fictitious force

From Physics wiki

A fictitious force is a an apparent force that is required in order for Newton's second law to hold in a non-inertial reference frame. Although no interaction is responsible for its appearance, the name is perhaps a misnomer, as inertial reference frames are by no means more fundamental than non-inertial ones, at least in the modern viewpoint.

Rotating coordinate systems

Recall that the rate of change of a vector $\mathbf{x}\,$ in a fixed frame $O\,$ is given by

$\left(\frac{d}{dt}\right)_O\mathbf{x} = \left(\frac{d}{dt}\right)_{O'}\mathbf{x} + \boldsymbol{\omega}\times\mathbf{x}'\,$ ,

where $\mathbf{x}'\,$ is the vector relative to a rotating frame $O'\,$ and $\boldsymbol{\omega}\,$ is the angular velocity of that frame. Suppose then that at some instant the axes of $O\,$ and $O'\,$ coincide, so that $\mathbf{x} = \mathbf{x}'\,$ .

Applying the above to the position of a particle,

$\mathbf{v}_O = \left(\frac{d}{dt}\right)_{O'}\mathbf{r} + \boldsymbol{\omega}\times\mathbf{r} = \mathbf{v}_{O'} + \boldsymbol{\omega}\times\mathbf{r}\,$ ,

and to obtain its acceleration,

$\mathbf{a}_O = \left(\frac{d}{dt}\right)_{O'} \mathbf{v}_O + \boldsymbol{\omega}\times\mathbf{v}_O\,$ ,

$\mathbf{a}_O = \left(\frac{d}{dt}\right)_{O'} \left( \mathbf{v}_{O'} + \boldsymbol{\omega}\times\mathbf{r} \right) + \boldsymbol{\omega}\times\left( \mathbf{v}_{O'} + \boldsymbol{\omega}\times\mathbf{r} \right)\,$ .

Evidently,

$\mathbf{a}_O = \mathbf{a}_{O'} + \frac{d\boldsymbol{\omega}}{dt}_{O'}\times\mathbf{r} + 2\boldsymbol{\omega}\times\mathbf{v}_{O'} + \boldsymbol{\omega}\times \left(\boldsymbol{\omega}\times \mathbf{r}\right)\,$ .

The apparent force in the rotating frame $O'\,$ is given according to Newton's second law:

$\mathbf{F}_{O'} = \mathbf{F}_O -2m \boldsymbol{\omega}\times\mathbf{v}_{O'} -m \boldsymbol{\omega}\times \left(\boldsymbol{\omega}\times \mathbf{r}'\right) -m \frac{d\boldsymbol{\omega}}{dt}_{O'}\times\mathbf{r}'\,$ ,