What happens to the bob when placed exactly between two parts of earth cut into two pieces?

Question

Take for example earth being cut into two equal pieces. If I leave a projectile then what will happen to this bob.

1) will it has an oscillatory motion in any direction

2) Will it spin at the center
e.g. Moon spins around earth and also rotates. But if we keep it between two pieces of earth it would only have a spinning motion.

It is open end answer. Please have a clear picture then only answer or mail me and ask for more clarity.

-Jaadu (Alias name)

Answer 1

If you place an object exactly at the centre between the two perfect halves of the Earth, the object will be in an "unstable" equilibrium: It will be equally attracted to each half, with a resulting force of exactly 0.

If you gave it spin when you placed it there, it will continue to spin.

The equilibrium is unstable, because if, for whatever reason, it were to drift ever so slightly towards one of the halves, the gravitational force of that half on the object would increase (because the distance decreases) and the attraction of the other half would drecrease (because the distance increases). This would create a net acceleration towards the closer half, accentuating the movement towards the closer half.
In other words, as soon as the equilibrium is broken, there is no oscillatory motion: the object simply falls towards one of the halves.

If the object is uneven (not a perfect sphere), then the tidal effects of each half would slow down the spin until the long axis is lined up with the line joining the barycentre of each half.

PS: There would be a plane (a 2-D sheet) of positions such that every point on that plane is equidistant from both halves of the sphere. An object placed anywhere on that plane would oscillate back and forth along a line through the barycentre of mass for the system. It the two halves of Earth are close enough (I'm not asking how you'd keep them that close), the oscilating period would be around 88 minutes.
However, this "orbit" would have the same instability as described above: as soon as the object deviates from the plane (could be the effect of the Sun or another planet) then it is bound to crash into one of the halves.