How Does the Value of an Inverse Cosine Function Change When the Unit of its Argument Changes?

How does the value of $\cos^{-1} \left ( 0.1 \, \textrm{cm} \right )$ differ from $\cos^{-1} \left ( 0.1 \, \textrm{m} \right )$? This is an interesting question, especially because the question of units in trigonometric and inverse trigonometric functions is rarely discussed in mathematics textbooks. It is discussed in physics textbooks, so you may … Read more

Why do Airplanes Fly?

In many physics textbooks, the explanation for lift on a flying airplane is that the top of a plane’s wing is longer than the bottom, and so air must travel faster across the top than the bottom, and therefore the pressure is lower above the wing than below. (This difference in pressure associated with different … Read more

Can Hot Water Freeze Before Cold Water? The Mpemba Effect

Here is a link to a fascinating post on this subject from earlier this year at Skulls in the Stars. The effect has apparently been observed for millennia (the post contains a reference by Aristotle!), but the conditions under which it occurs and the scientific explanation for the Mpemba effect are still unclear. Interested experimenters … Read more

Stupid Textbook Questions

One of the persistent complaints about mathematics and science textbooks, especially high-school textbooks, is that the questions tend to involve formula manipulation in a way that is not very meaningful. Such questions tend to be artificial. A case in point is the following question, taken from a chapter on special relativity in a high-school physics … Read more

We Teach Too Much “What to Think” And Not Enough “How to Think”

Way back when I was a little high-school student, I recall the pride I felt when working through the nuclear reactor unit of our Grade 11 physics course. The pride derived from the featured CANDU reactor. Wow! Our little Canada somehow produced the best nuclear reactor in the world. How did I know CANDU was … Read more

On Pulling Weeds

“A weed is a plant whose virtues have not yet been discovered,” said Ralph Waldo Emerson. This is a profound observation about judgements and the role of words in conveying judgements. Nowadays, with our quite extensive knowledge of plants and their benefits, we persist in calling quite useful plants weeds just because we don’t want … Read more

The Scientific Peer Review System: A Cautionary Tale

One of the brilliant strong points of the scientific enterprise is its self-correcting nature. Constant testing, constant criticism, constant reflection, constant doubt … they all lead to errors being caught and corrected, and fuzzy ideas being clarified. At its best it underlies the collaborative nature of science. The peer-review system seems to be essential; otherwise, … Read more

Resources For Superconductivity

I felt like learning more about superconductivity last week, so I spent some time reading from various sources. Not being an expert in the field, I cobbled together what I could find, and I thought I would list some items that might be of use to others who would like to learn about this very … Read more

Words, Episode 6: “… 2000 times as small …”

I’ve been reading the very fine book Superconductivity: A Very Short Introduction, by Stephen Blundell, about which I’ll have more to say in a subsequent post. The book is very well written, with only a very few editorial infelicities in the book, the most striking of which is the following phrase, which appears on page … Read more

Why Does A Propane Torch Get Cold When Used? Updated, With Refrigerators

I was fixing a leaky water pipe in my garage yesterday, and so I had occasion to use a propane torch to heat up the pipe and melt the solder into the joint. It’s not something I do often, but I do remember learning from my father (who was experienced in plumbing, and was somewhat … Read more