The description of tidal force was revised in the earth science textbook for the year 2023. The explanation using centrifugal force has been changed to an explanation as the difference in gravitational force between points on the earth and the center of the earth. Those who still use centrifugal force in the explanation of tidal force should change the explanation as soon as possible.

**Earth science textbook by Keirinkan (revised in 2023)**

In Japan, two publishers had been publishing high school earth science^{*} textbooks, but only Keirinkan published an earth science textbook in 2023, the year of periodic revision.
Now that I have the new textbook, let us look at the explanation of tidal force in the textbook after the revision.
(* Textbooks of "basic earth science" are not discussed here.)

I pointed out to the textbook publisher that using centrifugal force to explain tidal force is problematic. It was February 2022, when the publisher seemed to have almost finished its revision work. They replied to me that the editorial board would consider the revision. As a result, in the revised textbook for 2023, the explanation of tidal force was largely rewritten, and the difference between the gravity of the moon at a point on the earth and the gravity of the moon at the center of the earth was used to explain tidal force, which is orthodox but correct. The fact that the Japan Meteorological Agency had already revised the explanation may have been effective. Perhaps we should say that the explanation has reverted to the earlier explanation. This is because it is said that tidal force had been explained as the difference in gravity before the incorrect explanation using "centrifugal force of equal magnitude and direction" was introduced to textbooks several decades ago.

The following is the page on tides in the new textbook. The explanation of tidal force is given in the description of the figure:

Figure 22. Tidal forces due to the moon

The earth is under the gravitational pull of the moon. Since the magnitude of the gravitational force is inversely proportional to the square of the distance, the relation between the magnitudes of the gravitational forces (yellow arrows) exerted at the earth center O and points A and B on the earth isT_{B}<T_{O}<T_{A}. Considering A and B relative to O, the forceS_{A}=T_{A}−T_{O}acts on A, and the forceS_{B}=T_{B}−T_{O}acts on B (red arrows). In other words, at both A and B the force acts in the direction away from O. These are the tidal forces due to the moon.

**Comments**

In the new revised textbook, tidal force is explained as the difference between the gravity of the moon at each point on the earth and the gravity of the moon at the center of the earth. This is the simplest and clearest explanation of tidal force used in the Astronomical Dictionary (in Japanese) of the Astronomical Society of Japan.

Then, why do we take the difference in gravity? To explain this, the concept of inertial force is necessary. That is, when the earth is pulled by the gravity of the moon, there appears an inertial force acting on the whole earth in the opposite direction to the gravity of the moon, as seen from the earth. At the center of the earth, this inertial force just cancels the gravity of the moon, so that the moon's gravity becomes invisible. This is the same phenomenon that the gravity disappears in a free-falling elevator. However, outside the center, the force remains. This is the tidal force.

Although inertial force is taught in high school physics, it may have been avoided in the new textbook because it is too difficult to use in earth science. However, in order to understand the force that pulls up the sea surface on the far side from the moon, it would be easier to consider the inertial force which acts in the direction away from the moon. If we explain that the inertial force is the force that leaves things behind when viewed from the earth accelerating toward the moon, even a high school student could understand it intuitively. The two forces cancel each other at the earth center, but the moon's gravity outweighs on the near side, and the force to leave things behind outweighs on the far side.

By the way, the figure in the new textbook does not have an arrow indicating the force opposite to *T*_{O}, but it would be easier to understand if −*T*_{O} is included as shown in the figure below.

According to an article in the *Astronomical Herald*, a monthly magazine of the Astronomical Society of Japan, "centrifugal force" began to be used to explain tidal forces in textbooks around 1966.
At that time, many teachers had difficulty explaining the upward tidal force on the far side from the moon.
Then, the explanation using centrifugal force was introduced and widely accepted as an easier way to explain tidal force.
Since more than a hundred years ago, some astronomers called the uniform inertial force (translational inertial force) "centrifugal force of equal magnitude and direction," and no one noticed that it is a misnomer.
This is probably why the term "centrifugal force" was used in textbooks without any doubt.

In fact, as explained in the new textbook, tidal force is a difference in gravity, so it appears when the earth is simply subject to gravity. Neither revolution of the earth nor centrifugal force is necessary. Tidal force works even if the earth is heading straight for the moon or running away from the moon. The explanation in the new textbook is perfect in the sense that it can be used not only for the earth being pulled by the moon or a free-falling elevator, but for anything that is in motion under gravity. However, the difference in gravity alone may be difficult for high school students to understand. I would like teachers to supplement the textbook by explaining that the inertial force, i.e., the force that tends to leave things behind, appears in the opposite direction to the moon when viewed from the earth.

The animation below is a reoriented version of Fig. 2 in the main page of this article. It shows the motion of test particles placed on a spherical shell, pulled by the gravity of a celestial body. You can see that the spherical shell is stretched as a result of the gravitational pull on the side close to the celestial body and the inertial force on the far side of the spherical shell. This force is, of course, the tidal force. It is clear from this animation that tidal force works even without orbital motion. If you are not familiar with the inertial force that appears when viewed in the spherical shell, you may realize it by imagining the situation in the animation again and again.

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T. Fujiwara 2023/04