Fermilab

You asked:
If I hold a book at arms length with one hand on either side of the book and rotate around in a circle, does the book "spin" on an axis perpendicular to the plane of rotation. This is a very annoying problem. Can you please explain which frame of reference is to be used and why or why not?

This problem originally occurred when considering how the moon travels around the earth. Assuming that we only see one side only of the moon's surface (we actually see approx 54%)....and it is clear that the moon does spin on it's axis once for every revolution around the earth (that is why it keeps the same face towards us all the time.

Now, holding a book with both hands in front of you and spinning around in a circle......is the book behaving as the moon does......ie is the book spinning on its axis (and that is the reason that it has the same face towards us at all times. I am 50 years old and have been confounded by this problem for years. I love physics and have read extensively. I would really like to clarify this problem. It is to be viewed as you and the book are the only two objects in the universe. Please advise if you need any further clearification.Thanking you for you help, I remain

Yours truly

Frank

Dear Frank:

Instead of thinking of your two examples, earth-moon and you holding a book, think of you holding the hands of a beautiful woman, and the two of you start to dance. Extending your arms all the way, you start spinning around each other and you look into each other's eyes. How about that?

In this dance, there are two special ways of how you may do the spinning:

1) You stand in one spot and let the lady travels around you, and you turn simultaneously not to lose eye contact with her. Or:

2) The lady stands in one spot and you move around her. She would have to turn, too, to keep eye contact.

Let's look at the two possibilities, one at a time:

1) You stand in one spot and let the lady travel around you: In this situation, you, of course, think of yourself as spinning around an axis with you at its center, and you always see the same (pretty?) side of your lady.

2) The lady stands in one spot and you move around her: Now, of course, the lady thinks of herself as spinning around a central axis running through her body, and you move around her, always facing her.

This is exactly what earth and moon are doing. But which one stands in one spot?

Well, go back to the two dancers. What happens when they are dancing and spinning, and all of a sudden the floor, the whole world around them is gone?

Right, you cannot make out a specific spot. You are missing what physicists call a FRAME OF REFERENCE. Physicists, of course, are used to abstract reasoning and quickly draw a little coordinate system. They have their choice of fixing the frame of reference at the lady's feet or the gentleman's feet. BOTH FRAMES ARE EQUIVALENT, AND THE LAWS OF PHYSICS ARE THE SAME IN BOTH FRAMES. It is fair to say that the lady is spinning and that the gentleman is spinning, too. It only depends on your point of view or, as a physicist would say, your frame of reference. Some frame may be easier to use than others (think of people trying to explain the motion of objects in the sky when taking the earth as the center of the universe - impossible).

The answer to the dancing couple is then easily transfered to your example of holding a book and turning at the same time.

Now, to create the real earth-moon situation, you need to take an advanced level dance class. The figure we want to look at: The gentleman makes the woman spin around her own center in addition to her moving around him. As she does that, she will again and again see the gentleman's face, never his back. Again, both the gentleman and the lady are spinning (look at different frames of reference), with the lady spinning much faster. Exactly like earth (the lady) and the moon (the gentleman). The earth spins around its center once every 24 hours, and the moon spins around its center only once every four weeks.

Next, we need to talk about how things spin around other things, not just their own center. The moon spins around the earth once every four weeks - or you can say, the earth spins around the moon every four weeks. (It is accidental that this is the same time it takes the moon to spin around its own axis. Moons of other planets spin at a different rates around their own center than they spin around their planet.) You can pick the frame of reference that seems the most convenient. The easiest frame to do quantitative physics computations is the "center of mass" system. In the case of a spinning couple it is close to the point where their hands join, a little shifted towards the heavier person.

The statement I made a few paragraphs above, "Both frames are equivalent, and the laws of physics are the same in both frames," is the crucial point in answering all your questions. The abstract reasoning leading to this remarkable statement goes back to Galileo. In his honor, physicists call the transformation from one frame of reference to another frame of reference a Galilean transformation, if the transformation only requires simple rotation and/or simple shifting in one direction. (Transformations from one frame into an accelerated frame of reference, like a falling elevator, are not Galilean transformations. Physical processes seem to look different in those different frames. Studying general relativity, however, Einstein realized that those frames are also equivalent if the change in gravitational force is taken into account.) Many years after Galileo's research, the physicist Newton was able to put Galileo's reasoning into mathematical and physical equations.

The last paragraph might have been a little too technical, but I wanted to show you that great physicists pondered over your question, and it has led to important laws that every physics student learns about in college. Maybe we should teach them in high school?

Best wishes,

Kurt Riesselmann

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