel like it was wobbling.