1  Motion in a Rotating Frame

1.1  Introduction to the Basic Ideas

At the park near my house, there is a small merry-go-round. A cartoon of the situation is shown in figure 1.

1) Joe is standing on the ground, observing things relative to an ordinary terrestrial reference frame. We approximate this frame as being nonrotating, even though the earth does have a measurable rate of rotation. Joe’s reference frame is indicated by the rectangular yellow area with red lines.

2) Meanwhile, the merry-go-round is in motion. Moe is riding the merry-go-round, observing things relative to a rotating reference frame. Moe’s reference frame is indicated by the circular blue area with black lines.

Figure 1: Rotating and Non-Rotating Coordinate Systems

The following is common knowledge among the children in the neighborhood: Suppose you are riding the merry-go-round, not sitting near the pivot. You set a baseball on the floor of the merry-go-round and hold it stationary (relative to the rotating frame) for a moment. When you let go, the ball does not remain stationary; it accelerates outward. The initial acceleration is directly outward, radially away from the pivot.

The technical name for this is centrifugal acceleration. Please don’t tell me there is no such thing as centrifugal acceleration. It is a readily-observable physical fact. It is characteristic of rotating reference frames.

The magnitude and direction of the centrifugal acceleration varies from place to place in the rotating frame. Therefore we call it the centrifugal field. It is zero at the pivot. The farther we go from the pivot, the stronger it gets. It is everywhere directed radially outwards from the pivot.

It must be emphasized that the centrifugal field is associated with the rotation of the reference frame. It exists in rotating frames and not otherwise. In particular when Joe observes the motion of the baseball, he can (and should) explain everything he sees without any contribution from the centrifugal field.

The centrifugal field is not necessarily associated with the rotation of any material object. (A reference frame is something of an abstraction, and we can have a reference frame that is not tied to any material object.)

According to Einstein’s principle of equivalence, at each point in space, the gravitational field is indistinguishable from an acceleration of the reference frame. Einstein explained this in terms of the famous “elevator” argument. Please don’t try to tell me that Einstein’s argument is valid for elevators but not valid for merry-go-rounds; it’s the same physics in both cases. We conclude from this that the centrifugal field is as real as the gravitational field. Forsooth, at each point, the centrifugal field is indistinguishable from the gravitational field.

As we shall see in section 2, the centrifugal field is not the only contribution to the equations of motion in the rotating frame.

In an introductory physics class, it is common practice to not bother analyzing motion relative to a rotating frame. It is perfectly reasonable to say that such an analysis is “beyond the scope of the course”. In contrast, it would be completely unreasonable to say that such an analysis cannot be done. It would be completely incorrect to say that the centrifugal field does not exist. It exists in a rotating frame and not otherwise.

It is a matter of opinion whether we should say the centrifugal acceleration corresponds to a centrifugal force, or whether we should call it an unforced acceleration. See reference 1 for details on this. There is no point in debating the matter, because the equation of motion is the same no matter what we say about it.

If somebody asks about centrifugal force, it is usually best to answer in terms of centrifugal field and centrifugal acceleration. If the answer cannot be expressed in those terms, there is probably something wrong with the question.

The tricky question of how and why you feel the effects of gravity is discussed in reference 2.

2  The Equations of Motion

We now move toward a more quantitative understanding of motion in a rotating frame. The set of equations of motion we use in such a frame is different from the set of equations we would use in a non-rotating frame. The two sets of equations are consistent with each other, and indeed each can be derived from the other, as we now demonstrate.

For additional details about the laws of motion, see reference 3.

2.1  Position

Suppose we have a pointlike object whose position is described by the abstract vector P. This vector has components P_J relative to the nonrotating laboratory frame, and components P_M relative to the rotating frame.

Let’s see if we can express P_M in terms of P_J (and in terms of the other “givens”). We can get a good start just by turning the crank.

Let’s start with plain old position. To connect the position in one frame with the position in the other frame, we write

where γ₁ and γ₂ are any two orthonormal vectors in the plane of rotation, and θ is the angle of rotation of the rotating frame relative to the laboratory frame, and r() is a rotor that generates a rotation in this plane, as discussed in section 5.3.

2.2  First Derivative

Now we can calculate the velocity, based on the position we just calculated. To do that, we differentiate both sides of equation 1. For simplicity we assume the plane of rotation is unchanging.

where the “dot” denotes differentiation with respect to time.

This can be put into a nicer form if we factor out γ₁ γ₂ in a couple of places:

  
The first two terms on the RHS of equation 3 look tantalizingly similar, and indeed using equation 19 we can combine them to yield:

where ρ_M is defined to be the projection of P_M onto the plane of rotation, that is:

  
Equation 4 can be made slightly more symmetric-looking by use of equation 21:

The second term on the RHS of equation 6 can be interpreted as a rotated version of the “ordinary” velocity, namely the time-derivative of position, in close analogy to the LHS of the equation. Meanwhile, the first term on the RHS is an additional contribution, a “tangential velocity” term due solely to the rotation of the frame. It lies in the plane of rotation, and is 90 degrees “ahead” of ρ_M.

2.3  Second Derivative

Let’s continue to turn the crank. To form the second derivative, we differentiate both sides of equation 6.

which can be simplified by collecting like terms, i.e. combining the 2nd and 3rd lines, and combining the 5th and 6th lines:

and that can be further simplified by some simple algebra:

and put into more conventional form:

where we have introduced ω := θ^• to denote the rate of rotation.

The four terms on the RHS of equation 10 can be given the following interpretation: The last term is a duly rotated version of the “ordinary” acceleration, i.e. the second derivative of the position, in close analogy to the LHS of the equation. It is independent of position and velocity. The third term represents the Coriolis effect; it is proportional to ρ_M^• (the velocity in the plane of the rotation) and also proportional to ω (the rate of rotation). It is independent of position. It lies in the plane of rotation, and is perpendicular to the velocity. The second term is conventionally called the centrifugal effect; it is proportional to the square of the rotation rate, and also proportional to ρ_M (the distance from the axis of rotation). It is independent of velocity. The first term accounts for changes in the rotation rate; any observer attached to the rotating frame will be “swatted” as the rotation rate increases or decreases.

2.4  Discussion

Equation 10 is not the most general expression, because we have assumed that the plane of rotation is not varying. In a two-dimensional world, this is obviously OK, but in three dimensions it is perfectly possible to be rotating around an axis while the orientation of the axis is changing. I’ve never actually seen the general expression. I suspect it would be quite a bit messier than equation 10.

Textbooks commonly present an expression that is even less general than equation 10, because they assume steady rotation, so that the ω^• term (the swat term) does not appear.

The key equations – equation 1, equation 6, and equation 10 – give a complete description of the position, velocity, and acceleration of the object in the rotating frame, in terms of the position, velocity, and acceleration in the lab frame. Of course they also depend on the plane of rotation and rate of rotation, and other “givens” of the situation.

It is worth remarking, and perhaps worth emphasizing, that these are not force equations. There are no forces appearing in any of these equations, and no masses. These equations are in no way dependent on Newton’s laws of motion. There is a Coriolis term in equation 10, but it cannot be called a Coriolis force. Similarly the centrifugal term in equation 10 cannot be called a centrifugal force.

Equation 10 has the elegant property of having the P_J-related term on one side, and all the P_M-related terms on the other side. This makes clear the role of these equations; they are simply the mapping from one coordinate system to another.

We now sacrifice elegance by rearranging the equation to isolate P_M^•• on one side:

Remark: The word “centrifugal” means, literally, fleeing away from the center. The rearrangement leading to equation 11 changed the sign of the ω² ρ_M term, so that now it more clearly deserves the name “centrifugal”, in the sense that ρ_M is directed radially outward, and the centrifugal term ω² ρ_M is also directed radially outward, making an outward contribution to the acceleration P_M^••.

There are various possible reasons why you might want to use equation 11. For example, suppose you already know P_J^•• somehow, and you wish to solve for P_M by timestepping the equation of motion. Equation 11 will tell you the current value of P_M^••, which you can integrate to get the next value of P_M^•, and so forth.

In particular, we might know P_J^•• from Newton’s second law of motion, equation 14. In its usual form, this law is not valid in rotating frames, but for present purposes that’s no problem. We simply apply the law in the nonrotating frame. This gives us the acceleration-components P_J^•• in terms of the force-components F_J.

Plugging in, we immediately obtain:

  
At this point we may, optionally, apply the distributive rule in reverse to pull a factor of (1/m) out of every term on the RHS. Doing so means that each of the terms within braces now has dimensions of force:

3  Conservation of Momentum

Momentum is conserved.

Momentum is conserved, no matter whether it is being observed from the laboratory frame, observed from a rotating frame, or not observed at all.

In the laboratory frame, momentum is equal to mass times velocity (i.e. velocity relative to the lab frame).

In the rotating frame, momentum is not equal to mass times velocity (i.e. velocity relative to the rotating frame). This is obvious if you consider a free particle moving slowly through the rotating frame. Its momentum will change direction completely, again and again, as the rotating frame goes around and around.

In the rotating frame, momentum is conserved, but MV is not (in general) conserved. The general expression for momentum is more complicated than MV.

4  Gravity and Centrifugity

There is a profound analogy between gravitational acceleration and centrifugal acceleration.

Details coming soon.

5  Basic Laws + Lemmas

In this section we review some basic laws and lemmas, partly to establish notation, and partly to make this document more self-contained.

5.1  Newton’s First and Second Laws of Motion

Newton’s second law of motion may be written as:

where m is the mass of the object, a is the acceleration of the object, and F_imp is the force impressed upon the object by the surroundings. It must be emphasized that this equation is valid only in Newtonian “inertial” frames, not in rotating frames.

Usually equation 14 is usually written as simply F=ma, where it is understood that F is shorthand for F_imp. However, for present purposes it is worth being explicit about the distinction:

The force F_exp is a perfectly ordinary force, it just doesn’t happen to be particularly suitable for use in equation 14.

In principle it would be straightforward to reformulate the laws of motion in terms of F_exp instead of F_imp, but this would be highly unconventional. Non-experts would be well advised to adhere to the established convention.

The first law of motion can (in retrospect) be considered a simple corollary of the second law, and need not be discussed here.

5.2  Newton’s Third Law of Motion

Newton’s third law of motion may be stated in various ways. Perhaps the most general and most powerful way is to say that momentum is conserved. That is, momentum obeys a strict local conservation law.

The following is another (essentially equivalent) way of expressing Newton’s third law:

This is consistent with conservation of momentum because force is momentum per unit time.

5.3  Rotors

Rotations can be expressed by multiplying something on one side by a rotor [denoted r()] and on the other side by the reverse of the rotor [denoted r^∼()]. In the case of a rotation in the γ₁ γ₂ plane, we have

  
The rotor angle θ/2 is half of the rotation angle θ. For details, see reference 4.

5.4  Multiplying by γ₁ γ₂

In three dimensions, any vector P_M can be expanded in terms of γ₁, γ₂, and γ₃, such that

Then we find that

which tells us that γ₁ γ₂ commutes with any vector entirely perpendicular to the γ₁ γ₂ plane, but anticommutes with any vector lying entirely in the plane.

In two dimensions, the foregoing is trivial. The generalization from three dimensions to higher dimensions is straightforward.

5.5  90 Degree Rotations in the Plane

Given any vector ρ lying in the γ₁ γ₂ plane:

we can represent a 90 degree rotation in any of several ways:

5.6  Frequency Modulation

We are accustomed to seeing periodic functions written in the form f(ω t). If ω is constant, that’s fine ... but otherwise it’s a trap for the unwary.

If you consider ω to be a function of time, and differentiate f(ω t) with respect to time, you get two terms:

where the second term is unphysical. We define f^’() to be the derivative of f() with respect to its argument.

The better approach would be to start over, and write f(θ) instead of f(ω,t). Then the derivative is:

where we have simply defined ω to be the derivative of θ.

Note that the RHS of equation 23 is the same as the RHS of equation 22, without the nasty (ω^• t) term.

Running the same argument backwards, we find

which is equal to (ω t) if ω is constant, but generally not otherwise.

The physical significance of this can be appreciated with the help of figure 2, which compares ∫  ω dt to ω t. In both plots, ω=1 at early times, and ω=2 at later times. You can see that changing ω in the expression ω t produces a large unwanted “leverage” effect, with a lever-arm of length t.

Figure 2: Leverage: ω t

This issue arose in section 1. The same issue arises in many other contexts, including the frequency modulation that one encounters in radio electronics and other signal-processing applications.

6  Acknowledgements

This document incorporates many insights and suggestions from the members of the phys-l discussion list.

7  References

1.

John Denker “Fictitious Force? Pseudoforce” ./fictitious-force.htm

2.

John Denker “Can You Feel Gravity?” ./gravity-perception.htm

3.

John Denker “The Laws of Motion” [Chapter of See How It Flies] ../how/htm/motion.html

4.

John Denker “Rotations” ./rotations.htm