This post derives newton’s law in a rotating frame of reference using the more generalized framework of coordinate frames and rotation matrices. It has a nice intermediate result which is a general formula for computing the acceleration of a point in a new reference frame. It also has the benefit of starting from a simple and intuitive vector equation: rp0=r10+rp1.

Notation

  • ωcba : angular velocity of c with respect to b represented in the coordinates of frame a.
  • rcba: vector from b to c represented in the coordinates of frame a.
  • Tba: matrix that rotates vectors from frame a into frame b

System Setup

This section explains the geometric picture. Define two frames, frame 0 and frame 1, and say frame 0 is an inertial frame and frame 1 is rotating about some vector ω. Next, we define a point p. An observer in frame 1 is watching point p: rp11(t). He wants to apply newton’s law to understand it’s motion. He knows that the frame he is in is not inertial, so he cannot just apply F=ma.

Image is unavailable. Sorry.
Figure: Frame 0 is an inertial frame and frame 1 is rotating w.r.t frame 1. An observer in frame 1 is watching point p.

Forces in a Rotating Frame: Coriolis Force, Centrifugal Force, Euler Force

The vector from frame 0 to p equals the vector from frame 0 to 1 plus the vector from frame 1 to p. This is easy to reason and can be geometrically seen in the figure above. At this point, I want to emphasize that there are no assumptions about the motion of frame 0 or frame 1 or p.

rp0=r10+rp1

We can rewrite the equation in terms of specific basis, specifically in terms of frame 0 since we are after the point’s acceleration in an inertial frame. Notice that we are rotating the vector rp1 from frame 1 into frame 0. The motivation for this is because rp11 is what an observer in frame 1 sees.

rp00=r100+T01rp11

Next, we take the derivative twice to find the acceleration of point p w.r.t frame 0. This is just calculus.

˙rp00=˙r100+T01˙rp11+ω100×T01rp11¨rp00=¨r100+T01¨rp11+ω100×T01˙rp11+ω100×(T01˙rp11+ω100×T01rp11)+dω100dt×T01rp11¨rp00=¨r100+T01¨rp11+2(ω100×T01˙rp11)+ω100×(ω100×T01rp11)+dω100dt×T01rp11¨rp00=¨r100+¨rp10+2(ω100×˙rp10)+ω100×(ω100×rp10)+dω100dt×rp10

The last two steps were only expansion and then applying the rotation matrices. Already the familiar forms of the coriolis, centrifugal and euler forces are appearing. A few comments about this equation. First, there was absolutely no assumption made about the motion of the frames or the points. This is a very general equation. Second, the equation is saying that the acceleration of point p w.r.t. frame 0 is given by adding the acceleration of frame 1 w.r.t 0 (¨r100) plus the acceleration of the point p w.r.t. frame 1 (¨rp10) plus other terms due to the relative rotation.

Now we introduce two assumptions. First, we assume that frame 0 is an inertial reference frame. This means that we can use Newton’s law. Second, we assume that frame 1 is only rotating with respect to frame 0, and it is not moving/accelerating (e.g. ¨r100=0).

\[F_0 = m \ddot{r}_0^{p0} \\ F_0 = m \ddot{r}_0^{p1} + 2 m ( \omega_0^{10} \times \dot{r}_0^{p1} ) + m \omega_0^{10} \times (\omega_0^{10} \times r_0^{p1}) + m \frac{d\omega_0^{10}}{dt} \times r_0^{p1} \\]

This equation is true no matter what basis we work in, so we change it to frame 1 because that is the frame the observer is in and he will view and represent the coordinates of p and his own frame’s rotation in the basis of that frame. Mathematically though, you can multiply both sides by T10 and use the following fact: if a and b are vectors and M is an invertible matrix, then (Ma)×(Mb)=(detM)MT(a×b).

hint: remember that any rotation matrix TabSO(3), where SO(3) is the special orthogonal group. Look up the properties of this set!

\[F_1 = m \ddot{r}_1^{p1} + 2 m ( \omega_1^{10} \times \dot{r}_1^{p1} ) + m \omega_1^{10} \times (\omega_1^{10} \times r_1^{p1}) + m \frac{d\omega_1^{10}}{dt} \times r_1^{p1} \\]

The last thing to notice is that m¨rp11 is mass times the acceleration of point p w.r.t. frame 1. This is what we are after. We want to find a set of forces so Newton’s law holds in frame 1 and we can treat it as an inertial frame. Rearranging for m¨rp11:

m¨rp11=F12m(ω101×˙rp11)mω101×(ω101×rp11)mdω101dt×rp11 

This is the final, desired result. Recall the definitions of the coriolis, centrifugal and euler forces:

  • Corilois Force: 2m(ω101×˙rp11)
  • Centrifugal Force: mω101×(ω101×rp11)
  • Euler Force: mdω101dt×rp11

This equation tells us that in a rotating frame, mass times acceleration is equal to the applied force plus the three fictitious forces defined above.