**Ask a monkey a physics question thread**

[OngBonga]

Ok I've found another source and it makes a lot more sense.



You want L, the arc length.

You have C, which is the chord.

edit - I see you know the diameter (and hence radius), which is awesome and makes things simpler.

You'll need to know the angle of the sector that the arc is relative to. With C and R, you can do this by using this formula...



Now you have the angle, it's simple...

We essentially calculate the entire circumference, then multiply it by the angle (angle being a fraction of a whole, for example 90 degrees is 90/360, or 1/4, 0.25, one quarter... a 90 degree angle is one quarter of a circle)

2r*pi * a/360

That was a fun start to the day.

[OngBonga]

The angle is essential because it is what represents the ratio of the arc to the entire circumference. When you're looking for an angle in a circle, you're gonna have to get into inverse triganometry, which is not something I recall doing at school. I've never come across the arcsin function before, though my memory of Pythagorean triganometry from school means there's some basis to make sense of it.

[MadMojoMonkey]

Nice work, ong.

The method of recognizing that the triangle in ong's picture is isosceles is hidden in his explanation.

When you add a line from the center of the circle to the midpoint of c, you create a pair of congruent right triangles.
Now, we have bisected the angle a, so we'll keep that in mind.

We can now write out a sine equation for the angle a/2.
sin(a/2) = {opposite}/{hypoteneus}
sin(a/2) = (c/2)/R
sin(a/2) = c/(2R)
a/2 = arcsin(c/(2R))

a = 2arcsin(c/(2R))

as ong said.

and once you know the angle a, then the arc length is R*angle (where angle is measured in radians).

So the arc length is
R*a = 2R*arcsin(c/(2R))

[OngBonga]

We can now write out a sine equation for the angle a/2.
sin(a/2) = {opposite}/{hypoteneus}
sin(a/2) = (c/2)/R
sin(a/2) = c/(2R)
a/2 = arcsin(c/(2R))

a = 2arcsin(c/(2R))

Ok am I right in thinking that the arcsin is basically the inverse, or reciprocal, of sin?

Also...

So the arc length is
R*a = 2R*arcsin(c/(2R))

Where's pi? It's not c/2r because c here is chord, not circumference.

[Savy]

Ok am I right in thinking that the arcsin is basically the inverse, or reciprocal, of sin?

sin(30) = 1/2

arcsin(1/2) = 30

Multiplicative reciprocal would mean that multiplying them together would give 1 which isn't the case.

[OngBonga]

Actaully maybe pi is hidden in there... or, more accurately, a number which has the same ratio to pi as the chord does to the circumference.

Is that it?

[OngBonga]

The method of recognizing that the triangle in ong's picture is isosceles is hidden in his explanation.

Yeah I recongnise now we're using simple school triganometry (just using inverse functions that I'm unfamiliar with), in order to determine the angle, and hence ratio, of the sector, and hence arc.

My confusion arises when I see a solution to this and I don't see pi explicitly.

[MadMojoMonkey]

Yes, arcsine is the inverse of sine. It's the reflection of sine across the line y = x

arcsin(sin(theta)) = theta
Since sin(theta) is periodic, the inverse is not technically a function (it doesn't pass the single input -> single output test). The output of arcsin is always single valued by cropping it off between -pi/2 and pi/2 radians. This is most easily understood by looking at a graph of arcsin.



sin(arcsin(n)) = n, where -1 <= n <= 1

The limits make sense when you realize that sin(theta) is always a value between -1 and 1.

pi is the ratio of the circumference of a circle to the diameter of said circle.

It's hidden in the angle in this problem.

[Savy]

What you may not realise ong is that he's using radians not degrees.

0 degrees = 0 radians
180 degrees = pi radians
360 degrees = 2pi radians

[OngBonga]

What you may not realise ong is that he's using radians not degrees.

Yeah I had to look up radians, I don't remember learning about that at school either. It's clear now how pi is intricately related to the angle of a sector.

And yeah reciprocal is incorrect, I think I can see from mojo's equation solving what the arcsine is relative to the sine. It's what multiplied is to divide.

[Savy]

Yeah I had to look up radians, I don't remember learning about that at school either. It's clear now how pi is intricately related to the angle of a sector.

And yeah reciprocal is incorrect, I think I can see from mojo's equation solving what the arcsine is relative to the sine. It's what multiplied is to divide.

You won't have done, doesn't get taught till a-level. Is just a way of measuring angles that makes life a bit easier for more maths related things.

Radians are used in math and physics. Degrees and Grads (100 grads = 90 degrees) are more common in engineering.

Grads is essentially a % of vertical-ness scale.

I've never seen someone use gradians in practise.

[OngBonga]

You won't have done, doesn't get taught till a-level. Is just a way of measuring angles that makes life a bit easier for more maths related things.

I can see how it's a useful measure of an angle. It's just I naturally think angles in degrees as naturally as I think words in English.

[OngBonga]

I've never seen someone use gradians in practise.

I would imagine it's useful in engineering and construction.

[MadMojoMonkey]

I've never seen someone use gradians in practise.

I would imagine it's useful in engineering and construction.

I have seen it used in civil engineering when discussing the slope of a roadway. Even then, only among other civil engineers. I very much doubt they would describe - say - a truss bridge using grads.

When I was hiking all over the US, we used grads. It probably had something to do with using a contour map to plot the hiking path, but I'm guessing. If I'm right, that's a pretty strong link to why they use it in roadways.

That's about it, though, for using grads.

The mathematical definition of slope (rise / run) is used in plumbing. I.e. you need a drop of 1/8 inch per foot on a "horizontal" drainage line (more if the drain carries solids).

[MadMojoMonkey]

Radians are used in math and physics. Degrees and Grads (100 grads = 90 degrees) are more common in engineering.

Grads is essentially a % of vertical-ness scale.

[OngBonga]

Grads is essentially a % of vertical-ness scale.

There was an opportunity for you to say perpendicular, and you didn't take it.

I'll use that word wherever possible, it's a great word.

[boost]

Wow, I'm glad I opened this before heading out to work-- I don't completely get the equation, as I stupidly took the minimal math in high school (very little is required). However, I intuitively get how making a triangle with two radii and the chord is what I was missing and how the solution for the arc follows. When I get a chance I'll try to educate myself on sine and arcsin, and I may be back with questions.

Thanks guys.

[OngBonga]

Here's a UK road sign...



That's grads, right?

[MadMojoMonkey]

Here's a UK road sign...

That's grads, right?

Yep. Being used to describe the slope of a path / trail / roadway, too. Seems like the most common usage.

Cadwch is missing a vowel, right?
Ger is a suffix, not a word, but ger with an e-hat... epic. more words spelled with unit vectors should be happening right away.

Millter? Like some illegitimate lovechild between mile and meter?

[OngBonga]

The sign is in English and Welsh.

It's funny, most English people know only one Welsh word... ARAF

[oskar]

I found a video of a flat earther debunking the round earth claims of a young earth creationist. Where is your science now, Monkey? Where is your science now?

https://www.youtube.com/watch?v=PVeGpxnRqjg

[MadMojoMonkey]

I found a video of a flat earther debunking the round earth claims of a young earth creationist. Where is your science now, Monkey? Where is your science now?

https://www.youtube.com/watch?v=PVeGpxnRqjg

Science has been backing up the flat Earth hypothesis ever since Einstein taught us about space contraction.

You go past anything fast enough and that thing you pass is flat.

[OngBonga]

Science has been backing up the flat Earth hypothesis ever since Einstein taught us about space contraction.

You go past anything fast enough and that thing you pass is flat.

But it's not flat, just flatter, right?

[MadMojoMonkey]

But it's not flat, just flatter, right?

I don't think anyone is saying the Earth is a 2D surface. A shovel would disprove that one.

So IDK what you mean. Yes, it is flatter... and you can always go faster and then it would be even flatter.

If there is some finite thickness that meets your criteria of "flat," then there is some speed you could travel past the Earth which it would be flat.

[OngBonga]

I think discriminating against flat earthers should be the same as racism.

Stop mocking flat earthers, you racist bastard.

[oskar]

Ongs mom is so fat, if you move past her at 0.99c there is no observable difference in circumference.

[oskar]

I guess circumference wouldn't change. What's a scientific word for fat? cross section area?

[MadMojoMonkey]

Ongs mom is so fat, if you move past her at 0.99c there is no observable difference in circumference.

I guess circumference wouldn't change. What's a scientific word for fat? cross section area?

The circumference would change. The diameters of the Earth perpendicular to your direction of motion would not change. However, they wouldn't properly be diameters, since the Earth is not a sphere anymore, it's an ellipsoid. The semi-minor axis would be shorter than the 2 (equal) semi-major axes. So that one circumference which is the single biggest diameter and the only one which is a circle and not an ellipse - that one wouldn't change. All the rest of them would be shorter in length.

***
Oh, and it's fat. The scientific word for fat is fat, unless you mean volume, density or mass.

[OngBonga]

perpendicular

atta boy

[OngBonga]

Your mom is so fat, even light thinks she's fat.

That's probably the most concise way of getting that one across.

And I'm totally going to steal that.

[MadMojoMonkey]

Your momma so fat she can be used as a gravitational lens.

Your momma so fat she disrupts GPS.

Your momma so fat she's younger than you due to gravitational time dilation.

[oskar]

So I've spent my entire evening yesterday going through flat earth creationists claims... as you do. One is that when Eratosthenes measured the circumference of the earth by the shadow of two sticks some distance apart, what he really measured was the distance to the sun on a flat plane. That would turn out to only a few thousand kilometers. The fact that it doesn't change in observable size at different times in a day is due to the lens effect of the crystal dome that holds in the atmosphere.
That's a valid complaint!
So in order to show that the earth is round in that way, you first have to establish the distance to the sun. This is something I've wanted to do for a long time and I'm getting on it because it seems like a fun project. So naturally I've ordered a cardboard sextant and now I have to figure out a way to do this without any prior measurements.

[oskar]

You know what would help me a lot is if Eratosthenes way of measuring the circumference of the earth would work as a proof that it's round. There's a slight problem tho. If you measure the angle between the stick that casts no shadow and the stick that casts a shadow, you will get the radius of the earth, but only if you previously assume that it's a sphere. If you assume that it's flat you instead get the distance to the sun.
Can you think of any way of getting around this by just using sticks and a watch?

[OngBonga]

Can you think of any way of getting around this by just using sticks and a watch?

Throw the stick in the air, time how long it takes to fall, explain gravity to flat earther, then strike him on the head with the stick.

Gravity is the flat earther's nightmare, it seems. My understanding of their explanation is that the world is moving through space, and is constantly accelerating, much like going 0-60 in a car pins you back against the seat. But, of course, when you reach 60, and then cruise at that speed, you are no longer pinned back by acceleration. One must continue to accelerate to feel this force. And, of course, if one is constantly accelerating, then one will eventually reach light speed.

So why is the Earth constantly accelerating, where is the source of this acceleration, and why are we not seeing the rest of the universe become flatter and flatter as our relative velocity increases?

Also, they like to quote the bible, and the bible says that the Earth shall not be moved, or words to that effect. They use this biblical source to "prove" that the earth can't be rotating on its axis and orbitting the sun. But of course they're happy for us to go through space at an ever increasing velocity.

You can prove the world is round beyond any reasonable doubt by observing gravity. Once you have accepted this, then you don't need to worry about whether the sticks are telling the earth radius or the earth-sun distance, because it's obvious which one it is.

If you still insist on proving that the world is round in this way, well get two people to do the sticks experiment at the same time at different places in the world (with both at the same latitude at sea level). If your measurements are (nearly) identical, then you're measuring the earth radius. If your measurements differ by more than a very small amount, then you're measuring the earth-sun distance, which will be different from point A and point B on a flat model in which the sun is only thousands of miles away.

Once you've noticed that the measurements are the same, hit the flat earther with the stick.

[OngBonga]

To be fair, you don't need to have both measurements taken from the same latitude and sea level, you just need to factor these into your results, because the squished earth means that the radius is longer at the equator compared with the poles, while increasing topographic altitude also obviously increases the radius.

So long as you know how much higher above sea level you are from the other measurement, and how much further north/south from the equator, then you can still determine the world is round by taking these two measurements simultaneously.

edit - I wouldn't have a clue how to factor in the north/south aspect. It's not an obvious calculation like the altitude value. So, ensure both are at the same latitude, because otherwise you're getting into some filthy maths.

[OngBonga]

You could further refine this experiment by repeating it when there's a strong enough full moon for the sticks to cast a shadow.

Once you have you results, you have a value which is either a) the earth radius, or b) the earth-moon distance. Now compare them to your earth-sun results. Are they the same? They will be if you did the experiment properly. So, now we know either a) we measure the earth's radius, or b) the moon and sun are the same distance away.

Now, wait for a total solar eclipse, and note the sun and moon do not collide.

[OngBonga]

I think I'm finding a flaw with my simultaneous experiment method.

We need one of the sticks to not cast a shadow, right? Therefore, we need to do the experiment at midday between the tropics. The sun needs to be directly overhead. If that has to be the case, then we can't do it simultaneously across the same latitude, because it won't be midday in both locations, the sun won't be directly overhead in both locations.

The moon method works though. Wait until the moon is directly overhead and bright enough to cast shadows. We can still determine that either we're measuring the radius of a spherical earth, or that the sun and moon are the same distance away from us, and we know it's not the latter because of solar eclipses.

[OngBonga]

Furthermore, if the flat earthers counter by suggesting that the moon and sun are very close to one another but not on a direct collision course, fire radio waves at both the moon and sun, and note that only the moon is reflecting the radio waves back in any measureable manner. Note also that the time it took for the radio waves to go to the moon and back does not correspond with what we'd expect if the measurements we have supported flat earth theory, proving we have the earth's radius, not the sun/moon distance.

[OngBonga]

I want to argue this shit with a flat earther.

I'm going to youtube.

[MadMojoMonkey]

I'm not even sure if I should weigh in on this one.

[a500lbgorilla]

oooo Live and with a chill spacey soundtrack

[a500lbgorilla]

How do antenna work?

[MadMojoMonkey]

How do antenna work?

There are active and passive antennae. Active antennae are transmitters, passive antennae are receivers.

They are mostly the same thing, a long conductive wire attached to a circuit.

The circuit in a transmitter takes some input signal and amplifies it into a high-voltage AC signal. It outputs that signal to the long wire (usually in a vertical position, going up a tower). This signal pushes the electrons up and down the wire. Moving charges are electric currents. Changing currents create changing electromagnetic fields. The transmitter emits photons in the frequency that the AC signal is driving the wire. These are, unsurprisingly, called radio waves.

The range of the electromagnetic spectrum which is lower than visible photons is the radio wave band.

Play this backward in time and you get the receiver.

There is a long wire sticking up vertically, just waiting for a radio wave to be incident on it. Those waves are the changing electromagnetic fields, and those changing fields create changing currents in a conductor. The charges in the antenna are then moved up and down the wire, which the connecting circuit then inputs.

[OngBonga]

mojo posting fake NASA propaganda, obviously.

[OngBonga]

Where are the stars?

[a500lbgorilla]

Where are the stars?

If the universe is infinite, how come the night sky isn't white with the light of stars?

Infinite space says infinite stars says infinite light, yet it's dark at night. What's up with that?

[OngBonga]

If the universe is infinite, how come the night sky isn't white with the light of stars?

Infinite space says infinite stars says infinite light, yet it's dark at night. What's up with that?

Well done for proving that the universe isn't infinite in size and time.

[oskar]

Dark matter.
It's dark.
Next question.

[oskar]

https://en.wikipedia.org/wiki/Weakly...sive_particles

[a500lbgorilla]

https://en.wikipedia.org/wiki/Weakly...sive_particles

lol wimps

[OngBonga]

Even if we assume the universe is infinite in space, it's 14 billion years since the big bang or whatever. That's not long enough for an infinite amount of stars to form. We need an infinite amount of time. I don't think the universe is infinitely old yet.

[OngBonga]

Also, if there's an infinite numbers of stars being born, there's also an infinite amount of stars dying.

Infinity minus infinity equals zero.

So if the universe was infinite in time and space, there would be zero stars, not an infinite amount.

[OngBonga]

Oh there could be like 25 quadrillion chunks of space rock for every star out there, which would mean lots of shit in the way blocking all that infinite starlight.

[OngBonga]

And if you're still not satisfied, then dark matter.

[MadMojoMonkey]

It's called dark 'cause dark matter doesn't interact with light.

If it blocked light, then that would be an interaction.

Dark matter is invisible, not opaque.

[oskar]

It's called dark 'cause dark matter doesn't interact with light.

If it blocked light, then that would be an interaction.

Dark matter is invisible, not opaque.

Whoops. My mistake. Or really their mistake for calling it dark. supid astrophysicist.

[OngBonga]

It's just theoretical stuff, it can do whatever I want it to do, because you can't prove it doesn't.

So, dark matter.

[Savy]

It's just theoretical stuff, it can do whatever I want it to do, because you can't prove it doesn't.

So, dark matter.

Quite literally can.

[OngBonga]

Quite literally can.

Fine.

Prove I can't eat dark matter.

[MadMojoMonkey]

Fine.

Prove I can't eat dark matter.

Dark matter's only known interaction with other matter is through gravitation.

You don't eat via gravitation.

[OngBonga]

Dark matter's only known interaction with other matter is through gravitation.

You don't eat via gravitation.

I thought you knew what "prove" means?

[OngBonga]

This is sort of silly, I mean I wasn't even being serious when I said dark matter.

I'm pretty sure I covered why the night sky isn't ablaze with starlight when I pointed out the age of the universe.

[OngBonga]

If there's an infinite number of monkeys typing randomly on an infinite number of typewriters, won't there be an infinite amount of noise to fuck up the theory that they'll eventually write a Shakespeare play?

I would imagine an infinite amount of noise to be fatal to monkeys.

[a500lbgorilla]

Yup. Infinity is a funny idea, but it's just an idea. 1 monkey's work in infinity is essentially 0 monkey's work.

[OngBonga]

Infinity only exists in theory. Like circles. Circles only exist in theory. Show me a perfect circle.

[NightGizmo]

Infinity only exists in theory. Like circles. Circles only exist in theory. Show me a perfect circle.

Ever used a compass? That's just one of the simple ways to make a perfect circle.

[OngBonga]

Ever used a compass? That's just one of the simple ways to make a perfect circle.

Um... nope. I'll emphasise the word "perfect". A compass circle isn't even going to be close to perfect, I'd be able to find imperfections with a standard microscope.

At least force me to get to molecular level to refute you.

[NightGizmo]

Um... nope. I'll emphasise the word "perfect". A compass circle isn't even going to be close to perfect, I'd be able to find imperfections with a standard microscope.

At least force me to get to molecular level to refute you.

The definition of a circle is "a round plane figure whose boundary (the circumference) consists of points equidistant from a fixed point (the center)". So a "perfect circle" depends on your desired precision. If I need a circle down to the millimeter level, I'm not going to care if it's a few nanometers off.

By your absurd definition, you couldn't ever have a perfect circle because quantum mechanics.

[OngBonga]

Best I can come up with is a wave propagating through a vacuum, but of course there's no such thing as a perfect vacuum, which means the medium the wave is propagating through is not perfectly uniform. Furthermore, gravity will influence the wave's propagation, meaning that not only do you need a uniform medium, you need uniform gravity from every direction.

It's a lot, lot harder than you might think to make a perfect circle. I'm curious if mojo has any suggestions.

[MadMojoMonkey]

I'm curious if mojo has any suggestions.

In your definition, you have defined "perfect" as an unphysical property, so no, I can't prove you wrong.
(unlike the dark matter, which you took me too much at face value when I said we haven't observed it to interact in any way but gravitation. I should have said, we have observed that it does not interact electromagnetically - which includes chemically. It is through electromagnetic interactions that stuff has a surface to be grabbed and manipulated, which is how you move things into your digestive system and then digest it. Still can't eat dark matter in any meaningful way.)

I could take odds with the utility of your definition of perfect circle, as pertains to physics, but not as pertains to mathematics.
I wont do this, as it's stupid. The rigorous mathematical definition is so useful.

Mathematically, a circle is a perfect circle, 'cause that's the only kind of circle. Anything less than perfect is decidedly not a circle, no matter how circular.

The closest thing in physics would be something like the surface of a neutron star or the equator of a black hole's event horizon. The problem here is that you are kinda really talking about smoothness, and if you think circular has a tricky definition, then you'll find "smooth" to be even more fundamentally fun.

Gravitational orbits tend to become more and more circular over time. In general an orbit is elliptical. The measure of how much more of an ellipse it is than a circle is called the eccentricity. The orbit will shed eccentricity over many periods and will tend to circularize over time. This is, as you may have guessed, asymptotic behavior.

[OngBonga]

I think making a perfect circle is exactly as difficult as precisely calculating pi.

[NightGizmo]

I think making a perfect circle is exactly as difficult as precisely calculating pi.

There are multiple mathematical series that can be used to calculate pi without using a circle. Gregory-Leibniz series and Nilakantha series are two examples.

[OngBonga]

There are multiple mathematical series that can be used to calculate pi without using a circle. Gregory-Leibniz series and Nilakantha series are two examples.

Ok write down pi in decimal form, that's how hard it is to draw a perfect circle.

I would say that a perfect circle is a regular geometric polygon in which the diameter/circumference ratio is exactly pi.

A compass circle is not a perfect circle because it's not perfectly regular, just nearly, and as such its ratios will be very close to pi.

By your absurd definition, you couldn't ever have a perfect circle because quantum mechanics.

I've clarified my definition. Is it still absurd?

[NightGizmo]

I've clarified my definition. Is it still absurd?

Yes.

I would say that a perfect circle is a regular geometric polygon in which the diameter/circumference ratio is exactly pi.

Circles are not polygons. Absurdity still in effect.

[OngBonga]

I'll accept a perfect sphere as proof, by the way. A perfect sphere will be capable of projecting a perfect circle... that is, if it were viewed from above, its outline would be a perfect circle. So a perfect sphere is good.

Is the universe a perfect sphere? Maybe, but probably not.