/sci/ - Science & Math

>>6729464
8

>>6729444
Yeah, the force balancing equation is redundant and may be omitted.

>>6729426
Well, you can see what it will be. Considering that we are punching in <span class="math">p_i[/spoiler] and <span class="math">\delta(x)[/spoiler], the integrals are just constants, as are the terms <span class="math">p_j - p_i[/spoiler]. So each equation is just a linear equation in <span class="math"> F_i [/spoiler]. We have N linear equations for the N variables <span class="math">F_i[/spoiler]. Solving is simple.

>>6729388
I just wrote it out by hand.

>>6729381
Of course it's solvable, but whether we can write an analytically exact expression for it entirely depends on what <span class="math">\delta(x)[/spoiler] is.

>>6729380
Seriously, fuck this TeX in the tiny reply box. You get what I'm saying, yes?

>>6729378
A magnetic beam isn't really a thing. Generally opposing fields cause repulsion, aligned fields cause attraction.

>>6729376
Goddammit, I keep seeing small errors, that's what I get for copying and pasting.

Final results:

Force balancing:
<span class="math">\sum_{i = 1}{N} F_i = mg[/spoiler]

Torque balancing, end cases:
<span class="math"> g \int_0^{p_1} x \delta(x) dx = g \int_{p_1}^L x \delta(x) dx - \sum{i = 2}{N} F_i |p_1 - p_i| [/spoiler]

<span class="math"> g \int_0^{p_N} x \delta(x) dx - \sum{i = 1}{N-1} F_i |p_N - p_i| = g \int_{p_N}^L x \delta(x) dx [/spoiler]

Torque balancing for pins j = 2, 3, 4, ... N-1:
g \int_0^{p_j} x \delta(x) dx - \sum{i = 1}{j-1} F_i |p_j - p_i| = g \int_{p_j}^L x \delta(x) dx - \sum{i = j+1}{N}F_i |p_j - p_i|}

>>6729366
Oh balls, the sums should all be sums of <span class="math"> F_i (p_j - p_i) [/spoiler], not just <span class="math">F_i[/spoiler].

So you understand what I did, I just balanced the torques about each pin. So the torque from gravity acting on the beam, which is the integral term on each side of each equation, minus the torques from each other pin, which is the sum of forces-times-distances-from-central-pin that I wrote above, should be equal on either side. The first equation comes from the fact that they have to all add up to the total mass.

>>6729365
Fucked up that last typesetting:

<span class="math"> g \int_0^{p_j} x \delta(x) dx - \sum_{i = 1}^{j-1} F_i = g \int_{p_j}^L x \delta(x) dx - \sum_{i = j+1}^N F_i [/spoiler]

>>6729346
>>6729344
>>6729335

OK, here is my analysis. You get the following equations:

<span class="math"> \sum_{i = 1}^N F_i = mg [/spoiler]

<span class="math"> g \int_0^{p_1} x \delta(x) dx = g \int_0^{p_1} x \delta(x) dx - \sum_{i = 2}^N F_i [/spoiler]

<span class="math"> g \int_{p_N}^L x \delta(x) dx = g \int_0^{p_N} x \delta(x) dx - \sum_{i = 1}^{N-1} F_i [/spoiler]

and then for j = 2, 3, 4 ... N-1:
<span class="math"> g \int_0^{p_j} x \delta(x) dx - \sum{i = 1}{j-1} F_i = g \int_{p_j}^{p_N} x \delta(x) dx - \sum_{i = j+1}^{N} F_i [/spoiler]

From here it's just computation.

>>6729339
Oxygen itself doesn't just flame or explode. It is highly reactive with other chemicals in the environment. Oxygen is fairly stable on Europa due to the lack of anything for it to react with, whether it is exposed to heat or not.

>>6729335
This is a tough question. The answer is going to be a big system of equations with terms like <span class="math"> \int_{p_i}^{p_{i+1}} x \delta(x) dx [/spoiler], constrained by the balancing of torques. I could try and work it out, but it might take a little to get everything right. I'll get back to you.

>>6729325
This is the definition of an infinite sum. Any time you see sum to infinity written, you aren't actually plugging in infinity to n, you are taking the limit of value of the nth partial sum as n goes to infinity. Big N is just the outer limit of your sum, and is being used to explicitly write the infinite sum as the limit by which it is actually calculated.

>>6729327
Oh, I see. For a body under no external forces, any rotation is about the center of mass, so you actually won't see any wobbling about happening at all. However, this is thrown out the window if your gravitational field varies on the size scale of the object- if the field is stronger at the left end than the right end, you have a torque and essentially a different system of forces compared to if you had a point mass.

Ultimately this boils down to the fact that a point mass is a completely adequate approximation for a free falling body, if and only if the gravitation field varies slowly compared to the dimensions of the body.

>>6729303
I'm not a fusion engineer, so I can't really claim to know about what material properties are required in tokamaks. If the plasma-facing material is meant to absorb neutrons and radiate heat into a steam generator, I guess it would need to have a high neutron interaction cross section. If cooling is adequate I doubt that the temperature would be that much of a problem, so I'd go with beryllium?

>>6729309
Frankly there's not a lot of physics in finance, but research in physics has yielded and continues to yield results in nonlinear dynamics, chaos, probability, computer science, and other applied-math topics that are highly applicable to the study of finance.

>>6729307
That sum should be typeset as <span class="math">\sum_{n = -N}^{N}[/spoiler].

>>6729295
They don't. There are two ways that imaginary numbers come into geometry in physics. One is the use of geometry to describe complex numbers, using the complex plane. This is a way of representing an imaginary number <span class="math"> z = a + bi[/spoiler] as an x-y pair <span class="math">(a, b) [/spoiler] of the real and imaginary parts. Representing complex numbers in this way yields many beautiful and non trivial results that are invaluable for certain computations.

The other relationship is when complex numbers are used to represent geometry. The distance between two points in Euclidean space is the sum of squares of the differences in their cartesian coordinates, or for a 4D space, <span class="math"> d^2 = x_1^2 + x_2^2 + x_3^2 + x_4^2[/spoiler]. But we live in what is called a Minkowski space, and the fourth coordinate, time, actually contributes negatively to distance in spacetime: <span class="math"> d^2 = x^2 + y^2 + z^2 - t^2[/spoiler]. One thing that is entirely non physical and unrelated to math is to do a coordinate substitution <span class="math"> \tau = it [/spoiler], which makes the distance equation look more like Euclidean space: <span class="math"> d^2 = x^2 + y^2 + z^2 + \tau^2[/spoiler]. This can occasionally simplify calculations, but before any usable results are obtained you have to substitute back in the real <span class="math">t[/spoiler] coordinate.

>>6729300
I see. Well it's pretty much what I said. The first infinite sum gives you infinity, as you found. The second one can be found by observing that every 8 values of n contribute <span class="math">2 + 2 \sqrt{2}[/spoiler] to the sum, so (not in general, but within the limit as N goes to infinity) you can rewrite <span class="math">\sum_{n=-N}{N} | \sin (n \pi / 4) | = (2 + 2\sqrt{2}) (2N + 1) / 8 [/spoiler].

>>6727543
1. I second starting early. Read the problem set when you get it and mull it over for a while before you actually break into it.

2. Collaborate. Working with other people makes the problem sets easier, leads to better understanding of the material, and makes it a more enjoyable, social thing.

3. Figure out your attention span before your productivity tails off, cut that in half, and work in spurts of that amount of time. I usually work for 15 minutes solid, then get up and run around or whatever for another 15, then go back. Rinse and repeat.

4. Figure out how you recharge. Exercise? Reading the news? Bugging your housemates? Watching an episode of Adventure Time? Do what you have to do in your breaks to get yourself together.

5. When you find yourself at a complete loss, leave for a few hours and come back with fresh eyes.

>>6729277
>au.answers.yahoo.com/question/index?qid=20121203225405AAlUeXO
Can you post the figures referenced in the problem?

Also, I'd rather not do peoples' homework for them.

>>6729274
I still can't really provide input unless you explain the problem. Why are you summing values of the sines of multiples of <span class="math">\pi/4[/spoiler]?

I did overlook the absolute value sign, which changes my result slightly for the limit on the bottom. For every 8 values of n you are picking up <span class="math">2 + 2 \sqrt{2}[/spoiler] in the sum. So you can rewrite it as <span class="math"> \lim_{\rightarrow \infty} \frac{(2N+1)(2 + 2\sqrt{2})}{8(2N+1)} = (1 + \sqrt{2})/4 = 0.604[/spoiler]

>>6729258
I don't know what the physics problem is that you are talking about. I am just looking at the math on that page, and it is wrong. The limit as n goes to infinity of <span class="math"> sin(n \pi / 4) [/spoiler] is not infinity. Can you explain what the problem you are working on is?

>>6729244
No, lots of physics grad students are happy. I had a great time in grad school, and so did lots of my cohort, whether they ultimately went into industry, academia, finance, whatever. There were a few people who were totally miserable, and a couple people who didn't hate or love it but just were there to get through it, but that's the case anywhere you go.

>>6729239
Divergence is not the same as infinity. For no <span class="math">n[/spoiler] is <span class="math"> \sin (n \pi / 4) [/spoiler] going to be infinity.

What you want to do regarding the second equation is observe that <span class="math">sin(-x) = -sin(x)[/spoiler], so each negative term will cancel with each positive term in that summation. The only term left will be the <span class="math">n = 0[/spoiler] term, which will be divided by <span class="math">2N+1[/spoiler], so the limit is 0.

>>6729235
Sorry, my response to >>6729209 didn't really make sense the way I meant it to. What I meant to say is that the force on a body in a gravitational field is proportional to its mass, and force is mass times acceleration. Setting the force <span class="math">F = ma[/spoiler] equal to <span class="math">F = mg[/spoiler], where <span class="math">g[/spoiler] is the gravitational field strength, the mass terms cancel.

>>6729226
>>6729226
I don't know, people are miserable for all kinds of reasons. Maybe their advisor is a bastard, maybe they don't really enjoy physics.

But probably it has nothing to do with physics, and there's something else at play. Maybe they are ill, they are having family trouble, or they feel lonely, inadequate, or bored. Why don't you ask your friend and see what they say?