Of course they'll reach an equilibrium - that goes without saying - even on the Earth. The fact is that, if one end of the spring is attached to the 'outer' side of the ship, there will be a slight compression of the spring. If it is attached to the 'inner' side, there will be a small extension.But now, shouldnt the mass and a spring reach an equilibrium since the mass is pulled towards the outer wall?
Thanks in advance,
This is because the angular velocity is set by the angular velocity (omega) of the CM of the whole ship but omega squared times r will be greater for as r increases but the gravitational field will be reduced by a factor of 1/ r squared. The difference will cause 'microgravity'. If you had a massively long ship ( say a 10 km tube) - aligned along a radius, the 'microgravity' could, in fact, be very large for a low Earth orbit.
p.s. Actually. the mass isn't really "pulled towards the outer wall - the outer wall is pushing towards the mass less than necessary to maintain the appropriate orbit, keeping it in an 'unnaturally' large orbit because its omega is greater than it should be, for the gravity at that radius.