Fly High, Silver Bird

“TANSTAAFL!” Vinnie’s still unhappy with spacecraft that aren’t rocket-powered. “There Ain’t No Such Thing As A Free Lunch!”

“Ah, good, you’ve read Heinlein. So what’s your problem with Lightsail 2?”

“It can’t work, Sy. Mostly it can’t work. Sails operate fine where there’s air and wind, but there’s none of that in space, just solar wind which if I remember right is just barely not a vacuum.”

Astronomer-in-training Jim speaks up. “You’re right about that, Vinnie. The solar wind’s fast, on the order of a million miles per hour, but it’s only about 10-14 atmospheres. That thin, it’s probably not a significant power source for your sailcraft, Al.”

“I keep telling you folks, it’s not wind-powered, it’s light-powered. There’s oodles of sunlight photons out there!”

“Sure, Al, but photons got zero mass. No mass, no momentum, right?”

Plane-polarized electromagnetic wave in motion
Plane-polarized electromagnetic wave
Electric (E) field is red
Magnetic (B) field is blue
(Image by Loo Kang Wee and Fu-Kwun Hwang from Wikimedia Commons)

My cue to enter. “Not right, Vinnie. Experimental demonstrations going back more than a century show light exerting pressure. That implies non-zero momentum. On the theory side … you remember when we talked about light waves and the right-hand rule?”

“That was a long time ago, Sy. Remind me.”

“… Ah, I still have the diagram on Old Reliable. See here? The light wave is coming out of the screen and its electric field moves electrons vertically. Meanwhile, the magnetic field perpendicular to the electric field twists moving charges to scoot them along a helical path. So there’s your momentum, in the interaction between the two fields. The wave’s combined action delivers force to whatever it hits, giving it momentum in the wave’s direction of travel. No photons in this picture.”

Astrophysicist-in-training Newt Barnes dives in. “When you think photons and electrons, Vinnie, think Einstein. His Nobel prize was for his explanation of the photoelectric effect. Think about some really high-speed particle flying through space. I’m watching it from Earth and you’re watching it from a spaceship moving along with it so we’ve each got our own frame of reference.”

“Frames, awright! Sy and me, we’ve talked about them a lot. When you say ‘high-speed’ you’re talking near light-speed, right?”

“Of course, because that’s when relativity gets significant. If we each measure the particle’s speed, do we get the same answer?”

“Nope, because you on Earth would see me and the particle moving through compressed space and dilated time so the speed I’d measure would be more than the speed you’d measure.”

“Mm-hm. And using ENewton=mv² you’d assign it a larger energy than I would. We need a relativistic version of Newton’s formula. Einstein said that rest mass is what it is, independent of the observer’s frame, and we should calculate energy from EEinstein²=(pc)²+(mc²)², where p is the momentum. If the momentum is zero because the velocity is zero, we get the familiar EEinstein=mc² equation.”

“I see where you’re going, Newt. If you got no mass OR energy then you got nothing at all. But if something’s got zero mass but non-zero energy like a photon does, then it’s got to have momentum from p=EEinstein/c.”

“You got it, Vinnie. So either way you look at it, wave or particle, light carries momentum and can power Lightsail 2.”

Adapted from image by Josh Spradling / The Planetary Society

“Question is, can sunlight give it enough momentum to get anywhere?”

“Now you’re getting quantitative. Sy, start up Old Reliable again.”

“OK, Newt, now what?”

“How much power can Lightsail 2 harvest from the Sun? That’ll be the solar constant in joules per second per square meter, times the sail’s area, 32 square meters, times a 90% efficiency factor.”

“Got it — 39.2 kilojoules per second.”

“That’s the supply, now for the demand. Lightsail 2 masses 5 kilograms and starts at 720 kilometers up. Ask Old Reliable to use the standard circular orbit equations to see how long it would take to harvest enough energy to raise the craft to another orbit 200 kilometers higher.”

“Combining potential and kinetic energies, I get 3.85 megajoules between orbits. That’s only 98 seconds-worth. I’m ignoring atmospheric drag and such, but net-net, Lightsail 2‘s got joules to burn.”

“Case closed, Vinnie.”

~~ Rich Olcott

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The Big Chill

Jeremy gets as far as my office door, then turns back. “Wait, Mr Moire, that was only half my question. OK, I get that when you squeeze on a gas, the outermost molecules pick up kinetic energy from the wall moving in and that heats up the gas because temperature measures average kinetic energy. But what about expansion cooling? Those mist sprayers they set up at the park, they don’t have a moving outer wall but the air around them sure is nice and cool on a hot day.”

“Another classic Jeremy question, so many things packed together — Gas Law, molecular energetics, phase change. One at a time. Gas Law’s not much help, is it?”

“Mmm, guess not. Temperature measures average kinetic energy and the Gas Law equation P·V = n·R·T gives the total kinetic energy for the n amount of gas. Cooling the gas decreases T which should reduce P·V. You can lower the pressure but if the volume expands to compensate you don’t get anywhere. You’ve got to suck energy out of there somehow.”

Illustrations adapted from drawings by Trianna

“The Laws of Thermodynamics say you can’t ‘suck’ heat energy out of anything unless you’ve got a good place to put the heat. The rule is, heat energy travels voluntarily only from warm to cold.”

“But, but, refrigerators and air conditioners do their job! Are they cheating?”

“No, they’re the products of phase change and ingenuity. We need to get down to the molecular level for that. Think back to our helium-filled Mylar balloon, but this time we lower the outside pressure and the plastic moves outward at speed w. Helium atoms hit the membrane at speed v but they’re traveling at only (v-w) when they bounce back into the bulk gas. Each collision reduces the atom’s kinetic energy from ½m·v² down to ½m·(v-w)². Temperature goes down, right?”

“That’s just the backwards of compression heating. The compression energy came from outside, so I suppose the expansion energy goes to the outside?”

“Well done. So there has to be something outside that can accept that heat energy. By the rules of Thermodynamics, that something has to be colder than the balloon.”

“Seriously? Then how do they get those microdegree above absolute zero temperatures in the labs? Do they already have an absolute-zero thingy they can dump the heat to?”

“Nope, they get tricky. Suppose a gas in a researcher’s container has a certain temperature. You can work that back to average molecular speed. Would you expect all the molecules to travel at exactly that speed?”

“No, some of them will go faster and some will go slower.”

“Sure. Now suppose the researcher uses laser technology to remove all the fast-moving molecules but leave the slower ones behind. What happens to the average?”

“Goes down, of course. Oh, I see what they did there. Instead of the membrane transmitting the heat away, ejected molecules carry it away.”

“Yup, and that’s the key to many cooling techniques. Those cooling sprays, for instance, but a question first — which has more kinetic energy, a water droplet or the droplet’s molecules when they’re floating around separately as water vapor?”

“Lessee… the droplet has more mass, wait, the molecules total up to the same mass so that’s not the difference, so it’s droplet velocity squared versus lots of little velocity-squareds … I’ll bet on the droplet.”

“Sorry, trick question. I left out something important — the heat of vaporization. Water molecules hold pretty tight to each other, more tightly in fact than most other molecular substances. You have to give each molecule a kick to get it away from its buddies. That kick comes from other molecules’ kinetic energy, right? Oh, and one more thing — the smaller the droplet, the easier for a molecule to escape.”

“Ah, I see where this is going. The mist sprayer’s teeny droplets evaporate easy. The droplets are at air temperature, so when a molecule breaks free some neighbor’s kinetic energy becomes what you’d expect from air temperature, minus break-free energy. That lowers the average for the nearby air molecules. They slow their neighbors. Everything cools down. So that’s how sprays and refrigerators and such work?”

“That’s the basic principle.”

“Cool.”

~ Rich Olcott

Thanks to Mitch Slevc for the question that led to this post.

The Hot Squeeze

A young man’s knock, eager yet a bit hesitant.

“C’mon in, Jeremy, the door’s open.”

“Hi, Mr Moire. How’s your Summer so far? I got an ‘A’ on that black hole paper, thanks to your help. Do you have time to answer a question now that Spring term’s over?”

“Hi, Jeremy. Pretty good, congratulations, and a little. What’s your question?”

“I don’t understand about the gas laws. You squeeze a gas, you raise its temperature, but temperature’s the average kinetic energy of the molecules which is mass times velocity squared but mass doesn’t change so how does the velocity know how big the volume is? And if you let a gas expand it cools and how does that happen?”

“A classic Jeremy question. Let’s take it a step at a time, big-picture view first. The Gas Law says pressure times volume is proportional to the amount of gas times the temperature, or P·V = n·R·T where n measures the amount of gas and R takes care of proportionality and unit conversions. Suppose a kid gets on an airplane with a balloon. The plane starts at sea level pressure but at cruising altitude they maintain cabins at 3/4 of that. Everything stays at room temperature, so the balloon expands by a third –“

Kid drawing of an airplane with a red balloon
Adapted from a drawing by Xander

“Wait … oh, pressure down by 3/4, volume up by 4/3 because temperature and n and R don’t change. OK, I’m with you. Now what?”

“Now the plane lands at some warm beach resort. We’re back at sea level but the temp has gone from 68°F back home to a basky 95°F. How big is the balloon? I’ll make it easy for you — 68°F is 20°C is 293K and 95°F is 35°C is 308K.”

“Volume goes up by 308/293. That’s a change of 15 in about 300, 5% bigger than back home.”

“Nice estimating. One more stop on the way to the molecular level. Were you in the crowd at Change-me Charlie’s dark matter debate?”

“Yeah, but I didn’t get close to the table.”

“Always a good tactic. So you heard the part about pressure being a measure of energy per unit of enclosed volume. What does that make each side of the Gas Law equation?”

“Umm, P·V is energy per volume, times volume, so it’s the energy inside the balloon. Oh! That’s equal to n·R·T but R‘s a constant and n measures the number of molecules so T = P·V/n·R makes T proportional to average kinetic energy. But I still don’t see why the molecules speed up when you squeeze on them. That just packs the same molecules into a smaller volume.”

“You’re muddling cause and effect. Let’s try to tease them apart. What forces determine the size of the balloon?”

“I guess the balance between the outside pressure pushing in, versus the inside molecules pushing out by banging against the skin. Increasing their temperature means they have more energy so they must bang harder.”

“And that increases the outward pressure and the balloon expands until things get back into balance. Fine, but think about individual molecules, and let’s pretend that we’ve got a perfect gas and a perfect balloon membrane — no leaks and no sticky collisions. A helium-filled Mylar balloon is pretty close to that. When things are in balance, molecules headed outward approach the membrane with some velocity v and bounce back inward with the same velocity v though in a different direction. Their kinetic energy before hitting the membrane is ½m·v²; after the collision the energy’s also ½m·v² so the temperature is stable.”

“But that’s at equilibrium.”

“Right, so let’s increase the outside pressure to squeeze the balloon. The membrane closes in at some speed w. Out-bound molecules approach the membrane with velocity v just as before but the membrane’s speed boosts the bounce. The ‘before’ kinetic energy is still ½m·v² but the ‘after’ value is bigger: ½m·(v+w)². The total and average kinetic energy go up with each collision. The temperature boost comes from the energy we put into the squeezing.”

“So the heating actually happens out at the edges.”

“Yup, the molecules in the middle don’t know about it until hotter molecules collide with them.”

“The last to learn, eh?.”

“Always the case.”

~~ Rich Olcott

Thanks to Mitch Slevc for the question that led to this post.

Seesaw to The Stars

I look around the playground. “Where’s the seesaw, Teena?”

“They took it away. That’s good ’cause I hated that thing!”

“Why’s that, Sweetie?”

“I never could play right on it. Almost never. Sometimes there’d be a kid my size on the other end and that worked OK, but a lot of times a big kid got on the other end and bounced me up in the air. The first time I even fell off and they laughed.”

“Well, I can understand that. I’m sure you’ve been nicer than that to the littler kids.”

“Uh-huh, except for Bratty Brian, but he liked it when I bounced him. He called it ‘going to the Moon’.”

“I can understand that, too. If things go just right you come off your seat and float like an astronaut for a moment. I bet he held onto the handles tight.”

“Yeah, I just wasn’t ready for it the first time.”

“Y’know, there’s another way that Brian’s bounces were like a rocket trip to somewhere. They went through the same phases of acceleration and deceleration.”

“Uncle Sy, you know you’re not allowed to use words like that around me without ‘splaining them.”

“Mmm, they both have to do with changing speed. Suppose you’re standing still. Your speed is zero, right? When you start moving your speed isn’t zero any more and we say you’ve accelerated. When you slow down again we say you’re decelerating. Make sense?”

“So when Bratty Brian gets on the low end of the seesaw he’s zero. When I squinch down at my end he accelerates –“

“Right, that’s like the boost phase of a rocket trip.”

“… And when he’s floating at the very top –“

“Like astronauts when they’re coasting, sort of but not really.”

“… And then they decelerate when they land. Bratty Brian did, too. I guess deceleration is like acceleration backwards. But why such fancy words?”

“No-one paid much attention to acceleration until Mr Newton did. He changed Physics forever when he said that all accelerations involve a force of some kind. That thought led him to the whole idea of gravity as a force. Ever since then, when physicists see something being accelerated they look for the force that caused it and then they look for what generated the force. That’s how we learned about electromagnetism and the forces that hold atoms together and even dark matter which is ultra-mysterious.”

“Ooo, I love mysteries! What did Mr Newton tell us about this one?”

“Nothing, directly, but his laws gave us a clue about what to look for. Tell me what forces were in play during Brian’s ‘moon flight’.”

“Let’s see. He accelerated up and then he accelerated down. I guess while he was on the seesaw seat at the beginning the up-acceleration came from an up-force from his end of the board. And the down-acceleration came from gravity’s force. But the gravity force is there all along, isn’t it?”

“Good point. What made the difference is that your initial force was greater than gravity’s so Brian went up. When your force stopped, gravity’s force was all that mattered so Brian came back down again.”

“So it’s like a tug-of-war, first I won then gravity won.”

“Exactly. Now how about the forces when you were on the merry-go-round?”

“OK. Gravity’s always there so it was pulling down on me. The merry-go-round was pushing up?”

“Absolutely. A lot of people think that’s weird, but whatever we stand on pushes up exactly as hard as gravity pulls us down. Otherwise we’d sink into the ground or fly off into space. What about other forces?”

“Oh, yeah, Mr Newton’s outward force pushed me off until … holding the handles made the inward force to keep me on!”

“Nice job! Now think about a galaxy, millions of stars orbiting around like on a merry-go-round. They feel an outward force like you did, and they feel an inward force from gravity so they all stay together instead of flying apart. But…”

“But?”

“Mr Newton’s rules tell us how much gravity the stars need to stay together. The astronomers tell us that there aren’t enough stars to make that much gravity. Dark matter supplies the extra.”

~~ Rich Olcott

Conversation of Energy

Teena’s next dash is for the slide, the high one, of course. “Ha-ha, Uncle Sy, beat you here. Look at me climbing up and getting potential energy!”

“You certainly did and you certainly are.”

“Now I’m sliding down all kinetic energy, wheee!” <thump, followed by thoughtful pause> “Uncle Sy, I’m all mixed up. You said momentum and energy are like cousins and we can’t create or destroy either one but I just started momentum coming down and then it stopped and where did my kinetic energy go? Did I break Mr Newton’s rule?”

“My goodness, those are good questions. They had physicists stumped for hundreds of years. You didn’t break Mr Newton’s Conservation of Momentum rule, you just did something his rule doesn’t cover. I did say there are important exceptions, remember.”

“Yeah, but you didn’t say what they are.”

“And you want to know, eh? Mmm, one exception is that the objects have to be big enough to see. Really tiny things follow quantum rules that have something like momentum but it’s different. Uhh, another exception is the objects can’t be moving too fast, like near the speed of light. But for us the most important exception is that the rule only applies when all the energy to make things move comes from objects that are already moving.”

“Like my marbles banging into each other on the floor?”

“An excellent example. Mr Newton was starting a new way of doing science. He had to work with very simple systems and and so his rules were very simple. One Sun and one planet, or one or two marbles rolling on a flat floor. His rules were all about forces and momentum, which is a combination of mass and speed. He said the only way to change something’s momentum was to push it with a force. Suppose when you push on a marble it goes a foot in one second and has a certain momentum. If you push it twice as hard it goes two feet in one second and has twice the momentum.”

“What if I’ve got a bigger marble?”

“If you have a marble that’s twice as heavy and you give it the one-foot-per-second speed, it has twice the momentum. Once there’s a certain amount of momentum in one of Mr Newton’s simple systems, that’s that.”

“Oh, that’s why I’ve got to snap my steelie harder than the glass marbles ’cause it’s heavier. Oh!Oh!And when it hits a glass one, that goes faster than the steelie did ’cause it’s lighter but it gets the momentum that the steelie had.”

“Perfect. You Mommie will be so proud of you for that thinking.”

“Yay! So how are momentum and energy cousins?”

“Cous… Oh. What I said was they’re related. Both momentum and kinetic energy depend on both mass and speed, but in different ways. If you double something’s speed you give it twice the momentum but four times the amount of kinetic energy. The thing is, there’s only a few kinds of momentum but there are lots of kinds of energy. Mr Newton’s Conservation of Momentum rule is limited to only certain situations but the Conservation of Energy rule works everywhere.”

“Energy is bigger than momentum?”

“That’s one way of putting it. Let’s say the idea of energy is bigger. You can get electrical energy from generators or batteries, chemical energy from your muscles, gravitational energy from, um, gravity –“

“Atomic energy from atoms, wind energy from the wind, solar energy from the Sun –“

“Cloud energy from clouds –“

“Wait, what?”

“Just kidding. The point is that energy comes in many varieties and they can be converted into one another and the total amount of energy never changes.”

“Then what happened to my kinetic energy coming down the slide? I didn’t give energy to anything else to make it start moving.”

“Didn’t you notice the seat of your pants getting hotter while you were slowing down? Heat is energy, too — atoms and molecules just bouncing around in place. In fact, one of the really good rules is that sooner or later, every kind of energy turns into heat.”

“Big me moving little atoms around?”

“Lots and lots of them.”

~~ Rich Olcott

Conversation of Momentum

Teena bounces out of the sandbox, races over to the playground’s little merry-go-round and shoves it into motion. “Come help turn this, Uncle Sy, I wanna go fast!” She leaps onto the moving wheel and of course she promptly falls off. The good news is that she rolls with the fall like I taught her to do.

“Why can’t I stay on, Uncle Sy?”

“What’s your new favorite word again?”

“Mmmo-MMENN-tumm. But that had to do with swings.”

“Swings and lots of other stuff, including merry-go-rounds and even why you should roll with the fall. Which, by the way, you did very well and I’m glad about that because we don’t want you getting hurt on the playground.”

“Well, it does hurt a little on my elbow, see?”

“Let me look … ah, no bleeding, things only bend where they’re supposed to … I think no damage done but you can ask your Mommie to kiss it if it still hurts when we get home. But you wanted to know why you fell off so let’s go back to the sandbox to figure that out.”

<scamper!> “I beat you here!”

“Of course you did. OK, let’s draw a big arc and pretend that’s looking down on part of the merry-go-round. I’ll add some lines for the spokes and handles. Now I’ll add some dots and arrows to show what I saw from over here. See, the merry-go-round is turning like this curvy arrow shows. You started at this dot and jumped onto this dot which moved along and then you fell off over here. Poor Teena. So you and your momentum mostly went left-to-right.”

“But that’s not what happened, Uncle Sy. Here, I’ll draw it. I jumped on but something tried to push me off and then I did fall off and then I rolled. Poor me. Hey, my arm doesn’t hurt any more!”

“How about that? I’ve often found that thinking about something else makes hurts go away. So what do you think was trying to push you off? I’ll give you a hint with these extra arrows on the arc.”

“That looks like Mr Newton’s new directions, the in-and-out direction and the going-around one. Oh! I fell off along the in-and-out direction! Like I was a planet and the Sun wasn’t holding me in my orbit! Is that what happened, I had out-momentum?”

“Good thinking, Teena. Mr Newton would say that you got that momentum from a force in the out-direction. He’d also say that if you want to stand steady you need all the forces around you to balance each other. What does that tell you about what you need to do to stay on the merry-go-round?”

“I need an in-direction force … Hah, that’s what I did wrong! I jumped on but I didn’t grab the handles.”

“Lesson learned. Good.”

“But what about the rolling?”

“Well, in general when you fall it’s nearly always good to roll the way your body’s spinning and only try to slow it down. People who put out an arm or leg to stop a fall often stress it and and maybe even tear or break something.”

“That’s what you’ve told me. But what made me spin?”

“One of Mr Newton’s basic principles was a rule called ‘Conservation of Momentum.’ It says that you can transfer momentum from one thing to another but you can’t create it or destroy it. There are some important exceptions but it’s a pretty good rule for the cases he studied. Your adventure was one of them. Look back at the picture I drew. You’d built up a lot of going-around momentum from pushing the merry-go-round to get it started. You still had momentum in that direction when you fell off. Sure enough, that’s the direction you rolled.”

“Is that the ‘Conversation of Energy’ thing that you and Mommie were talking about?”

“Conservation. It’s not the same but it’s closely related.”

“Why does it even work?”

“Ah, that’s such a deep question that most physicists don’t even think about it. Like gravity, Mr Newton described what inertia and momentum do, but not how they work. Einstein explained gravity, but I’m not convinced that we understand mass yet.”

~~ Rich Olcott

A Momentous Occasion

<creak> Teena’s enjoying her new-found power in the swings. “Hey, Uncle Sy? <creak> Why doesn’t the Earth fall into the Sun?”

“What in the world got you thinking about that on such a lovely day?”

“The Sun gets in my eyes when I swing forward <creak> and that reminded me of the time we saw the eclipse <creak> and that reminded of how the planets and moons are all floating in space <creak> and the Sun’s gravity’s holding them together but if <creak> the Sun’s pulling on us why don’t we just fall in?” <creak>

“An excellent question, young lady. Isaac Newton thought about it long and hard back when he was inventing Physics.”

“Isaac Newton? Is he the one with all the hair and a long, skinny nose and William Tell shot an arrow off his head?”

“Well, you’ve described his picture, but you’ve mixed up two different stories. William Tell’s apple story was hundreds of years before Newton. Isaac’s apple story had the fruit falling onto his head, not being shot off of it. That apple got him thinking about gravity and how Earth’s gravity pulling on the apple was like the Sun’s gravity pulling on the planets. When he was done explaining planet orbits, he’d also explained how your swing works.”

“My swing works like a planet? No, my swing goes back and forth, but planets go round and round.”

“Jump down and we can draw pictures over there in the sandbox.”

<thump!! scamper!> “I beat you here!”

“Of course you did. OK, what’s your new M-word?”

“Mmmo-MMENN-tummm!”

“Right. Mr Newton’s Law of Inertia is about momentum. It says that things go in a straight line unless something interferes. It’s momentum that keeps your swing going.”

“B-u-u-t, I wasn’t going in a straight line, I was going in part of a circle.”

“Good observing, Teena, that’s exactly right. Mr Newton’s trick was that a really small piece of a circle looks like a straight line. Look here. I’ll draw a circle … and inside it I’ll put a triangle… and between them I’ll put a hexagon — see how it has an extra point halfway between each of the triangle’s points? — and up top I’ll put the top part of whatever has 12 sides. See how the 12-thing’s sides are almost on the circle?”

“Ooo, that’s pretty! Can we do that with a square, too?”

“Sure. Here’s the circle … and the square … and an octagon … and a 16-thing. See, that’s even closer to being a circle.”

“Ha-ha — ‘octagon’ — that’s like ‘octopus’.”

“For good reason. An octopus has eight arms and an octagon has eight sides. ‘Octo-‘ means ‘eight.’ So anyway, Mr Newton realized that his momentum law would apply to something moving along that tiny straight line on a circle. But then he had another idea — you can move in two directions at once so you can have momentum in two directions at once.”

“That’s silly, Uncle Sy. There’s only one of me so I can’t move in two directions at once.”

“Can you move North?”

“Uh-huh.”

“Can you move East?”

“Sure.”

“Can you move Northeast?”

“Oh … does that count as two?”

“It can for some situations, like planets in orbit or you swinging on a swing. You move side-to-side and up-and-down at the same time, right?”

“Uh-huh.”

“When you’re at either end of the trip and as far up as you can get, you stop for that little moment and you have no momentum. When you’re at the bottom, you’ve got a lot of side-to-side momentum across the ground. Anywhere in between, you’ve got up-down momentum and side-to-side momentum. One kind turns into the other and back again.”

“So complicated.”

“Well, it is. Newton simplified things with revised directions — one’s in-or-out from the center, the other’s the going-around angle. Each has its own momentum. The swing’s ropes don’t change length so your in-out momentum is always zero. Your angle-momentum is what keeps you going past your swing’s bottom point. Planets don’t have much in-out momentum, either — they stay about their favorite distance from the Sun.”

“Earth’s angle-momentum is why we don’t fall in?”

“Yep, we’ve got so much that we’re always falling past the Sun.”

~~ Rich Olcott

Swinging into Physics

A gorgeous Spring day, perfect for taking my 7-year-old niece to the park. We politely say “Hello” to the geese and then head to the playground. Of course she runs straight to the swing set. “Help me onto the high one, Uncle Sy!”

“Why that one, Teena? Your feet won’t reach the ground and you won’t be able to kick the ground to get going.”

“The high one goes faster,”

“How do you know that?”

“I saw some kids have races and the kid on the high swing always did more back-and-forths. Sometimes it was a big kid, sometimes a little kid but they always went faster.”

“Good observing, Sweetie. OK, upsy-daisy — there you are.”

“Now give me pushes.”

“I’m not doing all the work. Tell you what, I’ll give you a start-up shove and then you pump to keep swinging.”

“But I don’t know how!”

“When you’re going forward, lean way back and put your feet up as high as you can. Then when you’re going backward, do the opposite — lean forward and bend your knees way back. Now <hnnnhh!> try it.

<creak … creak> “Hey, I’m doing it! Wheee!”

<creak> “Good job, you’re an expert now.”

“How’s it work, Uncle Sy?”

“It’s a dance between kinetic energy, potential energy and momentum.”

“I’m just a little kid, Uncle Sy, I don’t know what any of those things are.”

“Mmm… Energy is what makes things move or change. You know your toy robot? What happens when its batteries run down?”

“It stops working, silly, until Mommie puts its battery in the charger overnight and then it works again.”

“Right. Your robot needs energy to move. The charger stores energy in the battery. Stored energy is called potential which is like ‘maybe,’ because it’s not actually making something happen. When the robot gets its full-up battery back and you press its GO button, the robot can move around and that’s kinetic energy. ‘Kinetic’ is another word for ‘moving.'”

“So when I’m running around that’s kinetic energy and when I get tired and fall asleep I’m recharging my potential energy?”

“Exactly. You’re almost as smart as your Mommie.”

“An’ when I’m on the swing and it’s moving, that’s kinetic.”

“You’ve got part of it. Watch what’s happening while you swing. Are you always moving?”

<creak … creak> “Ye-e—no! Between when I swing up and when I come down, I stop for just a teeny moment at the top. And I stop again between backing up and going forward. Is that when I’m potential?”

“Sort of, except it’s not you, it’s your swinging-energy that’s all potential at the top. Away from the top you turn potential energy into kinetic energy, going faster and faster until you’re at the bottom. That’s when you go fastest because all your potential energy has become kinetic energy. As you move up from the bottom you slow down because you’re turning your kinetic energy back into potential energy.”

<creak> “Back and forth, potential to kinetic to potential, <creak> over and over. Wheee! Mommie would say I’m recycling!”

“Yes, she would.”

<creak> “Hey, Uncle Sy, how come I don’t stop at the bottom when I’m all out of potential?”

“Ah. What’s your favorite kind of word?”

M-words! I love M-words! Like ‘murmuration‘ and ‘marbles.'”

“Well, I’ve got another one for you — momentum.”

“Oh, that’s yummy — mmmo-MMMENN-tummmm. What’s it mean?”

“It’s about how things that are moving in a straight line keep moving along that line unless something else interferes. Or something that’s standing still will just stay there until something gives it momentum. When we first sat you in the swing you didn’t go anywhere, did you?”

“No, ’cause my toes don’t reach down to the ground and I can’t kick to get myself started.”

“That would have been one way to get some momentum going. When I gave you that push, that’s another way.”

“Or I could wear a jet-pack like Tony Stark. Boy, that’d give me a LOT of momentum!”

“Way too much. You’d wrap the swing ropes round the bar and you’d be stuck up there. Anyway, when you swing past the bottom, momentum is what keeps you going upward.”

“Yay, momentum!” <creak>

~~ Rich Olcott

The Pretty-good Twenty-nine

Time for coffee and a scone. As I step into Al’s coffee shop he’s taking his Jupiter poster down from behind the cash register.

“Hey, Al, I liked that poster. You decide you prefer plain wall?”

“Nah, Sy, I got a new one here. Help me get it up over the hook.”

A voice from behind us. “Ya got it two degrees outta plumb, clockwise.” Vinnie, of course. Al taps the frame to true it up.

Teachers, click here to download a large-format printable copy.

“Hey, Sy, in the middle, that’s the same seven units we just finished talking about — amps for electric current, kelvins for temperature, meters for length, kilograms for mass, seconds for time, moles for counting atoms and such, and that candela one you don’t like. What’s all the other bubbles about? For that matter, what’s the poster about, Al?”

“What it’s about, Vinnie, is on May 20 the whole world goes to a new set of measurement standards, thanks to some international bureau.”

Le Bureau International des Poids et Mesures.” It’s Newt Barnes in from the Physics building. “The bubbles in that central ring are the BIPM’s selections for fundamental standards. Each one’s fixed by precisely defined values of one or more universal physical constants. For instance, a ruler calibrated on Earth will match up perfectly with one calibrated on Mars because both calibrations depend on the wavelength of radiation from a cesium-based laser and that’s the same everywhere.”

“How about the other bubbles and the rings around them?”

“They’re all derived quantities, simple combinations of the fundamental standards.”

“Hey, I see one I recognize. That °C has gotta be degrees centigrade ’cause it’s right next to kelvins. Centigrade’s the same as kelvins plus , uh, 273?”

“There you go, Al. What’s ‘rad’ and ‘sr’, Newt?”

“Symbols for radian and steradian, Vinnie. They both measure angles like degrees do, but they fit the BIPM model because they’re ratios of lengths and length is one of the fundamentals. Divide a circle’s circumference by its radius and what do you get?”

“Twice pi.”

“Right, call it 2π radians and that’s a full circle. Half a circle is π radians, a right angle is π/2 radians and so on. Works for any size circle, right? Anyone remember the formula for the area of a sphere?”

“4πr2, right?”

“Exactly. If you divide any sphere’s area by the square of its radius you get 4π steradians. Any hemisphere is 2π steradians and so on. Steradians are handy for figuring things like light and gravity that decrease as the square of the distance.”

Something occurs to me. “I’m looking at those bigger bubbles that enclose the derived quantities. Seems to me that each one covers a major area of physical science. The green one with newtons for force, pascals for pressure, joules for energy and watts for power — that’d be Newtonian physics. The red circle with volts plus coulombs for charge, ohms for resistance, farads for capacitance, siemens for electrical conductance — all that’s electronics. Add in henries for inductance, webers for magnetic flux and teslas for flux density and you’ve got Maxwellian electromagnetism.”

“You’re on to something, Sy. Chemistry’s there with moles and katals, also known as moles per second, for catalytic activity. How does your idea fit the cluster attached to seconds?”

“They’re all per-second rates, Newt. The hertz is waves per second for periodic things like sound or light-as-a-wave. The other three are about radioactivity — bequerels is fissions per second; grays and sieverts are measures of radiation exposure per kilogram.”

“Vinnie says you don’t like candelas, so you probably don’t like lumens or luxes either. What’s your gripe with them?”

“All three are supposed to quantify visible light from a source, as opposed to the total emission at all wavelengths. But the definition of ‘visible’ zeros in on one wavelength in the green because that’s where most people are most sensitive. Candelas aren’t valid for a person who’s color-blind in the green, nor for something like a red laser that has no green lightwaves. I call bogosity, and lumens and luxes are both candela-based.”

“These 29 standards are as good on Mars as they are here on Earth?”

“That’s the plan.”

~~ Rich Olcott

The Magnificent Seven

“Hey, Sy, you said there’s seven fundamental standards. We’ve talked about the second and the meter and the kilogram and the ampere. What’s left?”

“The mole, the kelvin and the candela, Al. They’re all kinda special-purpose but each has its charms. The mole, for instance, is cute and fuzzy and has its very own calendar date.”

“You’re pulling our legs, Sy. A cute unit of measure? No way.”

“Hear me out, Vinnie. How many shoes in a dozen pairs?”

“Huh? Two dozen, that’s twenty-four.”

“Sure, but it’s easier to work in dozens. How many hydrogen atoms in a dozen H2O molecules?”

“Two dozen, of course. Are we going somewhere here?”

“Next step. A mole is like a dozen on steroids, about 6×1023 whatevers. How many hydrogen atoms in a mole of H2O molecules?”

“Two moles, I suppose, or 12×1023.”

“You got the idea.”

“Cute.”

“A-hah! Gotcha for one.”

“Fair hit. How about the fuzzy part and the date?”

“The fuzzy has to do with isotopes. Every element has an atomic number and an atomic weight. The atomic number counts protons in the nucleus –all atoms of an element have the same atomic number. But different isotopes of an element have different numbers of neutrons. The ‘weight’ is protons plus neutrons, averaged across the isotopes. If you’re holding a mole of an element, you’re holding its atomic weight in grams. The fuzzy happens because samples of an element from different sources can have different mixtures of isotopes. You may have some special diamonds that contain nothing but carbon-12. A mole of those atoms masses exactly 12 grams. My sample is enriched with 10% of carbon-13. Mole-for-mole, my carbon is a tad heavier than yours. In fact, 6×1023 of my atoms mass 12.10 grams. That’s an extreme example but you get the idea.”

“Fuzzy, a little, OK. And the date thing?”

“June 23 is Mole Day, celebrated by Chemistry teachers everywhere.”

“What’s the kelvin about then?”

“Temperature. And most solid-state electronics. Zero kelvin is absolute zero, the coldest temperature something can get, when the maximum heat has been sucked out and all its atoms have minimum vibrational energy. From there you heat it up degree by degree until you get to where water can co-exist as liquid, solid and water vapor. It used to be the standard to call that temperature 273.16 K.”

“Used to be? Water doesn’t do that any more?”

“Oh, it still does, but the old standard had problems. It used five different ‘official’ techniques and 16 different calibration checks to cover the range from 3 K up to the melting point of copper. Some of those standards, like the melting pressure of helium-3, are not only inconvenient but expensive. Others led to measured intermediate temperatures that disagree depending on which direction you’re going. The defined standards did nothing for the plasma people who work above 1500 K. It was a mess.”

“So how does the new standard fix that?”

“It exploits new tech, especially in solid-state science. The Boltzmann constant, kB, is sort of the quantum of heat capacity at the microscopic level. The product kBT is a threshold energy. Practically everything that happens at the quantum level depends on the ratio of some process energy divided by kBT. If the ratio’s high the process runs; if it’s low, nothing. In-between, the response is predictably temperature-dependent. Thanks to a plethora of new solid-state thermal sensors that depend on that logic, we now have a handle on the range from microkelvins to kilokelvins and above.”

“Pretty good. What’s the last one?”

“It’s the one I’m least happy about, the candela. It’s a unit for how bright a light source is, sort-of. Take the source’s power output at all optical frequencies and ‘correct’ that by how much each frequency would stimulate a mathematically modeled ‘standard human eye.’ Isolate the ‘corrected’ watts at 555 nanometers, multiply by Kcd=683. It’s a time-hallowed metric that lighting designers depend upon, but it skips over little things like we actually see with rod cells and three kinds of cone cells, none of which match the standard curve. Kcd is just too human-centered to be a universal constant.”

“Humans ain’t universal. We’re not even on Mars.”

“Yet.”

~~ Rich Olcott