The pi’s, i’s, and integers were called “Oppenheimer factors” in California in the 1930s, for the physicist who famously used to leave them out. One can guess the future father of the atomic bomb was into dimensional analysis.
You derive the Bohr radius from dimensional analysis, but I noticed that you use the reduced Planck’s constant h/2pi. That thing didn’t get called h-bar till Dirac did so in 1930. Bohr called it Mo, and without the factor of 2pi, the radius of atoms comes out to be 2pi^2 too large. for knew that, and put in the two pie for reasons that are hard to explain. It’s basically numerology. Bohr postulated that electron angular momentum came in units of h-bar, not h. There was no good reason why it should work in 1913, and it wasn’t until 11 years later, when the wave nature of matter was suggested, that pi was connected to the wavelength along the *circumference* of an orbital. and indeed, HBR is not seen in quantum mechanics, unless circular spherical is symmetry is being invoked for a standing wave.
Did you know that dimensional analysis gives a pretty good figure for the yield of an atomic bomb, simply from the radius of the blast at a given time? The Oppenheimer factor in this case, turns out to be the square root of gamma, which is the ratio of heat capacities. Of course, it’s dimentionless. Newtons first guess for the speed of sound in gases was also missing precisely that factor. But how could he know it?
Dimensional analysis was one of the most useful things I ever learned in high school. I had a chemistry teacher who showed us how to use dimensional analysis to solve word problems, without any parentheses or brackets or anything.
If you take your example of swinging a mass on a string, he'd give us a question with actual numbers. "You're swinging a 10kg hammer at the end of a 2m rope, at 1 rev/sec. What is the tension in the rope?" We knew tension was in Newtons, so we'd write N=, and then draw a long line. We'd start with a formula we knew, like F=mv^2/r. That would let us start filling in numbers and units. Anything in the numerator of the formula went on top of the line, and anything in the denominator went on the bottom. Vertical lines separated different parts of the formula. Once it was set up, you just canceled units until you had the units you wanted. If the units didn't work out, there were only two possibilities: either use a conversion factor to get closer to the units you wanted, or you almost certainly had a mistake somewhere.
Later on, probably in an undergrad physics class, I realized the word problem approach could be used more generally with formulas themselves, as you've shown here.
That one skill carried me through my first two years of undergrad. I used it in all my chemistry, engineering, and physics classes. I watched other students struggle for hours trying to reason through a problem that I solved in minutes by just trusting the units. I was forever grateful to that teacher for showing us this strategy at such an important time in our lives.
The pi’s, i’s, and integers were called “Oppenheimer factors” in California in the 1930s, for the physicist who famously used to leave them out. One can guess the future father of the atomic bomb was into dimensional analysis.
You derive the Bohr radius from dimensional analysis, but I noticed that you use the reduced Planck’s constant h/2pi. That thing didn’t get called h-bar till Dirac did so in 1930. Bohr called it Mo, and without the factor of 2pi, the radius of atoms comes out to be 2pi^2 too large. for knew that, and put in the two pie for reasons that are hard to explain. It’s basically numerology. Bohr postulated that electron angular momentum came in units of h-bar, not h. There was no good reason why it should work in 1913, and it wasn’t until 11 years later, when the wave nature of matter was suggested, that pi was connected to the wavelength along the *circumference* of an orbital. and indeed, HBR is not seen in quantum mechanics, unless circular spherical is symmetry is being invoked for a standing wave.
Did you know that dimensional analysis gives a pretty good figure for the yield of an atomic bomb, simply from the radius of the blast at a given time? The Oppenheimer factor in this case, turns out to be the square root of gamma, which is the ratio of heat capacities. Of course, it’s dimentionless. Newtons first guess for the speed of sound in gases was also missing precisely that factor. But how could he know it?
Dimensional analysis was one of the most useful things I ever learned in high school. I had a chemistry teacher who showed us how to use dimensional analysis to solve word problems, without any parentheses or brackets or anything.
If you take your example of swinging a mass on a string, he'd give us a question with actual numbers. "You're swinging a 10kg hammer at the end of a 2m rope, at 1 rev/sec. What is the tension in the rope?" We knew tension was in Newtons, so we'd write N=, and then draw a long line. We'd start with a formula we knew, like F=mv^2/r. That would let us start filling in numbers and units. Anything in the numerator of the formula went on top of the line, and anything in the denominator went on the bottom. Vertical lines separated different parts of the formula. Once it was set up, you just canceled units until you had the units you wanted. If the units didn't work out, there were only two possibilities: either use a conversion factor to get closer to the units you wanted, or you almost certainly had a mistake somewhere.
Here's what it looks like for that example question: https://imgur.com/a/MEIwH7m
Later on, probably in an undergrad physics class, I realized the word problem approach could be used more generally with formulas themselves, as you've shown here.
That one skill carried me through my first two years of undergrad. I used it in all my chemistry, engineering, and physics classes. I watched other students struggle for hours trying to reason through a problem that I solved in minutes by just trusting the units. I was forever grateful to that teacher for showing us this strategy at such an important time in our lives.