# Fictitious force

A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame.[1] It is related to Newton's second law of motion, which treats forces for just one object.[2]

Passengers in a vehicle accelerating in the forward direction may perceive they are acted upon by a force moving them into the direction of the backrest of their seats for example. An example in a rotating reference frame may be the impression that it is a force which seems to move objects outward toward the rim of a centrifuge or carousel.

The fictitious force called a pseudo force might also be referred to as a body force. It is due to an object's inertia when the reference frame does not move inertially any more but begins to accelerate relative to the free object. In terms of the example of the passenger vehicle, a pseudo force seems to be active just before the body touches the backrest of the seat in the car. A person in the car leaning forward first moves a bit backward in relation to the already accelerating car, before touching the backrest. The motion in this short period just seems to be the result of a force on the person, it's a pseudo force. A pseudo force does not arise from any physical interaction between two objects, such as electromagnetism or contact forces. It's just a consequence of the acceleration a of the physical object the non-inertial reference frame is connected to, i.e. the vehicle in this case. From the viewpoint of the respective accelerating frame, an acceleration of the inert object appears to be present, apparently requiring a "force" for this to have happened.

As stated by Iro:[3]

Such an additional force due to nonuniform relative motion of two reference frames is called a pseudo-force.

Harald Iro in A Modern Approach to Classical Mechanics p. 180

The pseudo force on an object arises as an imaginary influence when the frame of reference used to describe the object's motion is accelerating compared to a non-accelerating frame. The pseudo force "explains," using Newton's second law mechanics, why an object does not follow Newton's second law and "floats freely" as if weightless. As a frame may accelerate in any arbitrary way, so may pseudo forces also be as arbitrary (but only in direct response to the acceleration of the frame). An example of a pseudo force as defined by Iro is the Coriolis force, maybe better to be called: the Coriolis effect;[4][5][6] The gravitational force would also be a fictitious force (pseudo force), based upon a field model in which particles distort spacetime due to their mass, such as in the theory of general relativity.

Assuming Newton's second law in the form F = ma, fictitious forces are always proportional to the mass m.

The fictitious force that has been called an inertial force[7][8][9] is also referred to as a d'Alembert force,[10][11] or sometimes as a pseudo force.[12] D'Alembert's principle is just another way of formulating Newton's second law of motion. It defines an inertial force as the negative of the product of mass times acceleration, just for the sake of easier calculations.

(A d'Alembert force is not to be confused with a contact force arising from the physical interaction between two objects, which is the subject of Newton's third law – 'action is reaction'.[13][14] In terms of the example of the passenger vehicle above, a contact force emerges when the body of the passenger touches the backrest of the seat in the car. It is present for as long as the car is accelerated.)

Four fictitious forces have been defined for frames accelerated in commonly occurring ways: