12.3: Rotating Reference Frame

Consider a rotating frame of reference which will be designated as the double-primed (rotating) frame to differentiate it from the non-rotating primed (moving) frame, since both of which may be undergoing translational acceleration relative to the inertial fixed unprimed frame as described in Figure \(12.2.1\).

Spatial time derivatives in a rotating, non-translating, reference frame

For simplicity assume that \(\mathbf{R}_{fix} = \mathbf{V}_{fix} = 0\), that is, the primed reference frame is stationary and identical to the fixed stationary unprimed frame. The double-primed (rotating) frame is a non-inertial frame rotating with respect to the origin of the fixed primed frame.

Figure \(\PageIndex{1}\): Infinitessimal displacement in the non rotating primed frame and in the rotating double-primed reference frame frame.

Appendix \(19.4.2C\) shows that an infinitessimal rotation \(d\theta\) about an instantaneous axis of rotation leads to an infinitessimal displacement \(d\mathbf{r}^{R}\) where

\[d\mathbf{r}^{R} = d \theta \times \mathbf{r}^{\prime}_{mov} \label{12.7}\]

Consider that during a time \(dt\), the position vector in the fixed primed reference frame moves by an arbitrary infinitessimal distance \(d\mathbf{r}^{\prime}_{mov}\). As illustrated in Figure \(\PageIndex{1}\), this infinitessimal distance in the primed non-rotating frame can be split into two parts:

1. \(d\mathbf{r}^{R} = d\theta \times \mathbf{r}^{\prime}_{mov}\) which is due to rotation of the rotating frame with respect to the translating primed frame.
2. \((d\mathbf{r}^{\prime\prime}_{rot})\) which is the motion with respect to the rotating (double-primed) frame.

That is, the motion has been arbitrarily divided into a part that is due to the rotation of the double-primed frame, plus the vector displacement measured in this rotating (double-primed) frame. It is always possible to make such a decomposition of the displacement as long as the vector sum can be written as

\[d\mathbf{r}^{\prime}_{mov} = d\mathbf{r}^{\prime\prime}_{rot} + d\theta \times \mathbf{r}^{\prime}_{mov} \label{12.8}\]

Since \(d\theta = \omega dt\) then the time differential of the displacement, Equation \ref{12.8}, can be written as

\[\left( \frac{d\mathbf{r}^{\prime}}{dt}\right)_{mov} = \left(\frac{d\mathbf{r}^{\prime\prime}}{dt}\right)_{rot} + \omega \times \mathbf{r}^{\prime}_{mov} \label{12.9}\]

The important conclusion is that a velocity measured in a non-rotating reference frame \(\left(\frac{d\mathbf{r}^{\prime}}{dt}\right)_{mov}\) can be expressed as the sum of the velocity \(\left(\frac{d\mathbf{r}^{\prime\prime}}{dt}\right)_{rot}\), measured relative to a rotating frame, plus the term \(\omega \times \mathbf{r}^{\prime}_{mov}\) which accounts for the rotation of the frame. The division of the \(d\mathbf{r}^{\prime}_{rot}\) vector into two parts, a part due to rotation of the frame plus a part with respect to the rotating frame, is valid for any vector as shown below.

General vector in a rotating, non-translating, reference frame

Consider an arbitrary vector \(\mathbf{G}\) which can be expressed in terms of components along the three unit vector basis \(\hat{\mathbf{e}}^{fix}_i\) in the fixed inertial frame as

\[\mathbf{G} = \sum^{3}_{i=1} G^{fix}_i \hat{\mathbf{e}}_i^{fix} \label{12.10}\]

Neglecting translational motion, then it can be expressed in terms of the three unit vectors in the non-inertial rotating frame unit vector basis \(\hat{\mathbf{e}}^{rot}_i\) as

\[\mathbf{G} = \sum^{3}_{i=1} (G_i)_{rot} \hat{\mathbf{e}}^{rot}_i \label{12.11}\]

Since the unit basis vectors \(\hat{\mathbf{e}}^{rot}_i\) are constant in the rotating frame, that is,

\[\left(\frac{d\hat{\mathbf{e}}^{rot}_i}{dt}\right)_{rot} = 0 \label{12.12}\]

then the time derivatives of \(\mathbf{G}\) in the rotating coordinate system \(\hat{\mathbf{e}}^{rot}_i\) can be written as

\[\left(\frac{d\mathbf{G}}{dt}\right)_{rot} = \sum^3_{i-1} \left(\frac{dG_i}{dt}\right)_{rot} \hat{\mathbf{e}}^{rot}_i \label{12.13}\]

The inertial-frame time derivative taken with components along the rotating coordinate basis \(\hat{\mathbf{e}}^{rot}_i\), Equation \ref{12.11}, is

\[\left(\frac{d\mathbf{G}}{dt}\right)_{fix} = \sum^3_{i-1} \left(\frac{dG_i}{dt}\right)_{rot} \hat{\mathbf{e}}^{rot}_i + (G_i)_{rot} \frac{d\hat{\mathbf{e}}^{rot}_i }{dt} \label{12.14}\]

Substitute the unit vector \(\hat{\mathbf{e}}^{rot}\) for \(\mathbf{r}^{\prime}_{mov}\) in Equation \ref{12.9}, plus using Equation \ref{12.12}, gives that

\[\left(\frac{d\hat{\mathbf{e}}^{rot}}{dt}\right)_{fix} = \omega \times \hat{\mathbf{e}}^{rot} \label{12.15}\]

Substitute this into the second term of Equation \ref{12.14} gives

\[\left(\frac{d\mathbf{G}}{dt}\right)_{fix} = \left(\frac{d\mathbf{G}}{dt}\right)_{rot} + \omega \times \mathbf{G} \label{12.16}\]

This important identity relates the time derivatives of any vector expressed in both the inertial frame and the rotating non-inertial frame bases. Note that the \(\omega \times \mathbf{G}\) term originates from the fact that the unit basis vectors of the rotating reference frame are time dependent with respect to the non-rotating frame basis vectors as given by Equation \ref{12.15}. Equation \ref{12.16} is used extensively for problems involving rotating frames. For example, for the special case where \(\mathbf{G} = \mathbf{r}^{\prime}\), then Equation \ref{12.16} relates the velocity vectors in the fixed and rotating frames as given in Equation \ref{12.9}.

Another example is the vector \(\mathbf{\dot{\omega}}\)

\[\mathbf{\dot{\omega}} = \left(\frac{d\omega}{dt}\right)_{fix} = \left(\frac{d\omega}{dt}\right)_{rot} + \omega \times \omega = \left(\frac{d\omega}{dt}\right)_{rot} = \mathbf{\dot{\omega}} \label{12.17}\]

That is, the angular acceleration \(\dot{\omega}\) has the same value in both the fixed and rotating frames of reference.