When an object simultaneously rotates about a point and moves relative to that point, an acceleration results from this. This acceleration is called Coriolis acceleration.

To illustrate this acceleration, consider a particle

From the equations from Curvilinear motion:

By applying Newton’s Second Law in the direction of

where

To gain an intuitive understanding of the acceleration

Since

Therefore there is an acceleration

Thus, the particle

Note that this restraining force

Now, let’s define a reference frame that has origin at point

For example, let’s say we have a merry-go-round rotating with a constant angular velocity. On the merry-go-round there is a ball rolling outwards from the center with a radial velocity

Now, let’s say there is no restraining force (

The figure below shows the motion of the ball relative to the merry-go-round, as the merry-go-round rotates between positions 1 and 2. The figure illustrates two cases: The first case is where the restraining force

The black arrow shows the motion of the ball relative to an inertial reference frame (ground) for the case where there are no external forces acting on the ball in the plane of the merry-go-round.

In the above figure:

The starting position of the ball is denoted by ‘1’

In the presence of a restraining force

If the restraining force

As mentioned already, the green line represents the motion of the ball relative to the merry-go-round for the case where

For the situation where

From the point of view of an observer sitting on the merry-go-round (and moving with it), the Coriolis force is the apparent force that appears to be acting on the ball, causing it to veer to the right.

More specifically, the Coriolis force acts in a direction that is perpendicular to both the angular velocity vector and the relative (linear) velocity vector.

Referring to the figure above this would mean that the (fictitious) Coriolis force acts perpendicular to the blue curve (where

The Coriolis force acting on the ball is indicated by the black arrows, shown below.

The Coriolis force explains why the ball tends to curve in on itself, relative to an observer sitting on the merry-go-round (and moving with it).

The black arrow shown in the previous figure (with starting position '1') represents the path traveled by the ball relative to ground (an inertial reference frame), for the situation where no external forces are acting on the ball in the plane of the merry-go-round (i.e. there is no friction with the merry-go-round surface). In this situation, the ball would travel a straight line (relative to ground) indicated by the black arrow (due to Newton’s First Law). However, relative to the merry-go-round, the ball would follow a curved path, similar to the blue curve.

The Coriolis force is not the only fictitious force that acts on the ball from the point of view of an observer sitting on the rotating merry-go-round. There is also a Centrifugal force, which appears to be pushing the ball radially outwards from the center of rotation

Now, if the merry-go-round has an angular acceleration, this would introduce yet another component of motion for the ball (relative to the merry-go-round). This component of motion points in the circumferential direction of the merry-go-round and has the effect of contributing to the deflection of the ball in the circumferential direction of the merry-go-round. This would add yet another fictitious force called the Euler force. From the point of view of an observer sitting on the merry-go-round, the Euler force appears to be pushing on the ball in the circumferential direction. This force is explained in more detail here.

Therefore, there are three fictitious forces acting on the ball for the general case where the merry-go-round is rotating with a given angular velocity, and with a given angular acceleration. These fictitious forces are illustrated in the figure below.

The black arrows represent the Coriolis force, which acts perpendicular to the path of the ball represented by the blue curve.

The green arrows represent the Centrifugal force, whose line of action passes through the center of rotation

The orange arrows represent the Euler force, which acts in the circumferential direction, and is perpendicular to the Centrifugal force.

These three fictitious forces all combine to produce the relative motion of the ball relative to the merry-go-round.

Understanding fictitious forces can be useful for understanding everyday phenomenon. For example, the rotation of cyclones on Earth is influenced by the Coriolis force, which results from the combination of Earth’s rotation and the direction of wind flow.

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