Bound states of spin-dependent potential

1. The problem statement, all variables and given/known data



Hi! My issue here is that I need to find the bound states (if any) of the potential:

[tex] U(r)=-C\frac{s_1\cdot \hat{r}\, s_2\cdot \hat{r}-s_1\cdot s_2}{r}.[/tex]



Here [itex]s_1[/itex] and [itex]s_2[/itex] are the spins of the two spin-one particles involved in this interaction. The two particles have the same mass.





3. The attempt at a solution



My initial reaction here is to try to diagonalize the numerator, since once this is done we essentially just have the hydrogen atom potential, so the bound states would be trivial to find given the solution to the hydrogen atom.



The second term is easy to do, we can define [itex]J=(s_1+s_2)[/itex] to get [itex]s_1\cdot s_2=\frac{J^2-s_1^2-s_2^2}{2}[/itex]. The other term has been giving me more trouble though. It's possible my approach here is completely wrong.



Any suggestions? I'd prefer an analytic solution, but if it has to be done numerically that's okay too.

Derivative Maxwell boltzmann distribution

1. The problem statement, all variables and given/known data

i need to show that the peak of the maxwell boltzmann distribution is equal to 1/2 kt.



2. Relevant equations

maxwell boltzmann distribution according to modern physics 3rd edition by kenneth kramer.



ill try to do my best with this



[itex] N(E)= \frac{2N}{√∏} \frac{1}{(kT)^\frac{3}{2}} E^\frac{1}{2} e^\frac{-E}{kT}[/itex]



N is the total number of molecules while N(E) is the distribution function (with units energy to the -1) defined so that N(E) dE is the number of molecules dN in the energy interval dE at E. dn=N(E)dE

3. The attempt at a solution



so i need to take the derivative and set that equal to 0 and hope i get 1/2kt. im having trouble with the derivative itself. im taking the derivative with respect to E so everything else is considered a constant. so to try to make this easier i took all that junk in front of the E and said it is just some constant a. that allowed me to go through and do the product rule. after that, ive been trying to simplify it but am getting nowhere. need some advice on how to do this properly as i believe im not.



thanks a bunch

Astrophysics Exercise

Hi, I am having problems with the following question:



If the Sun subtends a solid angle Ω on the sky, and the flux from the Sun just above the Earth’s atmosphere, integrated over all wavelengths, is f(d_sun), show that the flux at the Solar photosphere is ∏f(d_sun)/Ω.



Does anybody have an idea on how to start? I thought following formulas could be useful:



Ω = A/d^2

L = f * 4 ∏ * d^2



I would truly appreciate some help!

Connecting mySQL to Ruby on NETBEAN 8.0

So... I hae this issue trying to connect mySQL to ruby on net bean... Has anyone connected it before?



The code:



#!/user/bin/ruby





require 'mysql'





begin

puts "hello World"

con = Mysql.new 'localhost', 'sqluser', ''

puts con.get_server_info

rs = con.query 'SELECT VERSION();'

puts rs.fetch_row



rescue Mysql::Error => e

puts e.errno

puts e.error



ensure

con.close if con

end



LoadError: no such file to load -- mysql

require at org/jruby/RubyKernel.java:1027

require at /Users/eugenelemon/Library/Application Support/NetBeans/8.0/jruby/lib/ruby/shared/rubygems/custom_require.rb:36

(root) at /Users/eugenelemon/NetBeansProjects/RubyApplication1/lib/main.rb:9

a query

Guys this is really urgent and might b silly but what do you actually call the part which looks like a eccentric cam(egg shaped) I've seen it in some old generators were shaft is mounted at centre and there is a lever connected at other end of the part. When the lever is rotated it runs the shaft too.. Can anyone tel me the name of the component... Hurry plz..

Experiment on piston cylinder, issues with Ideal gas law

Hello everyone,



I have an rather large piston-cylinder containing gas that is compressed when the piston extends. Using three sensors I am measuring the pressure, temperature and volume of the gas while I force the piston out using a winch. The system is closed and no gas is released or added to the system while doing the experiments.



From the data collected I am calculating the gas-mass using the ideal gas law (PV=mRsT). I was expecting the gas-mass to be more or less constant, however the results show that the mass seems to change during the gas-compression phase, and slowly crawl back to the initially calculated mass from before the gas-compression phase was started.



I've attached the state variables that i logged during my experiment. I started with the piston fully retracted, then pulled it out using a winch and kept it there for the rest of the experiment.



















Could someone, with a bit more brains than me, enlighten me as to why the mass is not constant, and maybe point me towards another approach to calculating the mass of the gas in the cylinder?



My own hunch tells me that it got something to do with the limitations of the ideal gas law, but my thermodynamics knowledge is not sufficient to fully understand it. :)

Quick question on Minkowski space diagram

Consider the lower right plot in this picture (or many similar ones).



I interpret the angle of the t' axis with respect to the t axis as: From the point of view of the stationary observer, all progress in time for the moving observer will be accompanied by the latter's spatial progress (to the right in this case), so the 'spatial baseline' for the moving observer is angled with respect to that of the stationary observer, as shown. Right? This seems straightforward.



I find it harder to interpret the angle of x' to x. I get that it 'works out' as in how the lightning flash at the carriage's front occurs in the past for the moving observer (red dotted lines). (If the angle of x' to x were the negative of the actual one, then the flash would occur in the future, so it would not be a good representation.) But is there an intuitive equivalent to the interpretation for the t' axis? "From the point of view of the stationary observer, all progress in space for the moving observer will be accompanied by the latter's temporal ..." Somehow I get confused here. Something about the latter's temporal slowing down...? I don't quite see how that speaks from the diagram.



Thanks!

Pulling a box up a ramp

1. The problem statement, all variables and given/known data



A girl of mass mg = 50 kg is pulling a sled up a slippery slope. The coefficient of friction between the girl's boots and the slope is µs = 0.155; the friction between the sled and the slope is negligible. The girl can pull the sled up the slope with a ≤ amax = 0.030 m/s2 before she begins to slip. Assume the rope connecting the girl to the sled is kept parallel to the slope at all times. The angle of the slope is θ = 8°.



2. Relevant equations







3. The attempt at a solution



I have already worked out the maximum value of the force of friction on the girl as:

50kg *9.8 m/s^2 = 490N

Mg(j-hat) = COS 8 * 490N = 485.23N

Fs = Us * Mg(jhat) = 0.155 * 485.23N = 75.21N

Gravity Force on Girl, down the ramp = Sin 8 * 490N = 68.19N



Net force unused by girl is75.21N - 68.19N = 7.02N

Net acceleration is 0.03m/s^2 so, using remaining force, 7.02N / 0.03m/s^2 = 234kg

Total box should be 234kg / sin 8 = 1681 kg but that isn't right. I know my total friction for the girl and I know it's right . . . what am I missing in my calculation of the box weight?



Thanks!

Rod

Packing fraction of spheres in a HCC structure

1. The problem statement, all variables and given/known data

Show that the ratio of atomic sphere to unit cell volume in HCP (hexagonal close packing) is 0.74.



2. Relevant equations

volumes of spheres, geometry



3. The attempt at a solution

I did the same problem for FCC and BCC and it was fine.

My unit cell structure is that shown below. I labeled the height between the two hexagonal planes by ##h## and the length of the equilateral triangles comprising the hexagon by ##2r##. If we then consider a 1/6 of this structure in the obvious way and orient it suitably so that one of the sides of the lower triangles coincides with the x axis say, then the volume of 1/6 of this structure is $$V = 2 \int_0^{r} \int_0^{\sqrt{3}x} \int_0^h dx dy dz = \sqrt{3}hr^2.$$ Multiply this by 6 to get the whole volume of the unit cell structure shown below.



I would like to try to relate the height of this structure to the radius ##r## of the spheres so that in the ratio, I get cancellation. I am assuming that the three spheres on the middle layer of the structure (labeled B in the sketch) are wholly contained within the structure?

Attached Images

HPC.jpg (41.9 KB)

Flow, pressure, and pipe diameter?

If I have water being pumped through two lengths of pipe, both at 50 psi, but one pipe is 1.25" diameter and the other is 10" diameter, approimately how many gpm will be flowing through each pipe? Thanks in advance!

Should I just drop my physics class?

Trying to keep this as short as possible, I'd like to be an engineer, and I'm currently taking an algebra-based physics course. It's not a class I need for my major (of course, I will need calc-based), but I've already passed the drop deadline, so I'll receive a "W" on my transcript. It's been four (going on five) weeks of school, and the workload is extremely overwhelming for a class I don't need. I think the problem with this is largely due to my professor—compared to the workload of the other people in my labs, their professors hardly give any work that goes beyond having a fundamental (mathematical) understanding of each section. Basically, all the other algebra-based physics professors are a world easier on their students when giving work than mine is.



Right now, I also have a variety of computer classes (largely programming) and a math class in my schedule. Those are classes I actually do need to take, and this physics class is hindering me from receiving the A grades I should be able to get.



Now, don't get me wrong, I find that I REALLY love the material in this physics course—I truly think it's all incredibly intriguing! But at the rate my professor is handing out work, I don't think I can keep up with all my classes the way I'd like to. Opinions?

integration test of dirac delta function as a Fourier integral

1. The problem statement, all variables and given/known data

Problem:

a) Find the Fourier transform of the Dirac delta function: δ(x)

b) Transform back to real space, and plot the result, using a varying number of Fourier components (waves).

c) test by integration, that the delta function represented by a Fourier integral integrates to 1



2. Relevant equations

So far I've done a) and b) and the delta function turns out to be [itex] \delta (x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} e^{i\omega x}d\omega [/itex]



I've plotted this and it seems to be correct, and I also asked some other students in class and they got the same result, so i don't think that's the issue.



3. The attempt at a solution

So to solve c) I try to integrate δ(x) from -∞ to ∞, but it shouldn't really matter as long as 0 is between the integration limits.

[tex]

\frac{1}{2\pi} \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{i\omega x}d\omega dx\\

\frac{1}{2\pi} \int_{-\infty}^{\infty}\left[\frac{1}{ix}e^{i\omega x}\right]_{\infty}^{\infty} dx\\

\frac{1}{\pi} \int_{-\infty}^{\infty}\frac{1}{x}\frac{1}{2i}(e^{i\infty x} - e^{-i\infty x}) dx\\

\frac{1}{\pi} \int_{-\infty}^{\infty}\frac{1}{x}\sin(\infty x) dx

[/tex]

Apparently I end up with an integral that's impossible to solve (without approximation), and the sine function has infinity as its argument... So I was hoping you would know where I went wrong.

Friction problem

A dockworker loading crates on a ship finds that a 33-kg crate, initially at rest on a horizontal surface, requires a 72-N horizontal force to set it in motion. However, after the crate is in motion, a horizontal force of 51 N is required to keep it moving with a constant speed. Find the coefficients of static and kinetic friction between crate and floor.



static friction___

kinetic friction___

Event-horizon from the blackhole metric

What is the most general method of obtaining the event-horizon from the given blackhole metric.



Let us consider Kerr blackhole in Kerr coordinates given by

[tex]

ds^2 = -\frac{\Delta-a^2sin^2\theta}{\Sigma}dv^2+2dvdr -\frac{2asin^2\theta(r^2+a^2-\Delta)}{\Sigma}dvd\chi-2asin^2\theta d\chi dr + \frac{(r^2+a^2)^2-\Delta a^2sin^2\theta}{\Sigma}sin^2\theta d\chi^2+\Sigma d\theta^2,

[/tex]

where

[tex]

\Sigma = r^2+a^2cos^2\theta\\

\Delta = r^2 - 2Mr+a^2.

[/tex]



How do we find the killing vector in a coordinate system?

Any hint or reference would be of great help.

wheel speed at tyre compared to at the centre

does anyone know the formula for calculating the speed a wheel has to be travelling to cover a certain amount of rotations within a minute eg 30rpm,



if 2m diameter implies a 3.142 perimeter (x 30)

therefore do the wheels move at 94 meters p m (x 60 minutes , in time)

= 5.64 kmph or 3.5 mph? essentially small wheels to require faster speed hence gearing etc

? if yes does anyone have a simpler calculation?



Also the calculation to turn an item such as a water wheel different size for different torques on the crank shaft



The leverage required at different size to achieve a desired rpm at different levels of torque?