"The Mass of Men Live Lives of Quiet Despiration"
A thought occurred to me today, which, being quite ironic, was that I didn't feel that I'd properly "thought" in months now. Luckily, that seemed to have been the proper shower to make the drought wane. So, a few paragraphs should suffice to summarize a few items of interest.
In continuing my assault on morality, it occurred to me that it would be a worthy proposition to determine a definition for morality. So often it seems that things are debated in common speech without a rigorous definition. Morality is the collection of morals to which a person subscribes. Morals are a person's individual beliefs of what is universally acceptable, inacceptable, or proper. This seems like a reasonable definition, but I haven't decided if it should be modified. It would be fallacious for me to deride the existance of all morality, which I don't, but I do believe that morality should be replaced a desire to make actions that are most beneficially primarily for the mass of other people, and secondarily for the individual. This is quite subtle. Take for example the case of individuality. This argument is in no way critical of individuality. Rather, it supports it because individuality is often better for the mass of people. There are many other intricacies of this view which I don't have time to articulate now. This is all very vague at this point, but compelling.
I talked to Zaraza about deriving the moments of inertia for arbitrary bodies that can be represented by integrals. It's quite interesting, and I think I'm just starting to understand it. Take the simplest case for example: a thin rod of length L that can be approximated along 1-dimension, and that rotates about one end. The moment of inertia for a particle can be represented by:
I = mr2
If we are looking at a body of mass M, then we can say that the mass M is equal to the sum of all its infinitely small component masses dm. Thus, we shrink I and m to the differentials dI and dm:
dI = r2 dm
Integrating gives:
∫dI = I = ∫r2 dm
Now here's where I got confused, and what I'm starting to fully understand... suppose the rod has a linear density of:
ρ = M/L,
then we could also say that the linear density is equal to the ratio of differentials:
ρ = dm/dr
setting equal and solving for dm:
dm/dr = M/L
dm = (M/L)dr
substituting back into the integral and evaluating from 0 to L (since the end is the axis of rotation):
I = ∫r2 (M/L)dr
I = (M/L) ∫r2 dr
I = (M/L) (1/3)(L3)
I = ML2/3
And lo and behold! That's the expression for moment of inertia. More complicated bodies require more complicated processes (shell method, etc.), but the idea is the same.
In other news, I went a to college visit by the University of Chicago. Not only was this an excellent way to escape the redundancy and pointlessness of Culpepper, but it was somewhat interesting. It seems like a pretty interesting school. They don't do engineering or architecture, so I'd have to go the uber-purist route for math and physics (already happening...). Apparently they're trying to dispel the kind of reputation that they've gotten from having the highest suicide rate of any university in the US, and the notoriety of never having a student graduate with a 4.0 gpa, despite the kind of students who go there. Nevertheless, it definitely seems like a place to consider: 80% of classes have less than 25 people, and 60% have less than 15, the only classes not taught by professors are 100-level foreign language classes (taught by native-speaking graduate students) and low-level math courses, and more Nobel Laureates are affiliate with U of C than with any other US university (although they may not have the same student to Nobel Laureate ratio as Caltech, per se). All this is for a modest $43,000/year. Maybe I should give Ben Zimmerman a call next year...
Argh! I'm out of time; must go work!
I enjoyed seeing your math progress -- it's nice to see that you're beginning to understand basic calculus.
On U of Chicago: I don't know if you realize I did my graduate work there. It's a great school, but the place sucks -- bad weather, slums, very dreary. You'd be much happier in a better climate, e.g. Stanford or Caltech. All the academic benefits you mention are true, but it would all be a waste if you commited suicide.
The calculus is the easy part, it's understanding the the physical representation of the math and why it works that is difficult. Luckily, I generally consider myself to be stable enough to not committ suicide, but you're right: with that kind of weather, all bets are off.
It matters not the weather--It matters the quality of education.
Don't go to Stanford. Stanford doesn't give any merit-based scholarships. I had been considering it before I found that out, but I know there is no way my parents would pay for that without any help.
Geez, that bites. University of Chicago apparently offers 30 full scholarships and 100 partial tuition scholarships each year. Culpepper was saying that most of the people that she's known who have gone to Stanford really haven't felt very challenged in their first 2 years, apparently.