Fictitious force
A fictitious force',' also known as an inertial force or pseudo-force, is a force that appears to act on an object when its motion is described or experienced from a non-inertial frame of reference. Unlike real forces, which result from physical interactions between objects, fictitious forces occur due to the acceleration of the observer’s frame of reference rather than any actual force acting on a body. These forces are necessary for describing motion correctly within an accelerating frame, ensuring that Newton's second law of motion remains applicable.
Common examples of fictitious forces include the centrifugal force, which appears to push objects outward in a rotating system; the Coriolis force, which affects objects moving relative to the rotating frame, such as a wind parcel on Earth; and the Euler force, which arises when a rotating system changes its angular velocity.
While these forces are not real in the sense of being caused by physical interactions, they are essential for accurately analyzing motion within accelerating reference frames, particularly in disciplines such as classical mechanics, meteorology, and astrophysics.
Fictitious forces play a crucial role in understanding everyday phenomena, such as weather patterns influenced by the Coriolis effect and the perceived weightlessness experienced by astronauts in free-fall orbits. They are also fundamental in engineering applications, including navigation systems and rotating machinery.
According to general relativity theory we perceive gravitational force when spacetime is bending near heavy objects, so even this might be called a fictitious force.
Measurable examples of fictitious forces
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 instance. 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 seems to be the result of a force on the person; i.e., it is a pseudo force. A pseudo force does not arise from any physical interaction between two objects, such as electromagnetism or contact forces. It is only a consequence of the acceleration 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:
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. An example of a pseudo force as defined by Iro is the Coriolis force, maybe better to be called: the Coriolis effect. The gravitational force would also be a fictitious force in 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 is also referred to as a d'Alembert force, or sometimes as a pseudo force. 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.
Four fictitious forces have been defined for frames accelerated in commonly occurring ways:
- one caused by any acceleration relative to the origin in a straight line ;
- two involving rotation: centrifugal force and Coriolis effect
- and a fourth, called the Euler force, caused by a variable rate of rotation, should that occur.
Background
Fictitious forces arise in classical mechanics and special relativity in all non-inertial frames.
Inertial frames are privileged over non-inertial frames because they do not have physics whose causes are outside of the system, while non-inertial frames do. Fictitious forces, or physics whose cause is outside of the system, are no longer necessary in general relativity, since these physics are explained with the geodesics of spacetime: "The field of all possible space-time null geodesics or photon paths unifies the absolute local non-rotation standard throughout space-time.".
On Earth
The surface of the Earth is a rotating reference frame. To solve classical mechanics problems exactly in an Earthbound reference frame, three fictitious forces must be introduced: the Coriolis force, the centrifugal force and the Euler force. The Euler force is typically ignored because the variations in the angular velocity of the rotating surface of the Earth are usually insignificant. Both of the other fictitious forces are weak compared to most typical forces in everyday life, but they can be detected under careful conditions.For example, Léon Foucault used his Foucault pendulum to show that the Coriolis force results from the Earth's rotation. If the Earth were to rotate twenty times faster, people could easily get the impression that such fictitious forces were pulling on them, as on a spinning carousel. People in temperate and tropical latitudes would, in fact, need to hold on, in order to avoid being launched into orbit by the centrifugal force.
When moving along the equator in a ship heading in an easterly direction, objects appear to be slightly lighter than on the way back. This phenomenon has been observed and is called the Eötvös effect.
Detection of non-inertial reference frame
Observers inside a closed box that is moving with a constant velocity cannot detect their own motion; however, observers within an accelerating reference frame can detect that they are in a non-inertial reference frame from the fictitious forces that arise. For example, for straight-line acceleration Vladimir Arnold presents the following theorem:Other accelerations also give rise to fictitious forces, as described mathematically [|below]. The physical explanation of motions in an inertial frame is the simplest possible, requiring no fictitious forces: fictitious forces are zero, providing a means to distinguish inertial frames from others.
An example of the detection of a non-inertial, rotating reference frame is the precession of a Foucault pendulum. In the non-inertial frame of the Earth, the fictitious Coriolis force is necessary to explain observations. In an inertial frame outside the Earth, no such fictitious force is necessary.
Example concerning Circular motion
The effect of a fictitious force also occurs when a car takes the bend. Observed from a non-inertial frame of reference attached to the car, the fictitious force called the centrifugal force appears. As the car enters a left turn, a suitcase first on the left rear seat slides to the right rear seat and then continues until it comes into contact with the closed door on the right. This motion marks the phase of the fictitious centrifugal force as it is the inertia of the suitcase which plays a role in this piece of movement. It may seem that there must be a force responsible for this movement, but actually, this movement arises because of the inertia of the suitcase, which is a 'free object' within an already accelerating frame of reference.After the suitcase has come into contact with the closed door of the car, the situation with the emergence of contact forces becomes current. The centripetal force on the car is now also transferred to the suitcase and the situation of Newton's third law comes into play, with the centripetal force as the action part and with the so-called reactive centrifugal force as the reaction part. The reactive centrifugal force is also due to the inertia of the suitcase. Now however the inertia appears in the form of a manifesting resistance to a change in its state of motion.
Suppose a few miles further the car is moving at constant speed travelling a roundabout, again and again, then the occupants will feel as if they are being pushed to the outside of the vehicle by the centrifugal force, away from the centre of the turn.
The situation can be viewed from inertial as well as from non-inertial frames.
- From the viewpoint of an inertial reference frame stationary with respect to the road, the car is accelerating toward the centre of the circle. It is accelerating, because the direction of the velocity is changing, despite the car having constant speed. This inward acceleration is called centripetal acceleration, it requires a centripetal force to maintain the circular motion. This force is exerted by the ground upon the wheels, in this case, from the friction between the wheels and the road. The car is accelerating, due to the unbalanced force, which causes it to move in a circle.
- From the viewpoint of a rotating frame, moving with the car, a fictitious centrifugal force appears to be present pushing the car toward the outside of the road. The centrifugal force balances the friction between wheels and the road, making the car stationary in this non-inertial frame.
In the rotating reference frame perceived on the surface of the Earth, a centrifugal force reduces the apparent force of gravity by about one part in a thousand, depending on latitude. This reduction is zero at the poles, maximum at the equator.
For someone in the map perspective only one force is sufficient to explain the motion: the red arrow: centripetal force. After release, the number of forces is zero. For someone in the spinning frame the object moves in a complicated way that needs a centrifugal force: the blue arrow.Note: With some browsers, hitting will freeze the motion for more detailed analysis. However, the page may have to be reloaded to restart.
The fictitious Coriolis force, which is observed in rotational frames, is ordinarily visible only in very large-scale motion like the projectile motion of long-range guns or the circulation of the Earth's atmosphere. Neglecting air resistance, an object dropped from a 50-meter-high tower at the equator will fall 7.7 millimetres eastward of the spot below where it is dropped because of the Coriolis force.