# 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?

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.

• So I will specify: Are the forces including the pseudo-forces equal to zero in the non-inertial frame of reference? Why don't they obey Newton's 3rd law? – Yalom Jun 25 '17 at 16:05
• In your example, we know that the contact force which pushes the sock outwards is equal to the centripetal force in the inertial frame of reference. But if we change to the rotational frame of reference of the drum, then we still have the contact force which points to the the inside of the drum and the centrifugal force which points in the opposite direction (and we now that this force is equivalent to the negative of the centripetal force). As result, they cancel each other out (in case we don't have friction and gravity). – Yalom Jun 25 '17 at 16:05
• @Yalom (first comment) The vector sum of contact force and centrifugal pseudo force is zero. I'd say that pseudo forces don't obey Newton's third law because they're not forces in the ordinary sense! – Philip Wood Jun 25 '17 at 17:00
• @Yalom (second comment) The contact force on the sock is inwards! The rest your comment seems correct to me. – Philip Wood Jun 25 '17 at 17:03
• About the direction of the contact force: I made a slip of the pen. About your first comment: Could you please elaborate then why you would say that they don't obey newton's third law? This really go me confused and I don't know how to interpret it. – Yalom Jun 25 '17 at 17:14

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.

• The speeding car in the frame of reference (which has the same velocity as the car) is stationary in it. So why would any force act on it at all? And about the passenger: I would say that there is a force which is the mass of the passenger times the negative of the acceleration of the car. This force pushes the passenger into the seat, and since Newton's 3rd law applies to non-inertial frames of reference as well, the seat pushes the passenger back with the same force. As result, the sum of the forces on the passenger is zero. – Yalom Jun 25 '17 at 15:55