And now it’s all this

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] The sixth way we’ll derive the formula for the center deflection of a uniformly loaded simply supported beam is the Myosotis method, which I wrote about over a decade ago. This is the method popularized1 by J.P. Den Hartog in his Strength of Materials textbook. Image from Wikipedia. Myosotis is the genus of the forget-me-not flower, and the idea behind the…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] The next technique we’ll use to derive the formula for the center deflection of a simply supported beam with a uniform load is the slope-deflection equation: MA=2EIL(2θA+θB−3ψ)−FEMA Let’s start by explaining where all the terms come from. Here’s a beam of length L with arbitrary end supports (could be simple, fixed, free, or sprung) and an arbitrary…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] The fourth way we’re going to derive the formula for the center deflection of a simply supported beam with a uniform load is the conjugate beam method. This is probably tied with the moment-area method for the simplest and fastest way to get the formula—at least if you’ve memorized the properties of parabolas. I wrote about the conjugate beam method last…

In Paul Krugman’s post today, he includes two charts. One is fine, the other isn’t. Here’s the first one: Generally, I’m not a fan of putting stuff in the margins that could be within the body of the plot itself, but at least it’s clear which label goes with which data series. Too bad that’s not the case with Chart 2: The two series end at almost the same spot, so putting the labels out in the margin means they can’t both be centered on their series. The little leader lines that run from the…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] Continuing our odyssey through various ways of calculating the deflection at the center of a simply supported beam with a uniform load, today we use the moment-area method. There are two moment-area theorems used to calculate the slopes and deflections of a beam. Here’s how they’re given in the textbook I used as an undergrad, the 3rd edition of Elementary…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] In the introductory post to this series, I mentioned that the shear is the derivative of the moment. It’s easy to see that for our specific problem, the simply supported beam with a uniform distributed load, but it’s also true in general. A further relationship is that the distributed load function—let’s call it q—is the derivative of the shear (with a…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] Here’s the first of the derivations for the center deflection of a simply supported beam with a uniform load. We start with the differential relationship between the bending moment, M, and the deflection, y: M=−EId2ydx2 The second derivative of y is the curvature of the beam (for small deflections, which is one of the fundamental assumptions of beam…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] A few years ago, I wrote a post describing how I asked ChatGPT to solve a couple of elementary beam bending problems and how its answers were persistently wrong, even after I told it the mistakes it had made. For the first problem, determining the deflection at the center of a simply supported beam with a uniform load, ChatGPT gave the correct…

I mentioned last week that some wording in a news article struck me as odd. A similar thing happened when I read this Scientific American piece (that’s an Apple News link—I don’t have a SciAm subscription). In this case, I didn’t even have to read the article; the oddity is right there in the headline: “A SpaceX rocket booster is on track to hit the Moon at several times the speed of sound.” I know what the writer means, of course. He’s telling us that the SpaceX debris will hit the Moon at…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] Earlier this week, I saw this article in Apple News. It’s from the San Francisco Chronicle and discusses a recent report on the safety of the Golden Gate Bridge. The report is one of several reports spurred by the Francis Scott Key Bridge collapse a couple of years ago. Image from Wikipedia. Spoiler: the report finds the Golden Gate Bridge safe—quite…

After yesterday’s post, I got an email from reader Ron Sprague outlining two ways to block sender addresses from the list view in iOS Mail. Both of these methods are efficient, and neither starts with choosing a command with a stupid name like View Contact Card. I knew about one of these methods but not the other; this post will describe both. (My first thought was that this would be an update to yesterday’s post, but I soon realized it was going to be too long for that.) The first method…

Complaints about Apple’s design choices usually involve transparency, color, and other legibility concerns. These criticisms are legitimate, but the poor design choices that usually set me off are the ones involving the mechanics of commands—what I have to do to get something done. I think of these as the “design is how it works” mistakes. Yesterday, for example, I was going through my mail on my iPhone and came upon a piece of spam. I put it in the Junk folder with the hope that future emails…

I meant to write this last week, shortly after I finished listening to Episode 612 of Upgrade. But things didn’t work out the way I hoped, so now I’m rushing to finish these few paragraphs before this week’s episode comes out (they’re recording as I type). I have nothing to say about the Tim Cook/John Ternus news that hasn’t already been said. I want to focus on Jason and Myke’s choices—made and unmade—in their Apple at 50 Draft. My favorite picks were the oddballs, the products that weren’t…

[Equations in this post may not look right (or appear at all) in your RSS reader. Go to the original article to see them rendered properly.] In this morning’s blog/newsletter,1 Paul Krugman included a couple of unusual graphs that I want to talk about. It took me a little while—longer than it should have—to figure out what he was doing and why, and I’m still not sure I agree with his plotting choices. Let’s go through them and you can decide for yourselves. The two graphs of interest are made…

During my seven-week Spotlight trial, I was reminded of how easy it is to make file and folder launchers in Keyboard Maestro. In case you’ve also forgotten, here’s a short post on how to do it. There are three items that I open quite often and that Spotlight was slow to find: My blog-stuff directory, which is where I keep all the text files, scripts, images, and other components that go into making blog posts. Each post, or set of posts, gets its own subdirectory in blog-stuff. My notebook…

I started using launchers shortly after returning to the Mac (from Linux) in 2005. The first one I used was the great Quicksilver. I’m sure I learned about it from Merlin Mann, who was Quicksilver’s biggest advocate, but I can’t point to which of his many posts on QS got me started. When Nicholas Jitkoff (Alcor) stopped developing Quicksilver in 2007 or so, I switched to LaunchBar, and that’s been my main launcher ever since.1 I gave Alfred a workout for a few months—inspired, I think, by this…

Yesterday’s post included some behind-the-scenes calculations that I figured were worth talking about. They were all done in Mathematica, and here’s the notebook I used:

In its continuing attempt to teach us how calendars work, Scientific American has an article (Apple News link) up today that goes through Gauss’s algorithm for calculating the date of Easter. The article was written by Manon Bischoff, who also wrote the Friday the 13th article I covered a few weeks ago.

Back in January, I complained about the Apple Watch’s Timer complication being too complicated. A couple of days ago, Dan Moren told me on Mastodon that my complaint had been addressed in watchOS 26.4, which was released earlier this week. After a surprisingly quick update of my watch (I had already updated every other device to 26.4 but somehow forgot to do the watch), I added a 3-minute timer as the bottom center complication and, miracle of miracles, it worked exactly as it should.

I’ve been working my way slowly through Reingold and Dershowitz’s Calendrical Calculations. This week I hit Chapter 11 on the Mayan and Aztec calendars and came across a notation for modulo arithmetic that wasn’t familiar to me.1 I figured I’d write about it here on the off-chance that any of you would find it interesting. Also, to make it stick in my head a little better.

You may have seen this article by Joe Rossignol in MacRumors yesterday. It’s about a new feature in macOS 26.4 that’s designed to keep us from pasting dangerous commands in Terminal. If a user pastes such a command, this warning pops up:

I’ve been thinking about the new Apple Studio Display lately, mainly in the context of Jason Snell’s review of it on Six Colors and his further discussion with Myke Hurley on Upgrade.