Barycenter: The Earth Does Not Revolve Around the Sun

"Eppur si muove."
— Galileo Galilei

Even though the famous utterance is likely apocryphal, we have, nonetheless, known for hundreds of years that the Earth revolves around the Sun. And while Earth’s orbit does completely encompass the  Sun, technically, the point around which the Earth revolves is not the Sun. Rather, it is the center of mass of the Solar System — called the barycenter.

Barycenter of the Solar System

All the objects in the Solar System have mass and, therefore, generate a gravitational effect on all the other objects in the Solar System. This means giant Jupiter and tiny Mercury (and even non-planet Pluto) all pull on the Sun, and each other. The combined effects of all the gravitational effects means that the center of mass of the Solar System is not the center of the Sun.

The barycenter is sometimes close to the center of the Sun, and, oftentimes, it is within the Sun. Depending on the position of the planets, however, it is frequently not actually located within the Sun. At these times, the Earth, in a manner of speaking, does not revolve around the Sun.

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Vodka vs. Beer in the Freezer

1412027953754You can store vodka in the freezer but you cannot do the same with beer, as it will explode. Why is that? The answer lies in the relative alcohol content of liquor versus beer. Vodka is about 40 percent alcohol (80 proof) and beer is four to eight percent alcohol (and 90+ percent water). Alcohol (ethanol) has a melting / freezing point of -173°F, while water has a freezing point of 32°F.

Most home freezers are kept between -5°F and 0°F, which will freeze the water in beer, but is not cold enough to freeze the water in vodka, which has much more alcohol dissolved in it. To make it worse. water expands when it is a solid, compared to when it is a liquid – resulting in the exploding beer in your freezer.

Bonus fact: The boiling point of ethanol is 173°F is lower than water (212°F) but is still high enough that your liquor will not boil away except when cooking it. Sometimes, however, having a freezing point well below water and a boiling point well above water can be useful. For example, automotive radiator fluid has a boiling point around 300°F and a freezing point well below 0°F. This allows a driver to use the same fluid in both the winter and the summer.

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Soliloquy vs. Monologue

Alas, poor Yorick! I knew him …

Both soliloquy  and monologue mean “single speech.” The first is from Greek and the latter from Latin. However, there is a functional difference between the two forms of speech: A soliloquy is delivered when the speaker is alone on stage, while a monologue is delivered by one speaker in the presence of others. In other words, a soliloquy communicates only to the audience, while a monologue communicates to other characters within the story itself.

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Why Can’t You See Through Walls?

Why can I hear you through a wall but cannot see you through a wall? Why does my radio work indoors, even though the view of the transmitter is blocked? Simply, both light and sound are waves; why does light get blocked by a wall, but sound does not?

The answer has to do with the wavelength of the wave relative to the medium through which the wave is traveling. An analogy is that a person running through a rainstorm will get from point A to point B. However, a small insect will get knocked down by the rain.

In this analogy, the rain is the wall, the person is the sound wave and the insect is the light wave. Sound and radio waves have very long wavelengths compared to the size of the atoms that make up the wall. On the other hand, light has a wavelength that is about the same size as the atoms that make up the wall, and are, therefore, blocked by the wall.

This explanation also works for why high-frequency (5 GHz) WiFi waves are blocked by walls to a greater degree than lower-frequency (2.4 GHz) WiFi waves. In addition, this explanation completely ignores the chemical structure of the medium. For example, not all walls block light — a glass wall will allow light to pass. Similarly, radio waves cannot pass easily through a lead barrier. These differences are caused by the electron structure of the medium interacting differently with different types of waves.

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The Back of the Dollar Bill

Have you ever wondered what is on the back (technically, the reverse) of a US dollar bill? The a salient feature of the reverse side of the bill is the Great Seal of the United States, which pictures an eagle on the obverse and an unfinished pyramid on the reverse.

Reverse of the US One Dollar Bill

The obverse of the Great Seal is the Coat of Arms of the United and features an eagle with thirteen arrows in its left talon and an olive branch in its right talon. It clutches a banner reading, E Pluribus Unum, meaning, “From One, Many”.

 

The reverse of the Great Seal is an unfinished, thirteen-layer pyramid, symbolizing the unfinished nature of our nation, with the eye of Providence watching over the United States. The base of the pyramid reads, MDCCLXXVI, or 1776. Two mottos also appear on the reverse of the Great Seal. First, the words Annuit coeptis, which means, “[He has] approved of our undertakings,” and, second, Novus ordo seclorum, meaning, “a new order for the ages.”

 

 

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Can You See Across Lake Michigan?

If you go to the Sears Willis Tower, can you see Michigan? We can use geometry to answer the question.

We can use the Pythagorean Theorem to find the length of the side of a right triangle: a^2+b^2=c^2. In this case the right triangle has sides of d and r and a hypotenuse of r+s. Therefore, the Pythagorean formula is r^2+d^2=(r+s)^2, where r is the radius of the Earth, s is the height of the Sears Tower, and d is the distance one is able to see.

Using FOIL to expand (r+s)^2 gives:

r^2+d^2=r^2+2rs+s^2

 

Since r^2 is on both sides, we can eliminate it.

d^2=2rs+s^2

d=\sqrt{2rs+s^2}.

The radius of the Earth is r=3,963 miles and the height of the Sears Tower in miles is s=0.275379 miles. Consequently, d=46.72 miles. The distance across Lake Michigan is just over 50 miles, so NO, we cannot see all the way to Michigan.

Note: This example assumes that one can only see in a straight line-of-sight. A mirror would allow you to see much farther. In fact, the sky, clouds, and sunlight can form a mirror, a meteorological phenomenon called refraction, and it is sometimes possible to see across Lake Michigan.

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How We Can Save the Elephants

Externalities are a type of market failure that are loosely defined as an action for which the full cost or benefit of that action is not reflected in the market price. The implication of which is that since the person doing the bad thing is not paying the full cost to society of their bad actions, there is too much of this bad action.

African elephants in front of Mount Kilimanjaro.

Conversely, since a person only receives the private benefits of their good actions, rather than the full societal value, there is an under-provision of good acts, and society misses out on the optimal level of a socially good action. In order to arrive at the socially optimal level, one needs to figure out a scheme to internalize the external costs or benefits. Most solutions to externalities center around regulation, taxation, or other forms of government intervention. A very smart man, however, by the name of Ronald Coase came up with a scheme that, with two small assumptions, changed how everyone thought about externalities and the case for government intervention. Oh, and it also won him the Nobel Prize. Continue reading

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The Contrapositive

Today’s lesson comes from formal logic: the contrapositive. The contrapositive is the a reversal of an if-then statement, that is also true. In fact, it is the only construction that is always true. It is constructed by reversing the two items in the if-then statement and taking the opposite of both.

For example, a person might say, “if you jump off a bridge, then you will die.” The contrapositive of this statement is, “if you are alive, then you did not jump off a bridge.”

If the original statement is true, the contrapositive is true, and if the contrapositive is false, then the original statement is false. NB: Did you notice that I used the contrapositive here too?

Formally, the contrapositive is written as:

Original: A \rightarrow B

Contrapositive: -B \rightarrow -A

Quiz: For the statement, “If you have read this far, then you have reached the end of the post,” what is the contrapositive?

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How to Use Comprise

This week I saw not one, but two (gasp!) misuses of the word comprise; therefore comprise is this week’s Pearl of Wisdom. Let us begin with the completely, utterly, and totally incorrect. Do not ever write “is comprised of.”

So, what is the proper use of the word comprise? The DLMC practice comprises Dispute Consulting and Legal Management Consulting. In other words, the whole comprises the parts; comprise is another way of saying, ” is made up of” or “consists of.” Another example is The United States comprises fifty states.

What you do not want to say is that the parts comprise the whole, e.g., fifty states comprise the U.S.A. You would not say fifty states is made up of the U.S.A., so do not say fifty states comprise the U.S.A.

What should you write in such a case? I prefer the word compose. It is perfectly fine to write three separate branches of government compose the federal government. Switch that around, and you might also write the federal government comprises three separate branches.

To sum up: the whole comprises the parts, and the the parts compose the whole.

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Bates Numbering

Most of my fellow readers spend many hours each day looking at Bates numbered documents, the individual identifying number found on each document turned over in discovery as part of a legal proceeding.

You might wonder where the term Bates number comes from. In fact, it is named after the inventor of the Bates stamp, Edwin G. Bates, who patented the Bates Automatic Numbering-Machine in 1891 and then again in 1901. The patent, titled Consecutive-Numbering Machine, was assigned to the Bates Manufacturing Company of New York City.

The Bates stamp was quite literally an ink stamp that advanced numbers after each depression of the stamp. The numbering ranged from 0000 to 9999. Higher-end models allowed for stamping in duplicate or triplicate. Eventually, the Bates stamp was expanded to seven digits in order to allow for larger legal productions.

Eventually, the Bates stamp was sold to the General Binding Corporation (GBC) in 1993. Today, however, most Bates stamping is done via PDF software.

 

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