The h Bar

Jake Rose

12:55 AM

Anybody give me a hand with this? Not exactly sure where the R-r comes from

Physics Meta

1:26 AM

1

Q: Is this question in the correct forum?

I have doubts about this question, because it is a question about a software of physics but I don't know if the question is in the correct forum.

support

John Rennie

5:44 AM

@JakeRose you've probably sorted this, but just in case: the centre of the small cylinder moves in a circle of radius $R-r$ so it's velocity is $(R-r)\dot\theta$

Since the outer rim of the small cylinder is stationary, the velocity of that edge relative to the centre is $-(R-r)\dot\theta$, and that velocity is $r\omega$.

That gives us $r\omega = -(R-r)\dot\theta$

Dammit, I've misspelled it's and now it's too late to edit.

user400188

6:01 AM

I've got a curious question. Did anyone ever resolved the apparent contradiction of Poincaré's recurrence theorem, and the second law of thermodynamics? (I say apparent, because I'm not sure if they do contradict another.)

I am aware that the recurrence theorem only holds for systems with closed state spaces. But it appears to me that an isolated system, (for which the second law holds) would behave very much like one with a closed phase space, if the particles within it all obey Newtons laws.

John Rennie

6:16 AM

@user400188 there is no contradiction. The second law is a statement of probability. It says the most probable behaviour of the system is to change in the direction of increasing entropy.

It isn't usually phrased like that because the probability that the entropy will spontaneously decrease is so ridiculously small as to be completely negligible in everyday life.

Likewise, and macroscopic system has a recurrence time so absurdly long that the probability we will see a recurrence is negligible.

user400188

But don't we assume the second law of thermodynamics in many different theories, and discard those that contradict it?

It would appear to me that we are discarding theories due to a probabilistic argument, while having proved there are cases where the argument doesn't hold.

John Rennie

@user400188 I'm not sure it's fair to say we discard theories because they contradict the second law.

If a theory contradicts the second law it will have more fundamental problems, and we'd discard it on the basis of those more fundamental issues. Contradicting the second law is just a symptom of deeper problems.

user400188

6:34 AM

That sounds more reasonable.
I got the impression of the former due to a particular Wikipedia quote, originally by Sir Arthur Stanley Eddington:
"The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations – then so much the worse for Maxwell's equations. If it is found to be contradicted by observation – well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics I can …

John Rennie

6:46 AM

@user400188 That's hyperbole. He's saying it to make a point.

But the reality is that any theory that does contradict the second law almost certainly has a flaw.

user400188

That clarifies things for me. Thank you @JohnRennie

bolbteppa

8:19 AM

John Rennie

She's an anti-toddler in the Dirac swimming pool

Slereah

Square root is specifically positive roots you swine

@user400188 by the way the Poincaré recurrence theorem doesn't apply in our universe

Thanks to a theorem of Tipler

Jake Rose

11:23 AM

@JohnRennie could you explain why the elocutions of the edge is minus the. Elocutions of the CoM?

John Rennie

@JakeRose What are elocutions?

Jake Rose

Wtf

i. Dang. Elocity

wtf

one sec my keyboard is wet

i meant velocity *

@JohnRennie also I think you have a sign error

the overall answer should be positive

John Rennie

Consider a cylinder rolling along a flat plane at some linear velocity $v$. By this we mean the centre of mass of the cylinder is moving at velocity $v$. Is this setup clear or shall I draw a diagram?

Jake Rose

Yeah it’s clear

John Rennie

If the cylinder is rolling without slipping then the bottom point of the cylinder must be (instantaneously) stationary because it's in contact with the stationary floor.

That means the difference in velocity between the centre of the cylinder and its perimeter must be $v$.

Jake Rose

11:33 AM

Mhmm is there another way to think about that?

John Rennie

@JakeRose this seems the simplest way. I'm not sure what other way you'd look at it.

Jake Rose

Mhm not sure I quite get it @JohnRennie

John Rennie

Shall I draw a diagram?

Jake Rose

What is wrong with the way I do it? I equate the distance rolled by the small cylinder to the arc linger of the big cylinder?

yes please

John Rennie

@JakeRose That's the setup. The bottom point is (instantaneously) stationary because it's in contact with the floor. That means the cylinder is instantaneously rotating around the bottom point. OK so far?

Jake Rose

11:44 AM

Yes

John Rennie

Now suppose we are in a car trundling along the floor at the same speed $v$ as the cylinder. Looking from the car all velocities get $v$ subtracted from them so for example the velocity of the centre is $v - v = 0$ i.e. it is stationary with respect to us.

@JakeRose This is what the motion looks like from the car.

Jake Rose

Yes

John Rennie

@JakeRose So relative to the centre the circumference is moving at a speed $v$. I'm using speed not velocity because of course the direction changes depending on where on the circumference you are.

Jake Rose

Okay I think I get it

@JohnRennie can you give me a hint to a quesiton

John Rennie

12:00 PM

@JakeRose yes, though I'm just answering a question in a another room so I'll be a minute or two

Jake Rose

I’m trying to think of the forces on an element

gravity, centrifugal (if I’m the rotating frame) and that’s about it

but not sure how equilibrium comes about

@JohnRennie cool thanks

John Rennie

12:14 PM

@JakeRose as I recall the way you do this is by equating pressures.

Jake Rose

What do you mean?

John Rennie

Do you know the formula for the pressure at some depth $x$ in a fluid with density $\rho$?

Jake Rose

Is it just $\rho x g$?

John Rennie

Exactly, yes. Where $g$ is the gravitational acceleration.

Now, in circular motion we get a centripetal acceleration $a = r\omega^2$.

Jake Rose

Yes

John Rennie

12:18 PM

So at a distance $r$ from the centre of the rotating vessel you get a pressure due to the liquid in between you and the centre being accelerated.

Emilio Pisanty

@JakeRose Why is your profile pic bugging out?

Jake Rose

@EmilioPisanty nit sure, works fine for me

Emilio Pisanty

ah, got it, it has a tracking script and my client is blocking it

:48625763 no, it's probably Privacy Badger doing its job

Jake Rose

@JohnRennie I’m still not quite sure how to obtain the result :(

Emilio Pisanty

(and also SE being lazy about protecting privacy when it comes to profile pictures)

John Rennie

12:22 PM

Can you see that the centripetal acceleration causes a pressure that depends on the distance from the axis, just like gravitational acceleration depends on the depth of the fluid? Actually calculating the "centripetal pressure" needs an integral so we'll get back to that, but can you see the basic principle?

Jake Rose

it’s as if you just turned it sideways (ignoring gravity)

I think so

Emilio Pisanty

btw @JohnRennie check it out

physics.stackexchange.com/help/badges/1219/…

John Rennie

@JakeRose Cool. What happens is that this pressure pushes the fluid surface up, and it pushes the water up until the centripetal pressure is matched by the gravitational pressure. If we call the cetripetal pressure $P_c$ then we'll get an equation $P_c = \rho g z$.

@EmilioPisanty cool :-)

I wonder where I am with that badge ...

Jake Rose

Isn’t the surface horizontal?

@JohnRennie

Emilio Pisanty

@JohnRennie you can see the progress on the badge tracker in your profile

@JohnRennie There, have a silver tag badge.

John Rennie

12:27 PM

@JakeRose no, when you spin a vessel of liquid the surface curves into a parabola.

@JakeRose actually wait - there's an easier way to do this.

Let me draw a diagram ...

Emilio Pisanty

@JohnRennie is diagra similar to viagra?

Jake Rose

How would the centripetal pressure push upwards? Shouldn’t it push sideways?

John Rennie

@JakeRose I've drawn a cross section through the spinning fluid. The axis of rotation is the vertical line at the left, and the surface curves upwards as we move away from the axis. OK so far?

Jake Rose

Yep

is the resultant for e perpendicular to the surface? If so how did you know?

isuppose if it wasn’t it wouldn’t be perpendicular

John Rennie

@JakeRose because the surface is a fluid. If the resultant was not normal to the surface that would mean there would be a non-zero sideways force on the fluid, and in that case the fluid would flow sideways.

The fluid can't flow down in the direction perpendicular to the surface because there is other fluid in the way. But it could flow sideways.

@JakeRose convinced?

Jake Rose

12:39 PM

Ahhh I see how you get it now

you can just get the rate of change from the tan of the angle

John Rennie

BOOM! :-)

You will get the same result using the pressure, but the calculation is less obvious so this way is better.

I have to go now. Back later today or failing that tomorrow.

Slereah

1:18 PM

Tom Lehrer - Who's Next - oldest Wikipedia version

►

Oh Tom Lehrer

Emilio Pisanty

A song of patriotic predjudice - Flanders and Swann

►

The Gas Man Cometh - Flanders and Swann

►

MartianCactus

1:41 PM

hey guys I have a question

how can emf be induced in a coil?

Slereah

The laws of electromagnetism

MartianCactus

like, how can there be a potential difference?

if its a coil potential difference should always be zero?

so how does faraday's law work?

Slereah

I mean i'm not sure how to answer that question

Faraday's law is a fundamental law of electromagnetism

it works because we observe it to work

MartianCactus

but isn't a potential difference obserced between 2 specific points?

how can you observe that in a coil?

John Rennie

1:56 PM

@MartianCactus a potential can only be defined that way if the potential is due to a conservative field.

Static electric fields are conservative, so for a static electric field the electrical potential is a well defined quantity.

However magnetic fields are not conservative, and as a result there isn't a simple potential assiated with them.

MartianCactus

ohh so whats the electric field in the coil? Varying? hmm..yeah probably

John Rennie

So a changing magnetic field can produce a potential that changes smoothly around a coil even though moving round a coil brings you can to your startig point.

MartianCactus

so it isn't conservative. Then how is potential difference defined in a varying electric field?

John Rennie

With care :-)

MartianCactus

hmm?

John Rennie

2:00 PM

The potential difference between two points is still the work needed to move a unit charge between those points, but the problem is that the work now depends on the path you take between the points. So you cannot define a single potential difference between the points.

MartianCactus

oh yeah

so by constant field, you mean constant with time right?

and not with distance

John Rennie

Yes. I mean an electric field that doesn't change with time.

MartianCactus

ohh

John Rennie

But I was just using this as a simple example. Electromagnetism gets very complicated very quickly once you have time dependent electric and magnetic fields. I must admit I kind of bluffed my way through it at college so my mastery of it is somewhat shaky.

MartianCactus

ohh

ZeroTheHero

2:07 PM

@kartikc.p maybe I can chime in on this. This is quite a good book but it's not very comprehensive: it runs maybe 100 pages but the pages are small. As a digest to accompany another textbook it's fantastic, but as standalone it's not so good to study from, especially if you need to or like to work through examples. For some weird reason it is typeset using something like the MSWord equation editor, which gives it a strange feel.

The more highly unusual aspect are the podcasts that one can play, available from the author's webiste: danfleisch.com/maxwell

Abcd

2:29 PM

@JohnRennie its a file from my mobile that i transferred to laptop. How to make the page of proper size/resolution?

John Rennie

@Abcd is that in Word?

Abcd

@JohnRennie yes

John Rennie

Go to the View menu and there are commands there to zoom it to full size.

Abcd

@JohnRennie Done thanks.

John Rennie

Not just physics, but IT support too :-)

Slereah

2:42 PM

New York, New York, it's a hell of a town

The Bronx is up, but the Battery's down

The mimes are food for the bums underground

New York, New Yoooork

Mo_

2:53 PM

Cambridge, Cambriiiiiiidge, it's a hell of a university

@JohnR did you get your bachelor's degree at Cambridge too?

John Rennie

@Mo_ yes. I did my degree at Peterhouse and the PhD at Jesus.

Mo_

@JohnRennie Jesus Christ

I'd love to get my PhD (or at least do a postdoc) at Cambridge

John Rennie

It was a lot easier to get into Cambridge in the 70s. It's ridiculously competitive these days.

Mo_

From what I usually hear, life should have been way easier back then

Anonymous

3:22 PM

@Mo_ Eeeh, no internet and no laptops! :P

Slereah

@Blue you could get a laptop!

just need a big lap

Anonymous

Loool :D

Slereah

although that was more the 80's

GodotMisogi

@JohnRennie Did you have some kind of entrance exams like STEP back then?

Slereah

GodotMisogi

3:24 PM

Oh wait, that's for undergrad, not for Ph.Ds.

Slereah

Apparently this is what passed for a laptop in the 70's

you're gonna crush your legs!

John Rennie

@GodotMisogi in general the UK universities didn't have entrance exams. They just required you to get certain grades or better in your A level exams. However Oxford and Cambridge ran their own exams. I had to sit the Cambridge exam to get into Cambridge.

GodotMisogi

@Slereah Crush 'em or burn 'em? I'd be more worried about heating issues

Anonymous

@Mo_ Imagine all those PhD students writing their theses by hand! It's a nightmare. (I think typesetting systems became mainstream by late 80's, not 70's?)

Slereah

@Blue 70's was more the era of typewriter thesis I think

Which was also very bad

GodotMisogi

3:26 PM

@JohnRennie Ah, so it was similar back then too. STEP is the undergraduate entrance exam required for entry into Cambridge, and I think Southampton adopted it as well (with lower requirements, of course)

John Rennie

@GodotMisogi Ah, I didn't realise there was now a more widely accepted entrance exam. It makes sense, and it has to be cheaper to administer.

Though a quick Google tells me STEP is only for mathematics.

GodotMisogi

@JohnRennie Right, I forgot it's only for that. My bad

John Rennie

@Blue my PhD was done using a mainframe app called GCAL

GodotMisogi

That just reminded me of INTERCAL. What a gentlemanly language

Mo_

@Blue I've seen typesetted PhD dissertations from the 60s

It should feel really special to do your PhD at an 810 years old university

Anonymous

3:39 PM

@JohnRennie Oh, nice! I remember you mentioning it once. :D

Anonymous

Typesetting in1985-86 is still believable. :P

Anonymous

@Mo_ Really? :O

Terry Bollinger

Hi all! Just curious, is the h bar still going on, or is that extinct? I got an odd message that it was supposed to have begun at 10 a.m. eastern time this morning, which seems wrong.

GodotMisogi

Isn't the discussion beginning in 20 minutes?

Anonymous

@TerryBollinger We don't have any special agenda for the chat sessions these days. But yeah, it's still going on. ;)

Anonymous

3:42 PM

Today's session starts in 17 minutes.

Mo_

@Blue apps.dtic.mil/dtic/tr/fulltext/u2/424413.pdf

Terry Bollinger

What is the time? The message I got came out at 8 a.m. and said it started in 2 hours. Does it start at eastern 10, or 11, or 12? Aiee!

Anonymous

@TerryBollinger 4pm UTC

John Rennie

@Blue this is a page out of my thesis:

Not too shabby for 1986 :-)

Mo_

Looks like LaTex

Any equations?

John Rennie

3:46 PM

TeX was in its very early days in 1986. The main problem with it was that the printers didn't exist. Early pixel based printers were very slow and expensive.

Anonymous

@JohnRennie Oh, wow. That's reasonably nice! Even by TeX standards.

Terry Bollinger

@Blue thanks that's what I thought, but the official message with 8+2 eastern time was very confusing.

Knuth... Amazing fellow...

John Rennie

@Mo_ this:

Anonymous

@Mo_ Whoa. I think, they used some primitive typewriter for that though.

Mo_

@JohnRennie apparently GCAL was as good as TeX

John Rennie

3:53 PM

@Mo_ I think the equations were done in a separate system, though I forget what that was. Then they were copied and pasted. Literally! You cut out the equations, stuck them onto the page of text and photocopied the result to make it look seamless.

2

GodotMisogi

@JohnRennie : s no $\cos, \sin$?

John Rennie

@GodotMisogi cos and sin weren't invented back then.

Terry Bollinger

@JohnRennie apologies for a different topic, but have you ever encountered a formulation of general relativity that uses Wigner space, that is, that includes momentum space as an equal partner with XYZ space?

Anonymous

@JohnRennie Lol. :P So at least some things have improved.

Anonymous

@TerryBollinger Hehe, we haven't seen you here in a while. ;)

GodotMisogi

3:56 PM

@JohnRennie I wonder how people managed without trigonometry before that

John Rennie

@GodotMisogi it was a lot easier when pi was equal to 3.

Mo_

interview starting in 3 minutes

GodotMisogi

The most unexpected answer to a counting puzzle

►

John Rennie

@TerryBollinger possibly in quantum gravity. I've never seen any attempt to formulate GR in momentum space.

Terry Bollinger

I'm not sure if I believe this exponentiation stuff even now. When I was a kid, a 4 banger calculator was all we needed!

John Rennie

3:58 PM

A calculator? You were lucky! We had to use an abacus.

Slereah

@TerryBollinger ADM formulation?

Terry Bollinger

There was an Abbey in my class, and she did like to cuss...

Slereah

ADM formulation uses both the (spatial) metric and its momentum

Terry Bollinger

@Slereah, ADM? I'll look it up...

Slereah

it's the basis for all the canonical formulations of quantum gravity

like canonical quantum gravity or loop quantum gravity

Terry Bollinger

4:01 PM

't Hooft's recent sequence of papers on a momentum space interpretation of particle in fall outfall in a black hole has piqued my interest in unified representations.

@Slereah, 't Hooft seems to be using momentum more at the level of band spaces in solid-state physics. Particles are converted into momentum waves that emerged in the other hemisphere of the black hole. I've been trying to convince him that you don't really need a singularity if you go that route, since the to become interconvertible and therefore redundant.

't Hooft is mostly trying to get rid of the firewall problem

John Rennie

Two attempts at spelling hierarchy - neither correct

Terry Bollinger

@JohnRennie so that's what they mean by the hierarchy problem in particle physics...

John Rennie

:-)

Terry Bollinger

@Slereah, thanks for the ref. See you all later, I'm busy babysitting my unbelievably delightful two-year-old granddaughter. We're watching Elmo in Grouchland!

vzn

@TerryBollinger hi are you in email communication with 't hooft? have cited him around here quite a bit but he doesnt seem to have a lot of fans in physics :| ... esp admire his CA/ digital-physics like directions, but think his relative isolation after a Nobel is even reminiscent of einsteins...

Anonymous

4:15 PM

@TerryBollinger Good day! :)

Emilio Pisanty

@JohnRennie I thought TeX primarily printed to vector output formats like postscript

I guess even a postscript-capable printer is asking too much for 1986?

John Rennie

@EmilioPisanty bear in mind that in the early 80s daisy wheel printers were at the pinnacle of non-professional printing technology. For Postscript and anything similar you need some form of dot matrix printer. These only started appearing with the first laser printers.

As I recall the first widely available such printer was the Apple laser printer, but it was scarily expensive. The university IT dept bought one, but they were out of reach of home users.

LaserWriter

The LaserWriter is a laser printer with built-in PostScript interpreter sold by Apple Computer, Inc. from 1985 to 1988. It was one of the first laser printers available to the mass market. In combination with WYSIWYG publishing software like PageMaker, that operated on top of the graphical user interface of Macintosh computers, the LaserWriter was a key component at the beginning of the desktop publishing revolution. == History == === Development of laser printing === Laser printing traces its history to efforts by Gary Starkweather at Xerox in 1969, which resulted in a commercial system called...

That was it. Released in 1985.

@vzn 't Hooft has been very candid that he is exploiting his position to work on areas that interest him but are considered fringe. I would guess the consensus is that we all wish him well but it feels a little quixotic.

bolbteppa

4:40 PM

@vzn t'Hooft is a nobel prize winner for results in qft, of course he has fans here haha

enumaris

weeeez

Terry Bollinger

@vzn somewhat to my own surprise, yes, I've received a few e-mails back from 't Hooft on some black hole horizon topics, but mostly I didn't send emails and assume that he reads them, since he clearly read some of my earlier ones.

An interesting trivium: the 't means "het", or in English, "head". His recent hyper-determinism paper to preserve full Einstein spatial locality bothers me a lot though, since the calculation density implications are pretty unbelievable. I think he has good arguments, but comes to the wrong conclusion. Still, I like the flexibility with which 't Hooft approaches some of these issues.

bolbteppa

"In spring 1969 Veltman was joined by a 22-year-old pupil, Gerardus ‘t Hooft, who had expressed the desire to study high-energy physics. After he had written a first, fairly brief dissertation ‘t Hooft was accepted as a doctoral student that autumn. His task was to help in the search for a method of renormalizing non-abelian gauge theories. ‘t Hooft succeeded beyond all expectation and in 1971 published two articles that represented an important breakthrough in the research programme."

Terry Bollinger

John Bell also seems to have at least considered hyper determinism as a possible resolution for locality, interestingly.

"mostly I didn't send emails" --> "mostly I just send emails"

I love the fact that into his most recent paper 't Hooft discards the need for string theory at the event horizon of a black hole.

A general premise which I am trying to write a paper from a computational theory perspective is that physics has overshot on complexity. E.g., the Sandard Model of particle physics does not need the complexity of Planck foam or Planck strings to explain fermions.

Thus, intriguingly, the emission of particles from an event horizon may be much more closely related to why the Standard Model exists than anyone has realized.

Fermions may, for example, be some form of direct resolution solutions for how the momentum-intensive space at the event horizon (see those recent 't Hooft papers) interacts with the flatter XYZ space that we live in.

bolbteppa

5:00 PM

"Fact is, they predicted that supersymmetric particles and/or large additional dimensions of space should become observable at the LHC. According to those predictions, this should have happened already. It did not.

The important thing is now that those demonstrably flawed methods were the only reason to think the LHC should discover something fundamentally new besides the Higgs. With this method of prediction not working, there is now no reason to think that the LHC in its upcoming runs, or a next larger collider, will see anything besides the physics predicted by the already known theorie…

Kind of unbelievable

John Rennie

Please, let's move on.

bolbteppa

This is from a new post from today

Terry Bollinger

I say that in part because there some extremely good captures a fermion properties are possible simply by interfacing two spaces, and representing from fermions as (gigantic, hadron-scsle) strings that cross the spaces and resonate in both.

Emilio Pisanty

@JohnRennie "daisy wheel printers"?

goodness

I'm pretty sure this is the first time I hear of those

John Rennie

@bolbteppa We now have a pretty good idea what Hossenfelder thinks of the current HEP scene. Let's just leave her to it and move on.

Emilio Pisanty

5:03 PM

wouldn't "computer-controlled typewriter" be a better name?

John Rennie

@EmilioPisanty to be fair the quality of the professional daisywheel printers was really, really good. See the snaps of my thesis I posted above.

Terry Bollinger

@EmilioPisanty I've used a daisy wheel printers, you just made me feel very old!

John Rennie

By contrast the early laser printers were 300 dpi at best and looked pretty rough.

Emilio Pisanty

@JohnRennie those formulas were printed by a daisywheel printer?

Terry Bollinger

I don't miss the noise of the local dot matrix printers, or of the giant chain printers.

Emilio Pisanty

5:05 PM

yeah, that's pretty impressive

John Rennie

@EmilioPisanty no, the formula were done on something else, though I forget what.

Emilio Pisanty

@JohnRennie ah

what about the figures?

John Rennie

But all the text was daisywheel.

bolbteppa

@JohnRennie yeah, it's like a bad habit / guilty pleasure reading these things

The best of the best still being motls.blogspot.com/2007/11/exceptionally-simple-theory-of.html

Emilio Pisanty

@TerryBollinger I kinda miss the re-use rolls of dot-matrix paper that I used to draw on when I was a kid

ACuriousMind

5:08 PM

@EmilioPisanty If you think about it, a typewriter is closer to the original conception of "printing with movable types" than any modern printer :P

Emilio Pisanty

meaning: daisywheel printers were long gone by that stage

Terry Bollinger

Anyone here ever use the IBM type balls, or whatever they call them?

John Rennie

I still have a stack of several thousand sheets of line printer paper. I keep it thinking it might come in useful one day :-)

Terry Bollinger

@Emilio, you're clearly a lot younger than I thought. For some reason I had you pegged as being about my age.

ACuriousMind

@JohnRennie In case the apocalypse comes and very specifically eradicates the last few decades of progress?

John Rennie

5:12 PM

@ACuriousMind we used to use it as rough paper for doing calculations on. If you ever ran out you just nipped over to computer room and printed off another few hundred pages :-)

Terry Bollinger

@ACuriousMind er... Zombie apocalypse?

ACuriousMind

...why wouldn't you just have taken the blank paper?

John Rennie

@ACuriousMind the operators got annoyed if you just nicked paper :-)

ACuriousMind

@TerryBollinger Well, if we wrap all the zombies in printing paper we can turn it into a mummy apocalypse, but I'm not sure that's "useful"...

Terry Bollinger

Later all, my granddaughter and I are doing call outs for Dora in a skatepark... much more important than revolutionary physics! :)

John Rennie

5:16 PM