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76: Joule Pay for This! (Energy)
703 Plays1 year ago
Join Sofia Baca and her guests Millicent Oriana from Nerd Forensics and Arianna Lunarosa as they discuss energy.
The sound that you're listening to, the device that you're listening on, and the cells in both the ear you're using to listen and the brain that understands these words have at least one thing in common: they represent the consumption or transference of energy. The same goes for your eyes if you're reading a transcript of this. The waves in the ears are pressure waves, while eyes receive information in the form of radiant energy, but they both are still called "energy". But what is energy? Energy is a scalar quantity measured in dimensions of force times distance, and the role that energy plays depends on the dynamics of the system. So what is the difference between potential and kinetic energy? How can understanding energy simplify problems? And how do we design a roller coaster in frictionless physics land?[Featuring: Sofia Baca; Millicent Oriana, Arianna Lunarosa]
This episode is distributed under a Creative Commons Attribution-ShareAlike 4.0 International License. Full text here: https://creativecommons.org/licenses/by-sa/4.0/
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Transcript
Introduction to Energy Concepts
00:00:00
Speaker
The sound that you're listening to, the device that you're listening on, and the cells in both the ear that you're using to listen and the brain that understands these words have at least one thing in common. They represent the consumption of or transference of energy. The same goes for your eyes if you're reading a transcript of this.
00:00:15
Speaker
The waves in the ears are pressure waves, while eyes receive information in the form of radiant energy, but they are both still called energy. But what is energy? Energy is a scalar quantity measured in dimensions of force times distance, and the role that energy plays depends on the dynamics of the system.
Potential vs Kinetic Energy Discussion
00:00:31
Speaker
So what is the difference between potential and kinetic energy? How can understanding energy simplify problems? And how do we design a roller coaster in frictionless physics land?
Podcast Overview and Hosts Introduction
00:00:41
Speaker
All of this and more on this episode of Breaking Math.
00:00:44
Speaker
Episode 76. You'll pay for this. I'm Sophia and you're listening to Breaking Math. With me, we have on the host of Nerd Forensics, another Santa Fe Trail Media podcast, Millicent Oriana. Millie, welcome.
00:01:03
Speaker
Oh, that's great being on here again. Um, and for those of you that listened to, uh, to breaking math and haven't heard nerd forensics yet, give it a listen. We have talks about everything, history, video games, nerd culture in general, wrestling. Oh yeah. It's a really varied podcast. We have about how many episodes out do you have? We have 24 episodes now.
00:01:28
Speaker
You had 24 episodes of various kinds. So go take a listen. It is more explicit. It's marked explicit. But if you like the kind of humor that we have on this podcast, I think there's a chance that you'll like the kind of humor that there'll be on that podcast. So yeah, go check that out. Give that a listen.
Ariana's Insights into Electricity
00:01:44
Speaker
And we also have on Ariana Luna Rosa. Ariana, welcome. Hi, thanks for having me. So Ariana, the podcast today is about energy and you work with a form of energy in a way, electricity. Do you want to talk about your relationship to electricity?
00:01:58
Speaker
That's right, I'm a licensed electrician and I also teach in an electrical apprenticeship program. Cool, cool. So we're going to talk about a bunch of different forms of energy.
Podcast Promotions and Updates
00:02:07
Speaker
But first I want to plug the fact that we got a new poster coming out very soon. We're in talks with printers. It's about different types of numeral and numeric systems throughout the world. It has about five different ones that it goes through. And it's a lot of fun if you like that kind of thing, which you're listening to this podcast, so you do.
00:02:26
Speaker
And you can follow us on social media, Facebook.com slash Breaking Math Podcast. You could also follow us on Twitter at Breaking Math Pod, or you could go to our Patreon and donate. If you donate a dollar or more, you'll get an ad-free version of the episode.
00:02:44
Speaker
And if you are wondering where the second interview went from that last episode on graphics, we interviewed somebody for an episode, but we forgot to press record, so we're regrouping on that series, but stay tuned.
00:02:59
Speaker
All right, so quick rundown. We're going to be talking about a quick review of calculus. We're going to be going through a few concepts related to energy, going through a few problems, and having a general discussion by energy. So is everybody ready? We're ready.
Calculus and Vectors in Physics
00:03:17
Speaker
All right, so we're going to start off with a little quick review of calculus. If you've heard us review calculus before, because we've done that on a few episodes of the show, we're going to play this music. And if you pass forward to the end of the music, we'll be done talking about our review of calculus. That said, we're going to be talking about it in a way that's relevant to this episode. So it might be worth sticking around. All right, so calculus. So a function is something that takes something or a few things in and spits something out.
00:03:47
Speaker
So a doubling function when applied to the number three would yield what? Six. Yep, and a function that adds three to something when you plug in four would get what? We would get seven.
00:04:01
Speaker
Yeah, and we can write this like f with parentheses 4 equals 7, where f of x is equal to x plus 3. So that's how we could do that, where x is just like no thing. So then we have a scalar, which is a magnitude. So like 3 is a scalar. Give me an example of the scalar. 42.
00:04:22
Speaker
negative 8, etc. You know, 3.7. So it's just like, you know, a number, a natural number is a type of scalar. Other types of scalars are real numbers. There's all sorts of things like that.
00:04:37
Speaker
All right, then we got a vector, and a vector is something that has direction and a magnitude to it. So the wind speed on a certain piece of earth could be a vector. For example, you have the direction of the wind and how fast the wind is going, right? Yeah. Let's see, more examples of vectors. Is that the jury would also be a vector, I assume?
00:05:00
Speaker
Actually, temperature would be a scalar because a temperature doesn't have a direction. But you can say that the flow of heat is a vector because if heat flows from a cold source to a warm source, you could have the vector pointing from cold to hot with a magnitude that corresponds to how quickly heat is transferring. But yeah, can either of you think of more examples of vectors, possible vectors?
00:05:26
Speaker
A really good example of vectors I'm familiar with is in the electrical field. We describe electricity as having a voltage level and an angle when we get into advanced AC theory.
00:05:38
Speaker
Oh yeah, and that's another thing about vectors is that a lot of times they're used to, you know, measure things in multi-dimensional quantities because you said current to voltage, right? Voltage, yes, and current. And current times voltage is what? That would be watts. Millie, can you think of an example of a vector? Absolutely. Almost every sport uses vectors, whether it be where your punches are aimed or where a ball is going, a poloda, a puck.
00:06:05
Speaker
Yeah, exactly. You need to know the vector from the ball coming in. It turns out that if you take the momentum vector of the ball coming in, and you add the force vector, I mean the momentum vector of the momentum transferred from the ball or the poloera thing or the whatever.
00:06:25
Speaker
Yeah, you get the resultant motion. Or specifically get the combination of the resultant motion and what happens to you. So yeah, that's another vector quantity. Absolutely. Another vector quantity would be like, if you've ever seen one of those elevation maps, have either of you seen those elevation maps with the different ridges on them? Yes. Oh definitely, I'm a bit of a hiker. Oh yeah, so a vector would point in the direction of the ascent in the magnitude of the steepness of that ascent.
00:06:53
Speaker
I assume that a textured globe also has vectors on it. Let's see. Yeah, a textured globe. Let's see the vector. Yeah, the vector in that case, you know, you do the slope with respect to the what's called the normal between the surface of the ideal ball, you know, like if it were smooth and the slope that it does have. And also, of course, the distance between bare bodies of land.
00:07:17
Speaker
Well, the distance between bodies of land, you could point from one body of land to the other with a vector that corresponds to a line segment, but the distance itself would be a scalar. Specifically, it's something that doesn't point. You know, if something can be measured in like, you know, like a feet or like, you know, like a...
00:07:33
Speaker
So here's a really good way to compare the two. Speed is a scalar while velocity is a vector. Because speed, if you go backwards, right? If you go backwards at 10 miles an hour, you're still going 10 miles an hour, right? If you go forward at 10 miles an hour, you're going 10 miles an hour. But if you look at velocity, if you reverse and you're going 10 miles an hour, you're going negative 10 miles an hour. So the direction actually matters in a vector specifically.
Understanding Derivatives and Integrals
00:08:01
Speaker
Okay then.
00:08:02
Speaker
So cool, sounds like everybody is up to speed on vectors. So now derivatives. Derivatives are a rate of change with respect to some variable. So for example, miles per hour is...no, no. So for example, speed is a derivative of position.
00:08:21
Speaker
Over time so that so basically the amount that somebody's position changes over a period of time is their speed Does that make sense intuitively? Yes Yeah, and so the derivative the derivative is like if you took like how much fast I was going like right now and how fast I was going in one one one infinity of a second so to speak and
00:08:42
Speaker
and divided it by the amount of time that elapsed which is what that one infinity is of a second and then we get a number. Of course we can't actually do that so use a concept called limits where we do like okay what's the time here versus the time five minutes from now and then it's like okay now you reduce it to one second then half a second and then so forth and then mathematically that gives us derivatives. For anybody interested the mathematical formula is limit as some variable h approaches zero of f of x plus h that is to say the function
00:09:12
Speaker
one teeny little bit above x minus f of x, so the difference between those two, all divided by h, or the amount that the variable has changed, and you just solve that. And we go back to the mountain example, the derivative of the elevation, specifically what's called the gradient, since it's a two-dimensional function, right? You got like
00:09:32
Speaker
latitude and longitude, right? Right. Yeah, so if you take the height, the derivative of the height, I mean, the gradient of the height with respect to the field, you know, the distance, what's called, we could call it D horizontal and D vertical or DH and DV or something. D just meaning tiny little, tiny little like, almost like derivation. I don't know exactly where it comes from.
00:10:01
Speaker
but it's like a little change in x. The derivative of elevation with respect to the planar coordinates gives a mountain slope, which we were talking about earlier. So this is a case where the gradient, which is the type of derivative, gives a vector from something that was a scalar. Because height is a scalar, right? Does that make sense? Yes. And then we have partial derivatives. So back to our mountain example, if we want to see how fast the mountain height increases, if we go straight to the left,
00:10:31
Speaker
or straight to the right, you know, straight horizontal, straight vertical, that would be a partial derivative with respect to the vertical or the horizontal. Does that make sense? Yes. So we also have what's called the product rule and the chain rule. Just to quickly cover those, a product rule says that the derivative of two functions multiplied together is equal to the derivative of the first times the second plus the derivative of the second times the first. And then the chain rule means that if we take the function of another function and take the derivative of all of that with respect to the inner variable,
00:11:01
Speaker
that is equal to the derivative of the inside times the derivative of the outside of the inside unchanged. And then finally we have the integral. So let's say at zero seconds, right, we're going, we're stopped, right, in our car. And then at one second we're going two miles an hour, two seconds we're going four miles an hour, three seconds we're going six miles an hour, and so forth. How can we use this information to find out how far we've gone? You'd measure speed and time.
00:11:30
Speaker
Yeah, you'd multiply... Exactly. And distance is speed times time. Is that apparent to everybody? Yes. Yeah, so like, you know, 20 miles an hour times half an hour would be 10 miles, obviously. So yeah, you would do that. You would sum it up over these tiny little intervals, right? And if you decrease the size of the intervals to basically zero to like one over infinity or, you know,
00:11:54
Speaker
the dx or whatever if we're using the x variable would be a better way to say it. We get what's called the integral, specifically the Riemannian integral. We won't talk about the Lebesgue integral here. But the integral of velocity with respect to time is position. That is to say the summing up of velocity with respect to chunks of time is equal to the position traveled. Does that make sense to everybody? That makes sense.
00:12:19
Speaker
So what would the, let's see, what would the integral of the rate of weight loss of somebody over time give us their current weight?
00:12:34
Speaker
the current difference in weight from when you start it and when you end it because you know it's about change so that is an important thing about integral is that it is sometimes is that um when you integrate you have to know where you start when you're in know where you start when you end but at least where you start and uh let's see what would the what would the integral of the number one be with respect to
00:12:57
Speaker
If you have x, y, and z being three perpendicular coordinates like my hand right here, x, y, and z, if we have the integral of x going from 0 to 2, y going from 0 to 2, and z going from 0 to 2, what would the integral of all of that be?
00:13:14
Speaker
Does anyone have a hazard to guess? Well one is the constant value of things within it, but weirdly enough you're on the right track because what we have to consider is the integral since it's constant if we have the amount of area in every... because obviously we're describing a box, right?
00:13:34
Speaker
So if we're talking about summing up, what we're talking about is the volume of the box. So if the integral from x goes from 0 to 2, y goes from 0 to 2, and z goes from 0 to 2, what would be the total volume of the box or the integral over the space?
00:13:49
Speaker
That would be eight units? Yep, because two times two times two. Yeah, zero to two being, you know, two across. Does that make sense? Yeah, makes sense. All right, cool. And as y'all noticed, when you do derivatives, like you derivative with distance with respect to time, meters with respect to seconds is meters per second, meters divided by seconds.
00:14:09
Speaker
whereas the derivative of meters per second with respect to time, with respect to seconds, is meters to seconds times seconds. So you multiply the units with the integral and you divide the first unit by the second with the derivative. And it's really hard to do integration closed form compared to derivation. Derivation is straightforward for basically any function. There are some functions that integrals are not known for, but they can always be approximated.
Exploring Kinetic and Potential Energy
00:14:43
Speaker
So now we're going to jump into energy finally. So we have a quote here from the Feynman lectures, which can be found on Caltech's website. Ariana, would you read this block for us? I would love to.
00:14:55
Speaker
Imagine a child, perhaps Dennis the Menace, who has blocks which are absolutely indestructible and cannot be divided into pieces. Each is the same as the other. Let us suppose that he has 28 blocks. His mother puts him with his 28 blocks into a room at the beginning of the day. At the end of the day, being curious, she counts the blocks very carefully and discovers a phenomenal law. No matter what he does with the blocks, there are always 28 remaining.
00:15:22
Speaker
This continues for a number of days until one day there are only 27 blocks, but a little investigating shows that there is one under the rug. She must look everywhere to be sure that the number of blocks has not changed. One day, however, the number appears to change. There are only 26 blocks. Careful investigation indicates that the window is open and upon looking outside, the other two blocks are found. Another day, careful count indicates that there are 30 blocks.
00:15:50
Speaker
This causes considerable consternation until it is realized that Bruce came to visit, bringing his blocks with him, and he left a few at Dennis' house. After she is disposed of the extra blocks, she closes the window, does not let Bruce in, and then everything is going alright, until one time she counts and finds only 25 blocks.
00:16:09
Speaker
However, there is a box in the room, a toy box, and the mother goes to open the toy box, but the boy says, no, do not open my toy box and screams. Mother is not allowed to open the toy box. Being extremely curious and somewhat ingenious, she invents a scheme. She knows that a block weighs 3 ounces, so she weighs the box at a time when she sees 28 blocks and it weighs 16 ounces.
00:16:33
Speaker
The next time she wishes to check, she weighs the box again, subtracts 16 ounces and divides by 3. She discovers the following. The number of blocks seen, plus the weight of the blocks, minus 16 ounces divided by 3 ounces is constant.
00:16:48
Speaker
So what this is talking about is all the different forms of energy in a system. We have kinetic energy, for example, which is the amount of energy there is in a moving thing. And it's equal to 1 half times the mass times the velocity squared.
00:17:10
Speaker
And the square of the velocity is always going to be a scalar, because specifically when you square a vector in a lot of equations, basically the length of the vector. And this whole constant deal is an issue with potential energy and kinetic energy, or the other energy in a system. So does anyone want to say what potential energy is? Milly, what's potential energy? Potential energy is the
00:17:41
Speaker
Is the possible outcome of the consumption energy, right? Well, specifically, it does relate to the outcome. But what it is, is it's how much energy you could kind of produce. Sorry.
00:17:57
Speaker
Yeah, exactly. It's how much you can kind of extract in a specific instance. So for example, gravitational potential energy says how much kinetic energy you can get out of something that is falling near Earth's surface. And so the higher it is, the more potential energy there is, right? Yes. Because the higher it is, the more potential there is for it to fall faster and so forth.
00:18:24
Speaker
And we'll show that we'll show why kinetic energy is 1 half mass times velocity squared a potential energy near the surface of the earth is Mass times the gravitational constant times the height for two bodies further apart from one another the it changes because you know the further you get away from the earth the less the gravity is right? Yes, right You have electrical energy
00:18:46
Speaker
which is produced by electron flow across a wire, billed often in kilowatt hours, which is 3.6 megajoules. We'll be talking about joules and things later. We also have other forms of energy, such as the strong force. And as we'll see in a second, the way that you can manipulate a system can tell you a lot about the energy.
00:19:07
Speaker
But before we jump into all that, let me just have a quick little talk about thermodynamics. So in any real process, energy is lost, right? There's no such thing as a perpetual motion machine. Yes, it's impossible.
00:19:26
Speaker
Oh, yeah, it's impossible. All right, so energy is always lost in any real process, right? So we're going to talk about, let's think about an example of when friction would be like a hindrance. So let's talk about a system where any slowdown would be a hindrance. So let's talk about a swing, right? So when you swing, you pull somebody up and they have a certain gravitational potential energy, right? Yes. Right. And as they swing to the bottom, that gravitational potential energy turns into what?
00:19:55
Speaker
velocity, velocity through kinetic energy. Yeah, because kinetic energy corresponds to velocity. But yeah, so it goes from potential to kinetic, then when they go to the up to the other side of the swing, what's happening to the energy? It's transferring the opposite direction. Yeah, this transferring from kinetic back to potential. Is that what you meant? Yeah. Yeah. And so why does it why is it that if you when you push somebody on a swing, that you have to ever give them a second push?
Energy Loss and Entropy
00:20:22
Speaker
Uh, because of the fact that eventually the, uh, no matter how, like how, like how well balanced they are, it will stop swinging. It can't continuously swing. Oh yeah. But let's, but remember we were talking about energy that neither created nor destroyed, right? So where, where does the energy go?
00:20:42
Speaker
It goes into the one into the chains and that creates friction which helps to slow it down Mm-hmm to it goes into It gets dissipated through wind resistance that too and Three a lot of it goes into the ground because your feet usually drag across the sand as you go through the swing Oh, yeah, that's another that's another Play and then why doesn't this sound the sand bounce around forever?
00:21:08
Speaker
Well, because the the more friction like are the more resistance the slower you start to go Oh, yeah, but specifically the sand that you kick and you lose energy on the energy that that sand gains. Where does that go? How comes the sand doesn't bounce around forever? Oh because it eventually just it hits the ground and it slows down and the rest of the sand around it absorbs the energy
00:21:31
Speaker
Yeah, the rest of the sand absorbs it. And then can you think of any other ways that the system loses energy? We definitely see energy loss in electrical systems over the distance of conductors.
00:21:43
Speaker
specifically for this example. Oh, in this example? Yeah. What I see in this example is that, yeah, the individual little grains of sand influence all the other little grains of sand and give give a little push to every piece they encounter. And eventually that energy has just been dissipated throughout the system. Exactly. And let's see what other ways it could energy be lost, lost by the system, the swinging system. Can you think of any more entropy?
00:22:12
Speaker
Well, what specifically entropy is, it's a measure of the amount of useful energy that you cannot get out of a system in a way. We're going to talk about that I think on our next episode. We're going to do a little thing about thermodynamics probably. So yeah, what entropy is, is the amount of disorder in a system.
00:22:30
Speaker
So something with a really high amount of entropy or maximum entropy can get no useful work extracted for it. And we're going to talk about how work is used in this. But I'll give you all a hint. Can you hear somebody on the swing? Yes. Yes. So we're talking about sound energy, which is measuring pressure waves. And pressure itself is the amount of kinetic energy collisions that are happening in a specific amount of space, right?
00:22:57
Speaker
because the more kinetic energy collisions are in this specific amount of space, it's like, you know, if I throw one billiard ball at a window versus throwing, you know, barrels of billiard balls at a window. By the way, that's where I got all my new things. Oh yeah, and there's also other ways that the system could lose energy, for example, you know, why doesn't the chain get so hot that it's burning after a while when somebody's been swinging on it?
00:23:23
Speaker
No, because I mean, that would take a lot of friction to do. Oh, yeah. But why does why does the chain ever get cooler at all? Like, why doesn't let why doesn't after a year turn like glowing red hot? Well, because it has the energy has to go somewhere. It doesn't just stay in the metal.
00:23:38
Speaker
Oh yeah, but technically, theoretically it could with some kind of exotic, weird matter. It doesn't technically violate any law of physics except for that the change of entropy is always greater than zero or the change of energy is always less than zero, whatever.
00:23:56
Speaker
But actually the reason why is because heat is radiated in what's called black body radiation. So when you heat something so hot it gets red hot, that's the energy radiating outward. You also have convection, like the wind taking away heat.
Heat Dissipation and Work-Energy Relationship
00:24:15
Speaker
and the heat is dissipated also into the air around the chain. So yeah, so that's where that energy goes. And as for sound energy, you know, that's just dissipated in the same way as heat and then as light and then as heat and light and you know, everything's keep bouncing back and forth, you know, over and over again. So, I mean, you could hear the echoes of the chain on a loud swing in some parks, right? Yes. Right.
00:24:42
Speaker
So work is force times distance. So the amount of work that you put into something is related directly to the amount of energy that it uses. So the force times distance thing is like, let's say that you get a glue whale, then you put it at the end of a very, very sturdy seesaw. How could you lift up that killer whale using gesture body?
00:25:10
Speaker
Uh, if you had a, if you had a good enough, uh, pulley and, uh, or like a good enough lever system. Oh yeah. But we got a lever. We got a seesaw lever. Well, we'd need a super long lever. Yeah. We'd need an incredibly long one then.
00:25:24
Speaker
Yeah, and it would be short on the other side, right? Where the whale is. And we would have to travel a lot further than the whale, right? We would have to like go down like a mile for it to go up like, uh, not that, probably a few feet. Yeah. We would need an incredibly long one then. Yeah. And it would be short on the other side, right? Where the whale is. And we would have to travel a lot further than the whale, right? We would have to like go down like a mile for it to go up like, uh,
00:25:47
Speaker
Not that probably a couple feet, probably a few feet. Yeah. Yeah. It turns out you're exactly right. Two feet for every mile that we go down on a lever. Wow. And, um, yeah. And the reason, and so work is the force times the distance. So, so remember when we said earlier that the gravitational, um, potential energy is mass times the gravitational constant times height. Yeah.
00:26:13
Speaker
The mass is measured in kilograms, and the gravitational constant is an acceleration constant. It's meters per second squared. So you have the quantity kilogram meters per second squared, or newtons. So then you take the mass times gravity and multiply it by the height, you get units of work, or force times distance. Now if you notice, these are the same units as energy, right?
00:26:31
Speaker
Right. Correct. And they are intrinsically linked together. So basically, the work energy theorem states this, that the amount of work that it takes to do something corresponds directly to the amount of energy that takes to undo it or vice versa.
Gravitational and Magnetic Potential Energy
00:26:49
Speaker
Well, so for example, the potential energy between two magnets, like let's say we have two positive sides of a magnet that are in a tube, right? And if you push them together, they're gonna resist it, right?
00:27:00
Speaker
Yes. So if we start with them infinitely far apart from one another, and we give them very small push, we would not need a very small push to get it started moving at all. So we move infinitely slowly. But let's say we do it a little closer, obviously, although we do use the first form in the integral form because you can integrate from infinity to infinity or zero to infinity sometimes.
00:27:24
Speaker
So if we push these together, it takes a certain amount of force and the force increases as you get them close together, right? Yes. Right. So if you take how much force you're pushing at each measurement of distance, so like 2 feet, 1.9 feet, et cetera, and you add them all together, but you do it infinitely slowly, you'll get infinitely many. You'll get the potential energy between two magnets and a tube. Okay.
00:27:49
Speaker
And so now for gravitational potential energy, it's similar. So if we need to hold something, so hold this. How much force are you applying on that right now? Up. How much upward force? About 16, 17 ounces.
00:28:11
Speaker
Yeah. 60, 17 ounces worth, which, which technically we'd be measuring force and slugs, but nobody does that. Everybody just use pounds. Um, because American units are a mess. And honestly, this is an international podcast, so I should be using metric more often, but, um, yeah, you're applying force to resist, uh, gravity is force, right? Yeah. So now if you, now when you lift, uh, now lift it at a constant speed, uh, for a couple of seconds.
00:28:40
Speaker
No, you lifted it about, let's say two feet, right? Yeah, about. So the amount of work that you would do in that case would be since it's, let's just round that to a pound, two foot pounds of energy.
00:28:57
Speaker
Of course, you do have to convert the units, but a better way to do it would be you lifted about a meter and it's about half a kilogram. So that's half a kilogram times the acceleration due to gravity. Acceleration due to gravity is about 10 meters per second squared, it's 9.81. So what's 10 meters per second squared times one meter times half a kilogram? So that would give us five.
00:29:25
Speaker
Yeah, so five joules is what it took you to lift at that distance.
00:29:33
Speaker
I don't use a jewel. I use a regular e-cig. Don't smoke kids. All right. Yeah. And the kinetic energy formula works a lot like this too. So let's say we're in a car and let's say we accelerated a constant acceleration. That is to say, you know, the speedometer is moving like at a constant speed, right?
Work and Energy Applications
00:29:52
Speaker
Okay. So when we're moving slowly, we're pretty close to the starting point, right? Yeah.
00:29:57
Speaker
But the faster we move, the further we get away. So the distance between 0 and 10 miles an hour is going to be a lot less than the distance between you going 40 and 50, right? Yes. If we want to measure how much kinetic energy something has, we could measure how much energy it takes to stop it or how much work it takes to stop it. So if we measure how much distance it takes to stop it, the distances would be one half times the acceleration times the time squared.
00:30:22
Speaker
which is equal to the velocity squared over two times acceleration. So in that time, the work that we've done turns out to be work, which is force times distance, which in this case is mass times acceleration times distance, which turns into mass times acceleration times the velocity squared over two times acceleration. And since you're dividing acceleration by acceleration, you end up with a familiar and famous formula. Work energy equals one half nv squared because work is proportional to energy.
00:30:51
Speaker
Any questions? I think I'm good. Oh, yeah. And similarly, the gravitational potential energy of something is how much energy you would slam into the gravitational potential energy of an object on the Earth with respect to just the rest of space.
00:31:07
Speaker
is how fast an object would be going when it hits the Earth. If you take the object infinitely far away from the Earth, remove everything else from the universe. Does anybody want to guess how fast something would slam into the Earth if they started infinitely far away?
00:31:27
Speaker
I mean, would it be infinitely fast? It would be infinitely fast, yeah. Turns out it's not infinitely fast, actually. Oh, it would be right below the speed of light because it can go any faster than the speed of light. And actually not even close to that. Believe it or not, it's the escape velocity of an object on Earth or like about seven miles per second. Well, don't make too good that exactly. That's interesting.
00:31:46
Speaker
Yeah, so it would be about 40,000 kilometers per hour or 25,000 miles per hour. And the reason why it's not infinite is because it's not accelerating at a constant acceleration the entire time. When it's infinitely far away from the Earth, technically, there's no acceleration on it. It can never start reaching the Earth. But because we're in xenoland, derivative, funny, infinity land, we can do that.
00:32:15
Speaker
and escape velocity is how fast something has to be going at the surface of the Earth to escape the Earth's gravity. But I was surprised too when I first heard this. I thought it was going to be... I remember calculating it myself. I remember saying, okay, like, what would it be? Would it be infinite? And then I was like, oh, no, it's that. And it turned out that it was also related to a couple of other concepts, which we'll cover on probably like an orbital mechanics podcast.
00:32:40
Speaker
All right, so let's say I have two weights. We don't know how much they weigh yet. And we have a triangle. The triangle is three high, four across, and five on the diagonal. Everybody have that?
00:32:55
Speaker
Yep. All right. So how heavy would the weights have to be so that they balance each other? And this is a frictionless slide, obviously. So no energy is lost to anything. The energy is all just balancing out with one another. So how heavy would the weight on the left and right hand side have to be with, you know, with, with respect to one another or whatever? I would assume that the weight on the right would have to be about at least a third smaller than the weight on the right. I mean, what weight on the left?
00:33:26
Speaker
That's a good guess. I'm thinking that they have to be the same weight because regardless of the slope, it's still the same force of gravity.
00:33:35
Speaker
All right, so now let's do a thought experiment. All right, so these are in equilibrium. So what happens when, let's say that we have this taut, right? And let's say we were holding on to the right weight, right? And let's say we pull that weight one unit down the horizontal. How far will the block on the left go up if we pull the string on the right by one unit?
00:34:05
Speaker
And this is frictionless. Yeah. Right. And it, but right now we're just dealing with pretty much the geometry of it because we're talking like is specifically if I, if I have it like gripped and I'm pulling it down by one unit, it would move it up one unit then. Yeah, correct. And that's, is that apparent Ariana? Yes. But how, how far down will the block on the right have gone if it moves across the diagonal one unit? Um, if it moves on its own,
00:34:34
Speaker
No, no if we move it or whatever just one unit well It will go to one unit on diagonally, but I mean specifically vertically up and down the way that gravity is pulling So let me diagram this out a little more that would be three-fifths of a unit, right? Yeah, right because because going down one unit what bras down one-fifth of the way right and then but
00:34:58
Speaker
We know that, so if we divide by five on that side, we get the height, which would be three-fifths. So yeah, it would be right there about, right? If we dragged it one unit from the edge. And this would go up a whole unit though. But the thing is, they have to balance each other out. So to balance each other out, they have to have the same amount of potential energy with respect to one another, right? Yes, right.
00:35:24
Speaker
So the one if they weigh the same and we move the one on the right up one unit and the one on the bottom and the one on the side moves down even further than one unit, it'll just keep pulling it to the side. Yeah. So it turns out that it turns out that let's say the weight on the left is a five pound weight. The one on the right would be a three pound weight. Yes. Does that make sense? Okay. And then they balance out.
00:35:53
Speaker
Yeah. Um, but there's even a better proof of this. Uh, do y'all want to see a proof in a picture? Sure. Yeah. This is the, we're looking at the image that's on the cover for the episode. So check it out. Uh, these balls weigh the exact same amount. No. Will the, will the balls move one way or the other? No, they won't.
00:36:15
Speaker
Yeah, and they can't because the balls on the bottom balance each other out on each side, right? There's seven. There's one exactly in the middle and three on either side. So it's not pulling the ones on the top either way, right? Right. Correct. And the ones on the top are just resting as they are. Correct.
00:36:34
Speaker
So yeah, that's how it turns out. You solve that using just pictures.
Physics Problems on Energy and Forces
00:36:39
Speaker
But you have the same concepts going on. You have the energy on both sides of the chain. If the chain were unbalanced so that we had two weights on one side and then the weights were way heavier on the other side, it would pull the chain down so that the heavy weights were on the bottom. If you take a heavy necklace and you put it on your shoulder, it could fall down to your chest.
00:37:04
Speaker
Yes. Makes sense. It also turns out that a rope, you know, you could just do this with rope. Like, you know, if you have a heavy rope that's frictionless, you know, some kind of amazing frictionless heavy rope, although I don't think that'd actually be useful for knots. But yeah, you just have to drape it across all the way up and all the way across or have an equal proportion of both sides. And as long as the two bottoms are at the same elevation, it'll balance itself out. All right. And this is the same principle as how those old school scales work.
00:37:33
Speaker
All righty, so this is the basic setup we have a frictionless loop-de-loop imagine made out of like, you know, the slipperiest ice in the world And slipperier than that we have we're on the moon or something. So there's no air resistance you know, the only thing we care about is kinetic energy and Gravitational potential energy. Is that clear?
00:37:54
Speaker
clear. Yeah. So we go around this loop to loop and specifically it's just a loop actually, but I think the loop to loop technically has two loops, but, um, no loop to loop is just one. Oh, is it? Okay. Yeah. So we're go, uh, we go around the loop to loop, which we're going to assume is the circle, even though I did not draw a good circle. And, uh, yeah. So the question is how tall can the loop to loop be, uh, with respect to the original height?
00:38:19
Speaker
How, well, can it be like an unlimited number of tall? Like, can it be infinitely tall? Well, specifically, how tall must it be? Oh, no, no. How tall it like, let's say that the loop, let's say the top of the loop in the floor to where we start, how tall can the loop to loop be compared to that? Oh, the loop compared to the slope. Yeah. They can be the exact same height if there's no friction.
00:38:42
Speaker
And why is that? Because it would use the amount, it would build up the right, the correct amount of momentum going down the slope that it would need to send around the loop if there's no friction or resistance or anything. Yeah. So you're saying that it, it, it maintains all the energy it needs. Yeah. They would just need to be equal height.
00:39:01
Speaker
How about you, Ariana? For some reason, I think it needs to be a little bit less because I'm picturing that if you did halfway through the loop-de-loop at the same height, the ball is going to come to a stop or whatever we have traveling this loop-de-loop. All right, we'll see what the deal is. And the way that we can do that is by just kind of starting to plug and chug formulas.
00:39:23
Speaker
So we're going to do a quick tangent and talk about centrifugal force. So the force is the derivative of the momentum with respect to time. And so basically the way that it happens is if you're going around a circle with a certain angular velocity. So angular velocity is measured in radians per second.
00:39:47
Speaker
So what that means is, um, if something goes two times PI radians per second, it goes around a circle once per second. Cause two PI radians is ones is 360 degrees. Yes. Yeah. And PI radians would be what? Uh, that would be the, uh, times 3.14 to whatever. Oh yeah. But specifically how many degrees would PI radians be if two PI radians is 360.
00:40:12
Speaker
180 hundred yeah exactly and So in there's called a radian because it takes two pi radiuses to go around the circle So a radius is just a radian a radian is just one radius wrapped around the circle
00:40:26
Speaker
And so centrifugal force can be measured in omega in terms of radians per second. And we get the equation that the position of t, if we're just looking for the acceleration, because remember, f equals ma, force equals mass times acceleration. Yes. So we got position is r times cosine of omega t and r times sine of omega t as a vector xy. So the cosine of an angle is how far you are from the center of the angle horizontally at a specific angle. Does that make sense?
00:40:56
Speaker
Yes. So at 45 degrees or pi over 4 radians, it would be square root of 2 over 2 because if you take something that goes diagonally and hits the circle, it would be square root of 2 on this side, square root of 2 on this side, and if you multiply those by themselves, add them together, you get the distance, which is 1, you know, 1 from the center.
00:41:20
Speaker
And you multiply by r because that's the radius. So then we take the derivative of that and we take the derivative of sine and cosine. Since it's a function, you take the derivative of the inside and multiply it on the outside. And since it's sine omega times t, and the derivative of omega times t is just omega,
00:41:36
Speaker
The reason why is because, you know, it's just the rate, if something is like, you know, if something does something like three things per second, then to see how fast do you do something, you just have to take how much they've done and divided by the amount of time. Is that kind of clear? Yes. Yeah. And then acceleration is just the derivative of that, which ends up being the same thing with negative, and then you end up with negative omega squared times r times cosine of a omega t, and then negative omega squared times r times sine of t.
00:42:04
Speaker
And the magnitude of this is going to give us our absolute acceleration, which we could use to measure the amount of centrifugal force, specifically. And that gives us the square root of those terms together, added together, or square root of quantity negative omega squared r times cosine of omega t n quantity squared plus quantity negative omega squared r times sine omega t n quantity squared.
00:42:28
Speaker
and square root, which is equal to omega squared times R, or since omega is just equal to velocity divided by radius, because you know it's how many radians you go per second, so velocity divided by radius, it would just be b squared over R. So we're gonna use that in a second. So it's clear that to keep the cart on the track, we're gonna need enough centrifugal force to keep the cart on the track, right? Right, yes. And how much centrifugal force is that?
00:42:55
Speaker
Well, okay, so the track's also gonna be completely flat. There's nothing physically keeping the sled on. Yeah, there's no track. It's like an ice slide. Yeah, but there's no barriers on the side or anything like that. Oh, no, yeah. We're talking that we do a straight shot so we never even touch the side. Okay, yeah, so it just goes down this thing.
00:43:20
Speaker
Well, if there's no resistance whatsoever against it, like we were talking about, like it's just this magical ice that's slicker than anything and there's no gravity, it would just need forward momentum with no, um, no sort of, uh, interference.
00:43:37
Speaker
Right, exactly. And to have that, so back to, let me just hand you another thing. How much force are you putting on that to keep it in place? Like I can't even measure it. Oh yeah, but you're going against to what? Gravity. Yeah. So the force of the centrifugal force that we get at the top has to equal the amount of gravitational force on the object. Yeah. Okay. So there is gravity. Like that was the other reason I was like saying earlier, I actually want to change my answer about the height.
00:44:07
Speaker
Ariana's completely right. It would have to at least be a couple feet higher. Oh, yeah. Yeah, I get you now. Yeah. Honestly, that's very good physics intuition to not assume anything I didn't tell you. Yeah. And yeah, it made sense why you thought it was a track too. But cool. Let's see if these predictions pan out. So the change in potential energy is the opposite of the change in potential energy, right? Correct. Right.
00:44:31
Speaker
And the change of potential energy is the mass times the gravity times the change in height. Makes sense. And so that means wherever we are in the height, you know, we could calculate our potential kinetic energy because the kinetic energy is negative mass times the gravity times the change of height, which is just equal to one half mv squared.
00:44:53
Speaker
And since once half mv squared is equal to that, we just divide m by both sides, and we get negative gravitational constant times the change in height is equal to 1 half times velocity squared. So then we solve that, and we get that the velocity squared is equal to negative 2 times the gravity times the change in height. And the change of height is equal to the initial height over height at the top of the loop, right? h top.
00:45:17
Speaker
And so now, since we know what the change in height is, we plug that into the formula, and we get that b squared is equal to the square root of 2g times quantity h top minus h n quantity. So now, since we know about h, and h is a circle, if this is the total height, h top, what would the radius be? Oh, twice the radius?
00:45:42
Speaker
Yeah, the height is twice the radius. Oh, OK, yeah. Yeah, because I think you might have been thinking of circumference. Yeah, I am thinking of circumference. Yeah, radius is the distance between the center of the wheel and the outside. So now we plug that in and we get a v loop. The velocity that you're going around the loop is equal to the square root of 2g times h to the top minus 2r. So now we have the centripetal force of the loop is equal to the mass times the velocity of the loop over r, which we calculated earlier, right?
00:46:09
Speaker
Right. The velocity of the loop squared, which is equal to mg, is equal to mg because of what you said earlier, right? Because mg is the force against gravity. Yes.
Principle of Least Action and Variational Calculus
00:46:20
Speaker
And by the way, G on like, you know, Jupiter would be 10 times as great as it is here because it's 10 times the gravity on the surface of Saturn. It would be a little bit less than it is here. Saturn could float in a bathtub, by the way. Well, wait a minute. Well, how is it in Dragon Ball Z that the planet that Goku trained on was smaller than Earth, but it had more gravity than us? Are you saying that? No, it's just denser. Oh, OK. Well, that show still also doesn't have any real science fact in it. So I just wanted to like, you know, be funny.
00:46:50
Speaker
Dragon Ball Z isn't real. I've never watched Dragon Ball Z. But it is very ridiculous. But fun. Very fun. Check out the anime intro episode on Nerd Forensics for more fun. Yeah. Check out our anime episode. Check out our episode about sports. You people don't just listen to math podcast. You have other interests. Easy. Don't browbeat the listeners.
00:47:17
Speaker
But anyway, so we divide m by both sides by m, right? Yes. So now we get velocity of the loop squared divided by r is equal to g. And the velocity of the loop, we already found the equation of that squared, and that's 2g times h top minus 2r all over r equals g.
00:47:34
Speaker
And then we keep solving that. We divide both sides by g to get 2h top minus 2r all over r equals 1. Distribute and move over the r and we get 2h top minus 4r equals r. So 2 times h top being the top of the loop is equal to 5 times r. So then we finally get the height at the top is
00:47:58
Speaker
Yeah, that sounds right. So yeah, that means that two fifths of each top, the top of the loop is equal to the radius of the loop-to-loop. Or what that means is that the height of the loop-to-loop can only be four fifths the height of the entire thing.
00:48:17
Speaker
What's cool about that is we didn't have to do integrals We didn't have to do you know like consider all possible loop-to-loops, you know We didn't have to do a lot of things all we had to do is know how much energy required to do this and that, right? Yes, and that's one of the ways that energy can simplify problems significantly You see this all over the place I mean a lot of quantum physics problems are essentially just solving boundary problems with energy
00:48:44
Speaker
All right, so real quick, we're going to talk about the principle of least action. So action is defined as the integral, remember the sum, from the start to the finish of the difference in kinetic and potential energy over time. And what the principle of least action states, it states that if you don't mess with the system and the system moves in a specific way, if you move the thing that you're talking about in any different way, it's going to use more energy.
00:49:10
Speaker
Okay, so as an example of this dangly arm by the side of the chair and Let it swing back and forth naturally on its own Doesn't take any energy right at first the pendulum. Yeah, but then slows down
00:49:24
Speaker
Oh yeah, it does slow down, but you have to maintain that the frequency, because each time it takes the same amount of time. And it only slows down and gets closer, but it takes the same amount of time each time. Now move your arm back and forth, but do it twice as fast. And now even faster.
00:49:43
Speaker
As you hear, there's chair squeaking going on. And the reason why is because it took a lot more energy to go kind of against the natural way of the system, right? Right. And this is, in essence, the principle of least action. And it's what brought Newton to his idea of the calculus of variations.
00:50:04
Speaker
Now, what the calculus of variations is, is it's about motivating. So, you know how a function goes from a variable to like another variable? Yeah. The function in the function can also be a function. So, for example, the integral is a function of a function or what's called sometimes a functor.
00:50:24
Speaker
because you take a function and you take a function of it and you get a number out, right? And that's specifically what a functor is, it gets a number out of a function. Specifically for taking the minimums of functors, you use the calculus of variations. Newton was working originally on a problem with minimal drag. The second problem he ever worked on was something he worked on with a bunch of other people. And he claims he did it overnight, but he was known for exaggerating. He was kind of full of himself, honestly. You can't trust everything that Newton says.
00:50:52
Speaker
But it was the Brachistochrone problem. And what that problem is about is, let's say we have a slope and rolling a marble down it. What shape, you know, we get the marble from the start to the finish, the quickest? A round slotted ramp. A round? What slotted?
00:51:11
Speaker
Like a round, like a round, like a slide, essentially like a tube. Oh yeah. But what's the shape of the tube though? And get it there, get it from point A over here to point B the quickest. Straight down. How about you, Ariana? What's the path? Instinctively. I'd want to say a straight line, but I'm sure that's not correct.
00:51:27
Speaker
It turns out that it actually isn't correct. Because think about this, if you go down really quickly and then you roll all the way, you're still going to have to roll all this way down. But if you go directly down like this, you're going to be going the same speed, but you might not get the initial oomph from starting out on going a quicker way at the beginning. But you'll build momentum.
00:51:46
Speaker
Exactly. And so what it is, is it turns out to be a cycloid. A cycloid being like if we take a wheel and we mark, when we put like a light on a wheel and we watch the shape that it takes as it goes down the freeway, it would look kind of like a bouncing ball.
00:52:02
Speaker
That's a cycloid. And that's the quickest way between two points is a cycloid. And you could also use this to find areas of minimal revolutions. So for example, the shape with the smallest surface area for the greatest volume is what soap bubbles stretch out to be because basically the amount of energy that it takes to maintain any other shape takes more energy because it has greater surface area.
00:52:29
Speaker
surface tension you could think of it as like a bunch of people holding hands with like forehands like you know in all directions and pulling so the less distance they have the less hard they have to pull right yes right kind of like cloth you know you stretch it etc but um yeah this is uh this is how we a functional not functors I meant functors I meant functionals when I meant functors earlier
00:52:52
Speaker
But yeah, the basic technique we'll talk about on its own episode.
Functionals and Minimal Energy Configurations
00:52:56
Speaker
But essentially, you assume that there's some function that fits your criteria. And then you consider every other function, and it has a function that you add to it that we call the error function. And you assume that the error added to the best function would be equal to zero. And using that information, you solve backwards. All right, then. It's a very fun, it's a very difficult technique, but extremely rewarding technique.
00:53:22
Speaker
Yeah, I was going to say, I'd have to do it with a scratch piece of paper probably. I can't just do it in my head. Oh yeah, I can't do it in my head yet. So we could also see this in other applications. So whenever we have a boundary condition, something you need to find the minimum energy for, you can find this sort of thing. So the shape of a soap bubble between two rings.
00:53:41
Speaker
For example, have either of you done that experiment? Like, you know, you take a soap bubble at, like, Explorer or something, and you pull it out of the water? Well, what you'll find is you'll get a specific shape. It's not a cylinder. It's a ring. But it's not... You won't get a cylinder, and it's not even a cylindrical ring. If you check it out, you get what's called a catenoid, and a catenoid curve between two. It kind of goes inward.
00:54:10
Speaker
Yeah, it's like an hourglass. Yeah, a little bit like a really loose hourglass. And it turns out that that curve that you get on either side is a curve that you would get by hanging a chain between two points. Incidentally, something that you could also solve with the calculus of variations. And yeah, we're going to be talking about this a lot more on a subsequent problem episode.
Mass-Energy Conversion and Nuclear Power
00:54:35
Speaker
So what is Einstein's famous formula for mass and energy?
00:54:41
Speaker
energy what is it energy equals mass
00:54:49
Speaker
God, what is it? Energy equals mass times something. Speed of light squared. Speed of light squared. Yep, exactly. Yeah, so that's how much energy, you know, you could theoretically convert a certain amount of mass into. And this equivalence is, there's actually a more nuanced formula, and that's instead of E equals mc squared, it's E squared is equal to momentum times the speed of light, all squared.
00:55:13
Speaker
plus the momentum at rest times the velocity squared, and we square that as well. And you can think about this as the diagonal along something that has the rest energy on one side and the momentum on the other.
00:55:29
Speaker
Which incidentally means that the energy that something has also depends on the frame of reference, right? Because in momentum, if I'm in a rocket ship going a million miles an hour and it has artificial gravity and I drop a penny, it's not going to go back at a million miles an hour, right? Yeah. Or if I throw a cracker from the back of the car to the front of the car, it's not going to fly backward or forward, right? Yes.
00:55:55
Speaker
So if we can convert mass completely and totally into energy, which is obviously impossible because there's going to be conversion issues with any form of energy, we'd be able to power South Korea with only 50 kilograms worth of matter per year. And if you consider that the ocean is made out of
00:56:16
Speaker
So yeah, there's six billions of pounds of water in the ocean So if we dried up the entire ocean we could theoretically if we convert all its energy into mass power South Korea until the end of the heat death of the universe and beyond But wouldn't we all just suffocate for South Korea's sake? Oh Yeah, but I mean I'm just saying I assume if we were doing all this energy we could probably mine water from from that blue cat planet
00:56:43
Speaker
Well, yeah, I mean, granted, they had plenty of water. That wouldn't have been a problem. Oh, yeah, I know. But I'm just saying that, like, you know, we could put a trillion pounds of water into South Korea and it would still last longer than the universe has ever existed so far. Oh, I know. You're just saying like sucking all the water out. Oh, yeah. I'm just saying that. Yeah, I'm just saying that, like, you know, that that it's that the energy and that it would take to power something if we could turn mass and energy would be just a drop in the ocean.
00:57:11
Speaker
Oh yeah, it'd be a drop in a bucket, absolutely. So it turns out that not all forms of energy are as efficient as black holes, because black holes store it with that energy, the volume, the weight, the mass. It converts it all into energy. It evaporates in the form, as we've covered in our three-part black hole series, using Hawking radiation, which is a form of heat radiation.
00:57:34
Speaker
But, so yeah, that means that 50 milligrams of energy is per kiloton of TMT. So it's about the weight of a mouse of energy could generate a kiloton of TNT if converted. Wow. The Hiroshima bomb, only 700 milligrams or about a weight of a Tylenol worth of energy was actually converted, of mass was actually converted into energy.
00:57:55
Speaker
And Castle Bravo by comparison was 670 kilograms of mass equivalent, which is about a dairy cow. And Tsar Bomba was 2300 kilograms of mass equivalent or a small elephant. Wow. Tsar Bomba was ridiculous. You know, to get an idea of the efficiency, what was the weight of the Tsar Bomba?
00:58:16
Speaker
So yeah, the Batara Bomba weighed about 27,000 kilograms. And the fact that, you know, the fact that about 10% efficient means that it was an insanely efficient thermonuclear bomb. Yeah. But yeah, its blast yield was 50 to 80 megatons. Its blast radius was... So the ball of fire would have a radius of about three and a half kilometers or like a little over a mile.
00:58:40
Speaker
From the center of the bomb and that would just be the fireball itself a complete and total destruction of everything would reach out to 35 kilometers or for about 20 miles from the center of the blast everything would be completely destroyed raised to the ground and not to mention that everything that would be killed by the massive fallout. Oh, yeah, and That did happen and I know that part of Russia
00:59:05
Speaker
So yeah, the test was done near the North Pole. Yeah. It was done in a testing place near the Arctic sea and it was a gigantic, uh, it was a gigantic blast. Uh, I mean, uh, Millie hasn't covered this on her show, but it would be the type of thing that she would cover on her show. Oh yeah. We should totally do an episode about the effects of it as well as the lucky dragon number seven and castle Bravo test. And Oh, he could even do, I mean, for all our listeners, there used to be a handheld nuclear weapon.
00:59:34
Speaker
Yes, there were multiple. There were experiments on nuclear hand grenades that couldn't be thrown like far enough. So they got rid of them. There was also the Davy Crockett, which is a surface to surface missile that had nuclear nuclear warhead in it that I've wanted to cover in an episode because it was by far one of the dumbest ideas ever. Yeah, basically almost blow yourself up using it. Yeah, you couldn't actually get out of the fallout radius.
01:00:00
Speaker
Oh, wow. Makes it the perfect nuclear bomb. Actually, you know, you're going to die to use it. No, I'm just kidding. No, there's many problems with that. So, yeah, I decided to total up the amount of nuclear tests done by various countries and how many pounds of mass equivalent they would be. And India has exploded about three kilograms worth of matter equivalent or about 70
01:00:28
Speaker
kilotons China about 1136 kilograms France about 600 kilograms The UK about 400 kilograms about the weight of a cow each of those The United States during the nuclear nuclear tests exploded about 9,000 kilograms worth of material, which is the size of like, you know
01:00:51
Speaker
So yeah, the United States did about 9,100 kilograms worth of energy equivalent explosions. And the USSR comes in number one at 13,800 kilograms worth of explosives, which is about the weight of Big Ben.
01:01:08
Speaker
and a sixth of that was Zar Bomba. Yeah, a whole sixth of that was Zar Bomba. That's incredible. And the total of all nuclear tests done has totaled up to about 25,000 kilograms worth of energy, worth of matter converted into energy, or about the weight of any computer or a 17-foot round backyard pool. Wow. Not much. Yeah, not as much as you'd think. So I guess we're lucky that backyard pools don't explode.
01:01:37
Speaker
Exactly.
01:01:40
Speaker
So how efficient are various fuel sources compared to black holes slash pure matter? So gasoline, it's about 10,000 calories per kilogram, or 45.7 megajoules, which is more than 6.276 megajoules. The amount of calories in one kilogram of gasoline is how much Michael Phelps would burn in a day of training or competition or whatever, or about 10,000 calories, which is about 4 thirds of a liter of gasoline, or about a third of a gallon of gas, which there are days that I don't burn that in my car.
01:02:09
Speaker
And so if you do the same calculation, 45.7 megajoules per kilogram divided by the speed of light squared over a kilogram is equal to 5.08 times to the negative 10th or 0.0000508 percent as efficient as pure energy.
01:02:27
Speaker
So definitely a lot more efficient, right? Yes. Now we're going to look at uranium, enriched uranium specifically. One kilogram of uranium will generate 8.64 times 10 to the 13th joules per kilogram, which would power Greenland for a month. So it would take 12 kilograms of uranium per year to power Greenland, which is not very much. So there's about somewhere around a milligram per kilogram of uranium in just the soil itself.
01:02:56
Speaker
So it would take about 12 million pounds of just random rock purified into uranium to get that, but of course there's ores that generate a lot more uranium. But yeah, there are places where you can extract uranium such as the Athabasca Basin, which contains millions and millions of pounds of uranium in the form of U-308, with the average grade of somewhere around 10 to 20%.
01:03:22
Speaker
So with this stuff, you do have a factor of 10 that you would lose in converting regular uranium to enriched uranium. So that would mean that it would only take about 1200 pounds worth of this ore to power Greenland for a year.
01:03:44
Speaker
So yeah, that just gives you an example of how efficient uranium is. A kilogram also is how many 360 people would eat in a lifetime or would power Albuquerque for six days. So 60 kilograms would power Albuquerque per year. And the efficiency compared to matter itself is about one one thousandth as efficient as matter, or 0.096%, which is very, very efficient. I mean, if you look at all these other things,
01:04:13
Speaker
I mean, this is one in, you know, because gasoline is what one two trillionth as efficient as pure energy. So this is one one thousandth. So it gives you an example of how much more efficient uranium is than pure matter. Oh, yeah.
Human Metabolism and Energy Balance
01:04:29
Speaker
And you could use these energy calculations, specifically with human fat in other ways, but the long and short of it is that you could treat the energy that you receive as being stored as fat. And yeah, that's honestly the whole concept of losing weight depends on this as well. Calories in, when calories out, as some people say.
01:04:56
Speaker
Energy is something that can't be created or destroyed, only moved around.
Closing Remarks and Episode Preview
01:05:00
Speaker
It's also something that's needed to make any change between two components of any system. The question then naturally becomes, what are the limitations of moving energy around? Find out more when we cover this on the next episode of Breaking Math, where we discuss thermodynamics. I'm Sophia, and this has been Breaking Math. With me, we head on, Ariana Luna Rosa. Thank you, Ariana. Thank you. And is there anything you'd like to plug, anything you want to say to our audience?
01:05:26
Speaker
No plugs, no plugs, no words of wisdom. You don't have to say no to drugs. Um, Millie, um, uh, thank you for being on the show. Um, would you like to plug anything? Yes. I would love to plug nerd forensics. Uh, anybody who's listening to breaking math right now that doesn't just spend their time listening to math podcast.
01:05:49
Speaker
If you have any interest in art, history, movies, comic books, video games, I'm sure you'll find something that's a lot of fun on Nerd Forensics. And if you don't, send me an email at nerdforensics at gmail.com. Tell me what you would like me to do an episode about. Otherwise, I just want my audience to know I love them.
01:06:11
Speaker
Sounds great. And yeah, you could check that out anywhere you if you got this podcast you can find another podcast I don't know how to end this episode. I just looking at the cats over there