Thursday, February 22, 2018

Aristotle's wheel paradox #2

A rolling wheel is not a simple motion. While rolling is rotation plus translation, only the latter really matters for the paradox. Still, it is easy to be lured by the complexity. For example:

1. A point on the perimeter of a wheel travels a cycloid path. A more inner point travels a curtate cycloid path. In the paradox the center's path is a straight horizontal line.

2. A point's velocity in the direction the center moves is faster than the center's velocity during the top half of a rotation. A point's velocity in the direction the center moves is slower than the center's velocity during the bottom half of a rotation.

3. Pure rolling occurs when the circular object travels one circumference along the ground for every for every full rotation it makes. Slipping occurs when rotation is faster than pure rolling. A paradigm case is a car wheel stuck in snow. Skidding occurs when rotation is slower than pure rolling. A paradigm case is a car wheel that skids on ice after the driver brakes hard.

Slipping and skidding so described affect the two circles in the same way. For example, if the part of the tire in contact with the road slips or skids, then the metal rim the tire is mounted on is affected the same way. The rim can't slip when the tire doesn't. Yet the rim slips while the tire doesn't is one "solution" to the paradox given on Wikipedia. It makes no sense except as a far-fetched metaphor. In other words, the rim "slips" but it doesn't really slip. The rim "skids" would be less far-fetched.

Sunday, February 18, 2018

Aristotle's wheel paradox #1

Wikipedia. I judge the article as poorly written, especially when it says the paradox is about two wheels. A comment on the Talk page agrees. The quote from Mechanica, written more than 2,000 years ago, describes the paradox as about two circles. Two circles can depict one wheel, e.g. like on a car or truck, with the smaller circle depicting a metal rim. Or the two circles can depict one tire -- not mounted on a rim -- with the smaller circle depicting its smallest circumference, the bead or lip. They can depict a roll of tape.

If the rigidly coupled circles are rolled a full revolution, then all points on both circles have the same position relative to their common center at start and end. Every point's translation vector has the same direction -- parallel to the horizontal surface -- and length as the center's translation vector. Such length is 2*pi*R, where R is the radius of the larger circle. This necessary fact about translation elegantly solves the paradox. Every point on the smaller circle must move 2*pi*R. This shows that the smaller circle's circumference 2*pi*k*R, where k is its circumference divided by R, is irrelevant for one rotation and the given setup. How far the smaller circle moves horizontally is dictated by its center.

Wednesday, February 7, 2018

Adam Smith quote

Adam Smith wrote The Theory of Moral Sentiments more than 250 years ago. Therein he identified the totalitarian mindset -- "the man of system" in his words -- with a great analogy.

"The man of system, on the contrary, is apt to be very wise in his own conceit; and is often so enamored with the supposed beauty of his own ideal plan of government, that he cannot suffer the smallest deviation from any part of it. He goes on to establish it completely and in all its parts, without any regard either to the great interests, or to the strong prejudices which may oppose it. He seems to imagine that he can arrange the different members of a great society with as much ease as the hand arranges the different pieces upon a chess-board. He does not consider that the pieces upon the chess-board have no other principle of motion besides that which the hand impresses upon them; but that, in the great chess-board of human society, every single piece has a principle of motion of its own, altogether different from that which the legislature might chose to impress upon it." VI.II.42

Sunday, February 4, 2018

Federal deficits rising

Wall Street Journal article reports that US federal deficits are rising.

Treasury yields are also rising, some investors think, in part because the supply of government bonds hitting financial markets is on the rise as budget deficits rise as a result of the Trump administration’s recent $1.5 trillion tax cut.

A group of private banks that advise the Treasury estimated the Treasury would need to borrow on net $955 billion in the fiscal year that ends Sep. 30, up substantially from $519 billion the previous fiscal year.

The article does not mention withholding for federal taxes, but I suspect that lower withholding, already implemented, is a significant contributor to the rising deficits. Link.

Like most politicians, President Trump doesn't care a whit about deficits. That was clear during his campaign and reinforced since then. Like most politicians, Trump criticized his political opponents about deficits and the debt, but that is as far as it went. I was all in favor of cutting the corporate tax rate, but not personal tax rates. (Income tax revenues are about 80% personal and 20% corporate.) I recognize that cutting corporate taxes without cutting personal tax rates would probably not have been politically feasible. However, the personal tax rate cuts could have been smaller and more offset by reducing or eliminating tax breaks. The timing for them was bad considering the economic situation.

Thursday, February 1, 2018

Flu epidemic

Wow. This season's flu epidemic is a brutal one. Many deaths in the USA are being reported. I got my flu shot, but this season's vaccine is generally less effective against the H3N2 virus.

Strangely enough, a flu pandemic occurred in the USA and around the globe 100 years ago. According to Wikipedia about 28% of the population in the USA became infected, and 500,000 to 675,000 died. The upper part of that range is more than the number who died because of the American Civil War. The USA's population was much higher during the flu pandemic -- a lot due to immigration -- than during the American Civil War, so the percent of the population who died from the flu was lower than the percent who died because of the American Civil War.