A rotating frame of reference is a special case of a non-inertial reference frame in which the coordinate system is rotating relative to an inertial reference frame. An everyday example of a rotating reference frame is the surface of the Earth.

Fictitious forces

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All non-inertial reference frames exhibit fictitious forces. Rotating reference frames are characterized by three fictitious forces

• the centrifugal force
• the Coriolis force

and, for non-uniformly rotating reference frames,

• the Euler force.

Scientists living in a rotating box can measure the speed and direction of their rotation by measuring these fictitious forces. For example, Léon Foucault was able to show the Coriolis force that results from the Earth's rotation using the Foucault pendulum. If the Earth were to rotate a thousand-fold faster (making each day only ~86 seconds long), these fictious forces could be felt easily by humans, as they are on a spinning carousel.

Relation between positions in the two frames

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To derive these fictitious forces, it's helpful to be able to convert between the coordinates $\backslash left(\; x^\{\backslash prime\},y^\{\backslash prime\},z^\{\backslash prime\}\; \backslash right)$ of the rotating reference frame and the coordinates $\backslash left(\; x,\; y,\; z\; \backslash right)$ of an inertial reference frame with the same origin. If the rotation is about the $z$ axis with an angular velocity $\backslash omega$ and the two reference frames coincide at time $t=0$, the transformation from rotating coordinates to inertial coordinates can be written

$x\; =\; x^\{\backslash prime\}\backslash \; \backslash cos\backslash omega\; t\; +\; y^\{\backslash prime\}\backslash \; \backslash sin\backslash omega\; t$

$y\; =\; y^\{\backslash prime\}\backslash \; \backslash cos\backslash omega\; t\; -\; x^\{\backslash prime\}\backslash \; \backslash sin\backslash omega\; t$

whereas the reverse transformation is

$x^\{\backslash prime\}\; =\; x\backslash \; \backslash cos\backslash left(-\backslash omega\; t\backslash right)\; -\; y\backslash \; \backslash sin\backslash left(\; -\backslash omega\; t\; \backslash right)$

$y^\{\backslash prime\}\; =\; y\backslash \; \backslash cos\backslash left(\; -\backslash omega\; t\; \backslash right)\; +\; x\backslash \; \backslash sin\backslash left(\; -\backslash omega\; t\; \backslash right)$

This result can be obtained from a rotation matrix.

Relation between velocities in the two frames

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A velocity of an object is the time-derivative of the object's position, or

$\backslash mathbf\{v\}\; \backslash equiv\; \backslash frac\{d\backslash mathbf\{r\}\}\{dt\}$

The time derivative of position in a rotating reference frame has two components, one from the time derivative in the inertial reference frame and another from its own rotation. These are related by the equation

$$

\left( \frac{d}{dt} \right)_{\mathrm{inertial}} = \left( \frac{d}{dt} \right)_{\mathrm{rotating}} + \boldsymbol\omega \times

where the vector $\backslash boldsymbol\backslash omega$ points along the rotation axis with the magnitude of the angular velocity. Therefore, the velocities in the two reference frames are related by the equation

$$

\mathbf{v}_{\mathrm{inertial}} \equiv \left( \frac{d\mathbf{r}}{dt} \right)_{\mathrm{inertial}} = \left( \frac{d\mathbf{r}}{dt} \right)_{\mathrm{rotating}} + \boldsymbol\omega \times \mathbf{r} = \mathbf{v}_{\mathrm{rotating}} + \boldsymbol\omega \times \mathbf{r}

Relation between accelerations in the two frames

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Acceleration is the second time derivative of position, or the first time derivative of velocity

$$

\mathbf{a}_{\mathrm{inertial}} \equiv \left( \frac{d^{2}\mathbf{r}}{dt^{2}}\right)_{\mathrm{inertial}} = \left( \frac{d\mathbf{v}}{dt} \right)_{\mathrm{inertial}} = \left[ \left( \frac{d}{dt} \right)_{\mathrm{rotating}} + \boldsymbol\omega \times \right] \left[ \left( \frac{d\mathbf{r}}{dt} \right)_{\mathrm{rotating}} + \boldsymbol\omega \times \mathbf{r} \right]

Carrying out the differentiations and re-arranging some terms yields the acceleration in the rotating reference frame

$$

\mathbf{a}_{\mathrm{rotating}} = \mathbf{a}_{\mathrm{inertial}} - 2 \boldsymbol\omega \times \mathbf{v}_{\mathrm{rotating}} - \boldsymbol\omega \times \boldsymbol\omega \times \mathbf{r} - \frac{d\boldsymbol\omega}{dt} \times \mathbf{r}

where $\backslash mathbf\{a\}\_\{\backslash mathrm\{rotating\}\}\; \backslash equiv\; \backslash left(\; \backslash frac\{d^\{2\}\backslash mathbf\{r\}\}\{dt^\{2\}\}\; \backslash right)\_\{\backslash mathrm\{rotating\}\}$ is the apparent acceleration in the rotating reference frame.

The three extra terms on the right-hand side result in fictitious forces in the rotating reference frame, i.e., accelerations that result from being in a non-inertial reference frame, rather than from any physical force. Using Newton's second law of motion $F=ma$, we obtain

• the Coriolis force

$$

\mathbf{F}_{\mathrm{Coriolis}} = -2m \boldsymbol\omega \times \mathbf{v}_{\mathrm{rotating}}

• the centrifugal force

$$

\mathbf{F}_{\mathrm{centrifugal}} = -m\boldsymbol\omega \times \boldsymbol\omega \times \mathbf{r}

• and the Euler force

$$

\mathbf{F}_{\mathrm{Euler}} = -m\frac{d\boldsymbol\omega}{dt} \times \mathbf{r}

where $m$ is the mass of the object being acted upon by these fictitious forces.

For completeness, the inertial acceleration $\backslash mathbf\{a\}\_\{\backslash mathrm\{inertial\}\}$ can be determined from the total physical force $\backslash mathbf\{F\}\_\{\backslash mathrm\{tot\}\}$ (i.e., the total force from physical interactions such as electromagnetism) likewise using Newton's second law

$$

\mathbf{F}_{\mathrm{tot}} = m \mathbf{a}_{\mathrm{inertial}}

Frames of reference | Classical mechanics | Introductory physics | Celestial mechanics | Surveying