Forces in a non-inertial frame of reference If we have a body which does not move with respect to a non-inertial frame of reference then is the sum of the forces always zero for that body in the non-inertial frame of reference?
I've been thinking that it makes sense if I can use the 2nd law of Newton, but we are not allowed to do that in a non-inertial frame of reference. In case that the above-stated statement is true, could someone please explain to me why?
 A: A speeding car looks stationary for a passenger in the car - but the forces on it are not balancing out. So no, the sum of forces do not always become zero.
A: Take the case of a sock being whirled round in the drum of a washing machine. For an observer standing outside the washing machine and therefore in an approximately inertial frame of reference, the sock is accelerating towards the axis of the drum, and the force, F needed according to Newton's second law is the contact force of the drum acting on the sock. [According to Newton's third law, the sock exerts an equal and opposite 'outward' force on the drum.]
Now suppose we consider things using a frame of reference rotating with the drum [and sock] – a non-inertial frame. In this frame the sock is at rest, so has zero acceleration, but the contact force from the drum still acts on it. So apparently, $F≠ma$, that is Newton's second law doesn't work in this frame. However, if there were another force, equal in magnitude to $mr \omega^{2}$ (with the usual notation), acting radially outward on the sock, then there would be zero resultant force on the sock and Newton's second law would seem to work in our non-inertial frame. This hypothetical extra force is called centrifugal force, and is an example of a fictitious force or pseudoforce. It doesn't qualify as a 'proper' force primarily because we can't find another body that exerts the force; remember: forces come in pairs in Newtonian physics: body A exerts a force on body B, and B exerts a force on A. This is the basis of Newton's third law.
So for the sock in the rotating drum, in the frame of reference of the drum itself, we either have to sacrifice Newton's second law, or we have to invoke a centrifugal force which doesn't obey Newton's third law. In other words, Newton's laws have to be modified for use in non-inertial frames. One can argue about how 'real' pseudo forces are; for my money their failure to obey Newton 3, and their unclassifiability as gravitational, electromagnetic, weak or strong makes them unreal.   
