4:47 AM
The Young's modulus is a fundamental property of a material and it relates the tension to the fractional change in length.
For springs we have a corresponding equation:
$$F = K \frac{\Delta\ell}{\ell_0}$$
Where $K$ is a constant that depends on the modulus but also on the geometry of the spring. It's quite a complicated thing to calculate, but as long as you keep everything except the length of the spring constant $K$ is constant.
So in the usual Hooke's law equation $F = k\Delta\ell$ the constant $k = K/\ell$.

5:03 AM
Oops, that should be $k = K/\ell_0$.

5:36 AM
@SirCumference
Despite rumors, Coolmath Games is not shutting down in 2020

5:48 AM
Jul 23 '17 at 3:40, by The Raiders of Las Vegas

6:03 AM
@JohnRennie I have a question. I have a 5 watt blue laser. I want to find a formula to calculate the heat of the dot at a set distance. Or even a combination of formulas I can use to get the end calculation. What formula or set of formulas do that?

@ScientistSmithYT if you know the area of the dot $A$ then you know the intensity of the energy being supplied because it's just $I = 5/A$ watts per square metre.
But what you don't know is how fast the energy is being lost due to cooling.
At the equilibrium temperature the cooling rate must also equal 5 watts, i.e. the enegy being lost per second is equal to the energy being supplied per second. But to find out what this equilibrium temperature is you need to know how the temperature affects the cooling rate.

@JohnRennie Ok
@JohnRennie I do know for sure the output light power going out of the aperture is 5.16 watts. I rounded it to 5 watts.

@ScientistSmithYT to be honest, we'd normally answer the question by doing the experiment and measuring the temperature. It's too hard to predict theoretically because so many factors are involved.
If the object being heated is a perfect black body you can use Stefan's law to calculate the temperature. That would give you a guideline.
Stefan's law tells you the rate of heat loss per unit area, $j$, is given by $j = \sigma T^4$ where $\sigma$ is a constant.
So if the area of your dot is $A$ the rate of heat loss will be $\sigma A T^4$ and just set that equal to 5.16 watts to get the temperature.
The Stefan–Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths per unit time j ⋆ {\displaystyle j^{\star }} (also known as the black-body radiant emittance) is directly proportional to the fourth power of the black body's thermodynamic temperature T: j ⋆...
But this will only be a guideline as heat can be lost by other mechanisms such as conduction and latent heat of melting or vaporisation as well. So the real temperature will be lower than this.

6:23 AM
@JohnRennie Ok thanks, I'll just take it as a guideline.

6:54 AM
@JohnRennie riddle me this: I have a large pot of boiling water, and I throw a chicken egg inside it... effectively boiling it in X minutes. Then I decide to throw an ostrich egg, whose radius is 3 times larger. How much longer will it take to cook?
Prof. threw this at me yesterday, i found it a good question ;)

You can estimate it by assuming that the thermal conduction inside the egg doesn't matter and all that matters is the heat flow through the shell. The mass of the ostrich egg is 27 times greater (3^3) than the chicken egg so the thermal capacity is 27 times greater.
I wonder if you could approach the problem using dimensional analysis ...

6

The egg boiling problem is a seemingly simple problem posted from the blog of UChicago's physicist Dam Thanh Son. A chicken's egg has a length of 5 cm and takes 6 minutes to boil. An ostrich's egg has a length of 15 cm. How much time does it take to boil an ostrich's egg? (Photo for il...

@kylecampbell It's a joke - Noether = No ether.
2

niice play on words

7:10 AM
mass and thermal capacity are 27 times larger yes
but the time to cook the egg is not 27X

@Gyromagnetic But the shell surface area is a factor of 9 larger so for a given temperature gradient 9 times as much heat per second flows through the ostrich egg shell. That reduces the difference to a factor of three.

true, but the answer is not 3X

questions like this beg the response "when am I ever going to use this?" :-)

@Gyromagnetic And finally the temperature gradient is the temperature difference divided by the radius, and the ostrich egg has a radius three times greater than the chicken egg so the thermal gradient is factor of three lower.

exactly
the time is 9X
that last term is the funkiest one imho

7:14 AM
@Gyromagnetic or in general the time is proportional to $r^2$

@JohnRennie I have a question

@Akash.B morning :-)

@skullpetrol , "when am I ever going to use this?" :-)... next time you boil an ostrich egg, ofc! haha :)

Morning

@skullpetrol also good conversation at dinner parties hahahaha

7:16 AM
@JohnRennie earth is a giant magnet right?

true dat @Gyromagnetic

mOrNiNg

@Akash.B among other things

So if I move a iron piece randomly

@Akash.B yes. The outer core is liquid iron and the flows in it cause the magnetic field.

7:20 AM
it should produce electricity
@JohnRennie right?

well, earths magnetic field is quite homogeneous
and to produce current you need d flux/dt
but in principle yea

If a conductor cuts a magnetic field it should produce electricity
@Gyromagnetic change in flux?

true, if you rotate a loop in earth constant field, you would get an AC current

@Akash.B this is exactly how an electric generator works.

but if you just move it around you wouldnt

7:26 AM
When you rotate the generator you are, in effect, waving a conductor (the wire) around.

yeaaaaah, but you get what im saying, you can also wave wire around and get nothing
but i think thats beside the point atm

Let @Akash.B finish asking his question @Gyromagnetic :-)

@Akash.B yeah sorry :(

7:49 AM
@skullpetrol YO WHAT

yup, looks like a vicious rumour

here i was wondering how i'd continue learning math

stick with what works for you

@skullpetrol i've been kidding about learning from there :P
the site's for elementary school math

:-)
Happy World Emoji Day!

8:13 AM
morning

good morning

mornin'

the heat death of the universe is poorly named
it doesn't die of heat at all
quite the opposite

it also can't die

World Emoji Day?
Good grief, that actually exists.
@Slereah it's the death of heat not the death of the universe
The universe and Cthulhu never die.
The difference between them is that Cthulhu is real.

8:25 AM
@JohnRennie hey, I still have a question on springs. When we cut a spring in $n$ parts, the force will still be the same since $\Delta l/\ell_0$ won't vary, but won't the stiffness increase by a factor of $n$? So shouldn't the force also increase by a factor of $n$?

suppose you cut a spring into $n$ pieces, then two things happen:
1. the stiffness of each subspring increases by a factor of $n$
2. the extension of each subspring deceases by a factor of $n$

but also $\ell_0$ is now $\ell_0/n$ which cancels the $n$ in $\Delta l/n$
I'm using $F=k\Delta l/\ell_0$

Suppose I take a 1m spring and I stretch it to 1.1m so the extension is 0.1m and $\Delta \ell/\ell_0 = 0.1$

if you cut it in $n$ parts you'll have $F_n=k_n\Delta l/\ell_0$ assuming the equal forces you have $k\Delta l/\ell_0=k_n\Delta l/\ell_0$

Now I cut the spring in half to give two springs with an original length of 0.5m that are stretched to 0.55m i.e. the extension is 0.05m

8:32 AM
wait so it appears $k_n=k$?

So for the two springs $\Delta \ell/\ell_0 = 0.05/0.5 = 0.1/1$ i.e. $\Delta\ell/\ell_0$ hasn't changed.

yes I am cool with that already
what have I done wrong in the last equation?

You need to be absolutely clear what constant you are taking about. When we talk about the force constant of a spring we usually mean the constant $k$ in $F = kx$. This constant $k$ depends on the original length of the spring so it changes if we cut the spring in half.

@JohnRennie But
strange eons
Also I don't actually care for Cthulhu

If you write the equation as $F = K x/\ell_0$ then this constant $K$ is independent of the length of the spring and it does not change if you cut the spring in half.

8:36 AM
I'm all about the Great Race of the Yith
look at those dorks

@Luyw I suspect what you are doing is mixing up the two constants when you say:
> won't the stiffness increase by a factor of $n$?
The constant $k$ will change but the constant $K$ will not.

But $F=(\frac{K}{\ell_0})\Delta l$ cut spring in $n$ parts, $F_n=(n\frac{K}{\ell_0})\frac{\Delta l}{n}$
oh yeah

Yes :-) The $n$s cancel!

yes and writing the new force as $F=k\Delta/n$, we consider $k$ as being $n$ times more than that before cutting
ok
thank you thank you

:-)

9:42 AM
How does one prove that a ball falls down using string theory

9:56 AM
I thought string theory isn't taken seriously anymore, it is not the case?

@Luyw I suspect Sam was joking :-)
He knows perfectly well how to derive general relativity as the low energy limit of string theory.
The fact that string theory gives us GR for free is one of the reasons it became so popular.
Although if I understand correctly any theory that includes supersymmetry as a gauge symmetry inevitably reproduces GR so it's less of an amazing achievement than you might initially think.

Yes yes, totally understanding you

10:23 AM
publications of string theory look the most inscrutable to me
students in my previous lab said string theory is more harmonious to people with background in quantum field theory while loop quantum gravity is more harmonious to people with background in general relativity

string theory is a swamp... that must be drained

10:41 AM
it's like some research groups designate their research area as string theory but don't actually work on string theory or do not only work on string theory; they work on various theories of gravity and quantum fields, like supergravity, higher derivative gravity, supersymmmetry gauge field theories, higher spin theories, bigravity, etc.; the reason they use string theory to designate their research area probably because these theories they are involved with can be derived from
string theory at certain limits.

11:26 AM
@JohnRennie you may be overestimating my power
@CaptainBohemian maybe putting string theory in the names gets you more grant money

Really. The string spectrum includes the states of a massless spin 2 QFT, and a spin 2 QFT produces a force like gravity. Perhaps neither you nor I understand all the details (I certainly don't) but the general principle is straightforward.

Well I know it in the broad sense
Knowing how to show it
That's a different matter
Also spin 2 stuff doesn't necessarily produce GR
You can add arbitrarily many terms to the Lagrangian
And indeed must for renormalization purpose
Although I'm not sure if adding terms in a non-trivial way produce proper QFTs
Since it makes the stress energy tensor not locally conserved in a classical theory
Not sure what that would do for a QFT
Unstable vacuum???

I recall reading a paper discussing whether a massless spin 2 QFT was equivalent to GR, and the conclusion was that unless you are being pedantic yes it is.

Slim chances of me not being pedantic

12:00 PM
One thing I'd like to do on my site (one of the many things, of course) is just doing solutions of simple problems in various theories
like free particles in Newtonian mechanics/Hamiltonian/Lagrangian/relativistic/GR/QM/QFT
etc
And the same for free fall, two particles in orbit, etc etc

@JohnRennie so if I just keep on moving an iron piece randomly in atmosphere it should produce an electric current right?

Although rly, free particles in string theory is quite awful already
bloody boundary conditions

1:03 PM
@Akash.B You'll get some current, but the Earth's magnetic field is rather weak, so you need a long piece of metal to get useful current. And you shouldn't move the metal randomly, you should move it in a path that maximizes the crossings of the magnetic field lines. This is impractical on the Earth's surface, but it's feasible to do it in orbit:
Electrodynamic tethers (EDTs) are long conducting wires, such as one deployed from a tether satellite, which can operate on electromagnetic principles as generators, by converting their kinetic energy to electrical energy, or as motors, converting electrical energy to kinetic energy. Electric potential is generated across a conductive tether by its motion through a planet's magnetic field. A number of missions have demonstrated electrodynamic tethers in space, most notably the TSS-1, TSS-1R, and Plasma Motor Generator (PMG) experiments. == Tether propulsion == As part of a tether propul...

1:19 PM
@Slereah yo
have you ever seen the normalization $G=(n-1)\omega_{n-1}T$
for $n+1$ dimensions

What
What is even $\omega$
volume form?

area of $S^{n-1}$ sphere...

oh
I see
I have not but I guess it's supposed to give you the proper Newtonian formula in the classical limit that way?

it's probablt there to normalize flux somehow

yeah
Usually factors of the EFE are either nothing or just SI units
or "some factor"
not a whole of in between

1:23 PM
this is the most complicated factor I've seen
@Slereah hmmm
there's an explanation
you know what the constant in the ADM mass should be so that it gives the right result for Schwarzschild
but since that's vacuum, you don't know what the constant in the EFE should be
but you want the ADM mass to be the integral of the energy density
so pretending the ADM mass integrand is the scalar curvature of the Cauchy hypersurface, that fixes the constant in the EFE
because $R"="\mu$

Ah yes

there's a factor of 1/2 there

I tried to prove Newton's formula from Schwarzschild once
without ever assuming Newton at all
it was a tricky thing

so the ADM mass is $$\frac{1}{2(n-1)\omega_{n-1}}\int (g_{ij,j}-g_{ii,j})n^j$$

Hard to keep track of where all the $c$'s should go

1:33 PM
and the integrand is $R$ modulo lower order terms
uhhh that's not true sorry
you integrate by parts first
then it's an integral of $R$ over the bulk modulo lower order terms
so you want $$\frac{1}{c_n}\int R=\int\mu,$$ which fixes the overall constant
where $\mu=T(\nu,\nu)$

$\nu$ timelike I s'ppose

its the normal to the Cauchy hypersurface

sounds timelike enough
you know I wonder if there's like
A dumb paper on string theory
like just something practical
like "scattering cross section of strings"

2:14 PM
it's like a lot of recruitment systems use their native languages, so if you want to read their information, you need to invoke google translate.

3:06 PM
2

3:18 PM
This was the moment of moments regarding JP

Zizek & JBP was good entertainment

Apparently it's still an open question whether LQG even has GR as it's classical limit, that can't be...

IMHO, Peterson is a good social psychologist

3:36 PM
One of the great stable geniuses of our time, skull.

Socially, yes.

3:58 PM
no JBP posting guys, we might have to lock you up in the hbar-gulag

why

just kiddin, its fine ;) <3

It's like, no, it's not.

some ppl in my work group have very strong feelings about him hahaha

It's not.

4:09 PM

Postmodern neo-Marxists, to be precise. And you have to be precise about that, bucko
Otherwise you will end up in deep, deep, trouble. And that's no joke.

cant tell if Balarka is serious

im doing a jbp impression

hahaha a good one also

that's step one

4:12 PM
thats the belly of the whale
roughly speaking

has JBP done an analysis of toy story 4

wut? rly hahahaha

its a question
molyneux probably has
he has actually gone full cult leader in the last few years

@RyanUnger has to

He always was with his de-fooing

4:17 PM
i wonder how things will play out... i have the feeling everything is very volatile atm tbh

@Slereah tricky? It's straightforward isn't it?

don't you get it from writing out the radial geodesics in coordinates or something

The geodesic equation gives you:
$${d^2 r \over d\tau^2} = - \frac{GM}{c^2r^2}\left(1 - \frac{2GM}{c^2r}\right) u^t u^t$$

where is equation (1)

Approximate $d\tau = dt$, $u^t = c$ and $2GM/c^2r \ll 1$ and the result drops out.

4:24 PM
$$\text{Here} \tag{1}$$

@RyanUnger oops, copy paste error :-)

$$\text{Pre-Here} \tag{0}$$

4:36 PM
@JohnRennie not terribly hard but I tend to lose some factors of $c$ along the way
Doing it in natural unit's a breeze tho :p

2 hours later…
7:03 PM
0

Referring to the previous question Can we report moderators for abuse of power? I ask, kindly, il link of the tool to report abuse of power by some moderators. Thank you.

7:27 PM
@ACuriousMind Question for you (and possibly follow-ups based on the answer) -- are you familiar with how faculty positions work in Belgium? (Is it similar to the German system?)

I have no idea how it works in Belgium, I'm afraid (and I wouldn't be sure about Germany, either, to be honest :P)

Heh, okay.
There's a faculty position open that intrigues me... but, I know how the US system works in terms of teaching/research loads, funding, tenure, etc..
But I have no idea how the various countries in Europe operate

Tenure definitely varies markedly between Germany and the rest
Funding variations are more determined by (sub)field than country, I think

I guess that's true in the US also -- the way aerospace engineering faculty need to work for money is different than math or philosophy or whatever else.

Since I didn't end up as a lecturer, I have no idea about teaching loads at all. Some lecturers taught much more than others, but I don't know whether that had anything to do with requirements or just personal preference

4 hours later…
11:32 PM
Hi, John good night. I was having some trouble to access the site yestoday night, I am in Asia and then went to bed, Sorry