The Couple Talks about the Volume of the Truncated Cone

On June 7, my wife and I joined a bus trip to Exhibition of Paintings from State Pushkin Museum and Flower Festival Commemorative Park. We had lunch at a French restaurant, whose building was a nationally designated Important Cultural Property. At the lunch table, a couple younger than my wife and I took the seats in front of us. Let's call them Mr. and Ms. N. A glass containing water from the well of the restaurant and some ice cubes was prepared for each person from the beginning of the lunch course (see the photo above).

At some stage of the lunch course, Mr. N took the glass and asked Ms. N if she could guess how to calculate the volume of the solid of such a form. She said, "Add a small cone to make it a large cone. Calculate the volume of the large cone and subtract the volume of the small cone from it." Mr. N replied that it would be very cumbersome. Then, he said that the volume can be obtained as the mean of volumes of three cylinders having the same height as the solid. His voice was so low that I was unable to hear the radii of the three solids, but from the movement of his hands, I supposed that he referred to the radii of the top and bottom circles of the truncated cone and a certain mean of the two. He additionally stated that we could obtain the formula by integration of the circular area.

I had never heard of the formula for the volume of the truncated cone, and thought it wonderful that Mr. N learned it and remembered it for some reason. However, I also wondered why he who spoke of a more complex method of integration said that his wife's simpler method was cumbersome. After returning home, I calculated the volume by Ms. N's method and easily found that the third radius mentioned by Mr. N was the geometric mean of the radii of the top and bottom circles.

To see the formula and the derivation of it, visit here. The explanation is in Japanese, but readers might easily follow equations by looking at a diagram included. The third method mentioned there by the use of Pappus-Guldin theorem (also known as Pappus' centroid theorem; the second theorem is relevant here), however, might be a little difficult to understand, if you have never heard of that theorem.

Yukawa's Utushi-e

When I read the English translation, by L. Brown and R. Yoshida, of Yukawa's autobiographical book Tabibito, I found many errors and sent a list of corrections to Brown. The list included shadow pictures as the translation of "うつしえ (utsushi-e)." Yukawa mentioned it together with ground-cherries and Kintaro wheat-gluten bars, etc. as things sold at street-stalls at a fair in his childhood. Utsushi-e meant both shadow pictures (写し絵) and transfer pictures (移し絵), but shadow pictures are played rather than sold. So, I thought the correct translation of utsushi-e should be transfer pictures but was not confident about this. In my childhood transfer pictures were surely popular among children. In the days when Yukawa was a child, however, sheets for shadow pictures (the equivalent to slides of the present days) might have been sold, and they might have enjoyed a picture show by passing lamplight through them. (This story was given in Part I, Chapter 1, of my book Passage through Spacetime.)

Recently I read Soseki's autobiographical novel Michikusa and found the following passage:

Of course, he was able to get whatever toys he wanted. Tools for utsushi-e (写し絵) were also included in it.

In Notes section at the end of Michikusa, the following description is given about utsushi-e:

The thing that reflects pictures drawn on glass onto a screen made of cloth or paper, in the dark; a magic lantern.

These indicate that the shadow picture was one of popular toys in Meiji period, to which childhood days of both Soseki and Yukawa belonged. Soseki was born in 1867; Yukawa, in 1907; and I in 1935. As for the year of birth, therefore, Yukawa and I are closer than Soseki and Yukawa. However, there was a big change of cultural environments when electric lights spread through the country [an almost full spread in Tokyo was achieved in 1912 (Ref. 1)]. In my childhood, I saw a projector of 8-mm film in a nearby home already being used instead of a magic lantern. Thus, Soseki and Yukawa had much in common in their childhood, and I have to admit that the translation, "shadow pictures," by Brown and Yoshida was quite right.

Reference

1. Chronology of electricity from Taisho to Showa, A Web page of the Federation of Electric Power Companies of Japan.

Boy of Age 16 Asks Me about Relativity, etc. 21. Does Time Really Exist? What Is Time?

Relativity of simultaneity: Event B is simultaneous with A in the green reference frame, but it occurred before in the blue frame, and occurs later in the red frame (Ref. 1). The original PNG file of the figure was created by Army1987; Acdx converted it to SVG. (GFDL or CC-BY-SA-3.0), via Wikimedia Commons.

A friend of mine on Twitter, Aaron (a pseudonym), is an overseas, 16-year old boy, who seriously admires Albert Einstein and wants to become a physicist. He continually writes me (Ted, also a pseudonym) questions about the theory of relativity and related topics, and I am sending answers. In this series of blog posts, those questions and answers are reproduced with modifications. I am not an expert in the fields of physics related to relativity. So, my answers might contain errors. If you find any error, please do not hesitate to write a comment for the benefit, not only of the boy and me, but also of other readers.

Aaron: Do we control in time? Or does it control in us?

Ted: I do not understand what you exactly mean by the words "control in" in your question. The question sounds like a philosophical one rather than that of physics. But I can say this: Time is one of physical dimensions connected to the Universe or Nature. Therefore, human being can do nothing to affect it. However, "psychological time" (duration of time one feels about a definite length of physical time under different situations) can be controlled by the adjustment of one's mind. Am I talking in the wrong direction than you expected?

Aaron: Sorry, I want to know if time really exists. In relativity, time also seems to be relative, right? A body that travels at a speed close to that of light can slow time, right? So, what is time?

Ted: I see, Aaron. Time is one of dimensions of the physical framework of the Universe, "spacetime," and is the measure of durations of events and the intervals between them. It has a definite meaning when we consider the movement of something. If there were nothing moving around in the Universe, time would be meaningless and could be said that it does not exist. However, the real Universe includes a lot of moving things. So, time is a meaningful and useful concept. Duration of events and simultaneity depend on the coordinate system (reference frame) on which it is measured (see the figure above), but this does not deny the reality of time.

By the way, the slowing-down of the passage of time (time dilation) occurs for the fast-moving body, as you mentioned, but this occurs for the moving system as a whole, i.e., your biological activity and ability also slow down. So, you cannot do much more thing during the high-speed flight in a rocket compared with what you can do on the earth in the same duration of time. You cannot be the master of time but remain to be its slave.

It would be another problem to ask if time is a fundamental concept. There is a growing movement to create a theory that shows spacetime is emergent, i.e., not fundamental (see for example Ref. 1). In this respect, time is one of things still mysterious.

References

1. "Time," Wikipedia: The Free Encyclopedia (February 19, 2013, at 06:41).
2. Graeme Stemp-Morlock, Melting Spacetime, Web site FQXi Community (April 30, 2012).

(Originally written on October 14, 2011)

Boy of Age 16 Asks Me about Relativity, etc. 20. Have You Heard about Naruto?

Cover of the first Japanese Naruto manga volume.

A friend of mine on Twitter, Aaron (a pseudonym), is an overseas, 16-year old boy, who seriously admires Albert Einstein and wants to become a physicist. He continually writes me (Ted, also a pseudonym) questions about the theory of relativity and related topics, and I am sending answers. In this series of blog posts, those questions and answers are reproduced with modifications. I am not an expert in the fields of physics related to relativity. So, my answers might contain errors. If you find any error, please do not hesitate to write a comment for the benefit, not only of the boy and me, but also of other readers.

Aaron: I just wanted to ask you if you have heard about Naruto.

Ted: What I think of from the word "Naruto" is tidal whirlpools in the Naruto Strait, Japan. However, this is probably not what you mean. Perhaps, you mean this (though it is not related to the physics of relativity): Naruto (ナルト), an ongoing Japanese manga series written and illustrated by Masashi Kishimoto. I don't know about it. So, I quote a passage from a Wikipedia page (Ref.1): "The plot tells the story of Naruto Uzumaki (note by Ted: Uzumaki means tidal whirlpools), an adolescent ninja who constantly searches for recognition and dreams to become the Hokage, the ninja in his village who is acknowledged as the leader and the strongest of all. The series is based on a one-shot comic by Kishimoto that was published in the August 1997 issue of Akamaru Jump."

Aaron: Yes, it's one of the best Japanese anime series. Ninjas can travel near the speed of light.

Ted: Oh, Naruto has then a relationship to relativity. Ha-ha!

Reference

1. Naruto, Wikipedia: The Free Encyclopedia (January 24, 2013 at 19:01).

(Originally written on October 7, 2011)

Boy of Age 16 Asks Me about Relativity, etc. 19. Do All the Forces in Nature Travel at the Same Speed as That of Light?

The Feynman diagram for the beta-minus decay of a neutron (n) into a proton (p) , due to the weak force, i.e., via an intermediate heavy W⁻ boson. One of down quarks (d) in the neutron decays into an up quark (u) to make a proton, emitting an electron and an electron anti-neutrino. By Joel Holdsworth (Joelholdsworth) [Public domain], via Wikimedia Commons.

A friend of mine on Twitter, Aaron (a pseudonym), is an overseas, 16-year old boy, who seriously admires Albert Einstein and wants to become a physicist. He continually writes me (Ted, also a pseudonym) questions about the theory of relativity and related topics, and I am sending answers. In this series of blog posts, those questions and answers are reproduced with modifications. I am not an expert in the fields of physics related to relativity. So, my answers might contain errors. If you find any error, please do not hesitate to write a comment for the benefit, not only of the boy and me, but also of other readers.

Aaron: Do all the forces in nature travel at the same speed as that of light?

Ted: There are four known fundamental interactions in nature to cause fundamental forces in the universe: gravitational, electromagnetic, strong nuclear, and weak nuclear interactions. Among these interactions, the mediators of three interactions, i.e., the gluon for the strong interaction, the photon for electromagnetic interaction and the graviton for the gravitational interaction, have (or assumed to have) zero mass. Therefore, the forces based on these interactions are transmitted by the speed of light. The remaining one interaction, weak interaction, is mediated by the heavy W and Z bosons and cannot be transmitted by the speed of light. However, the effective range of the weak force is quite short (around 10⁻¹⁷–10⁻¹⁶; Ref. 1), so that we can regard that the weak force is transmitted almost instantaneously for it actually to work.

Reference

1. J. Christman. The Weak Interaction, Physnet (Michigan State University, 2001) p. 2.

(Originally written on October 1, 2011)