# The Frame Game

A familiar footstep outside my office, “C’mon in, Vinnie, the door’s open.”

“Hi, Sy, how ya doin’?”

“Can’t complain. Yourself?”

“Fine, fine. Hey, I been thinking about something you said while Al and us were talking about rockets and orbits and such. You remember that?”

“We’ve done that in quantity. What statement in particular?”

“It was about when you’re in the ISS, you still see like 88% of Earth’s gravity. But I seen video of those astronauts just floating around in the station. Seems to me those two don’t add up.”

“Hah! We’re talking physics of motion here. What’s the magic word?”

“You’re saying it’s frames? I thought black holes did that.”

“Black holes are an extreme example, but frame‑thinking is an essential tool in analyzing any kind of relative motion. Einstein’s famous ‘happy thought‘ about a man in a free‑falling elevator—”

“Whoa, why is that a happy thought? I been nervous about elevators ever since that time we got stuck in one.”

“At least it wasn’t falling, right? Point is, the elevator and whoever’s in it agree that Newton’s First Law of Motion is valid for everything they see in there.”

“Wait, which Law is that?”

“‘Things either don’t move or else they move at a steady pace along a straight line.’ Suppose you’re that guy—”

“I’d rather not.”

“… and the elevator is in a zero‑gravity field. You take something out of your pocket, put it the air in front of you and it stays there. You give it a tap and it floats away in a straight line. Any different behavior means that your entire frame — you, the elevator and anything else in there — is being accelerated by some force. Let’s take two possibilities. Case one, you and the elevator are resting on terra firma, tightly held by the force of gravity.”

“I like that one.”

“Case two, you and the elevator are way out in space, zero‑gravity again, but you’re in a rocket under 1-g acceleration. Einstein got happy because he realized that you’d feel the same either way. You’d have no mechanical way to distinguish between the two cases.”

“What’s that mean, mechanical?”

“It excludes sneaky ways of outside influence by magnetic fields and such. Anyhow, Einstein’s insight was key to extending Newton’s First Law to figuring acceleration for an entire frame. Like, for instance, an orbiting ISS.”

“Ah, you’re saying that floating astronauts in an 88% Earth-gravity field is fine because the ISS and the guys share the frame feeling that 88% but the guys are floating relative to that frame. But down here if we could look in there we’d see how both kinds of motion literally add up.”

“Exactly. It’s just much easier to think about only one kind at a time.”

“Wait. You said the ISS is being accelerated. I thought it’s going a steady 17500 miles an hour which it’s got to do to stay 250 miles up.”

“Is it going in a straight line?”

“Well, no, it’s going in a circle, mostly, except when it has to dodge some space junk.”

“So the First Law doesn’t apply. Acceleration is change in momentum, and the ISS momentum is constantly changing.”

“But not in a straight line. Momentum is a vector that points in a specific direction. Change the direction, you change the momentum. Newton’s Second Law links momentum change with force and acceleration. Any orbiting object undergoes angular acceleration.”

“Angular acceleration, that’s a new one. It’s degrees per second per second?”

“Yup, or radians. There’s two kinds, though — orbiting and spinning. The ISS doesn’t spin because it has to keep its solar panels facing the Sun.”

“But I’ve seen sci-fi movies set in something that spins to create artificial gravity. Like that 2001 Space Odyssey where the guy does his running exercise inside the ship.”

“Sure, and people have designed space stations that spin for the same reason. You’d have a cascade of frames — the station orbiting some planet, the station spinning, maybe even a ballerina inside doing pirouettes.”

“How do you calculate all that?”

“You don’t. You work with whichever frame is useful for what you’re trying to accomplish.”

~~ Rich Olcott

# Turn This Way to Turn That Way

“I don’t understand, Sy. I get that James Webb Space Telescope uses its reaction wheels like a ship uses a rudder to change direction by pushing against something outside. Except the rudder pushes against water but the reaction wheels push against … what, the Universe?”

“Maybe probably, Al. We simply don’t know how inertia works. Newton just took inertia as a given. His Laws of Motion say that things remain at rest or persist in linear motion unless acted upon by some force. He didn’t say why. Einstein’s General Relativity starts from his Equivalence Principle — gravitational inertia is identical to mechanical inertia. That’s held up to painstaking experimental tests, but why it works is still an open question. Einstein liked Mach’s explanation, that we experience these inertias because matter interacts somehow with the rest of the Universe. He didn’t speculate how that interaction works because he didn’t like Action At A Distance. The quantum field theory people say that everything’s part of the universal field structure, which sounds cool but doesn’t help much. String theory … ’nuff said.”

“Hey, Moire, what’s all that got to do with the reaction wheel thing? JWST can push against one all it wants but it won’t go anywhere ’cause the wheel’s inside it. What’s magic about the wheels?”

JWST doesn’t want to go anywhere else, Mr Feder. We’re happy with it being in its proper orbit, but it needs to be able to point to different angles. Reaction wheels and gyroscopes are all about angular momentum, not about the linear kind that’s involved with moving from place to place.”

“HAH! JWST is moving place to place, in that orbit! Ain’t it got linear momentum then?”

“In a limited way, pun intended. Angular momentum is linear momentum plus a radial constraint. This goes back to Newton and his Principia book. I’ve got a copy of his basic arc‑splitting diagram here in Old Reliable. The ABCDEF line is a section of some curve around point S. He treated it as a succession of short line segments ABc, BCd, CDe and so on. If JWST is at point B, for instance, Newton would say that it’s traveling with a certain linear momentum along the BCd line. However, it’s constrained to move along the arc so it winds up at D instead d. To account for the constraint Newton invented centripetal force to pull along the Sd line. He then mentally made the steps smaller and smaller until the sequence of short lines matched the curve. At the limit, a sequence of little bits of linear momentum becomes angular momentum. By the way, this step‑reduction process is at the heart of calculus. Anyway, JWST uses its reaction wheels to swing itself around, not to propel itself.”

“And we’re back to my original question, Sy. What makes that swinging happen?”

“Oh, you mean the mechanical reality. Easy, Al. Like I said, three pairs of motorized wheels are mounted on JWST‘s frame near the center of mass. Their axles are at mutual right angles. Change a wheel’s angular momentum, you get an equal opposing change to the satellite’s. Suppose the Attitude Control System wants the satellite to swing to starboard. That’d be clockwise viewed from the cold side. ACS must tell a port/starboard motor to spin its wheel faster counterclockwise. If it’s already spinning clockwise, the command would be to put on the brakes, right? Either way, JWST swings clockwise. With the forward/aft motors and the hot‑side/cold‑side motors, the ACS is equipped to get to any orientation. See how that works?”

“Hang on.” <handwaving ensues> “Yeah, I guess so.”

“Hey, Moire. What if the wheel’s already spinning at top speed in the direction the ACS wants more of?”

“Ah, that calls for a momentum dump. JWST‘s equipped with eight small rocket engines called thrusters. They convert angular momentum back to linear momentum in rocket exhaust. Suppose we need a further turn to starboard but a port/starboard wheel is nearing threshold spin rate. ACS puts the brakes on that wheel, which by itself would turn the satellite to port. However, ACS simultaneously activates selected thrusters to oppose the portward slew. Cute, huh?”

~~ Rich Olcott

Mr Feder has a snarky grin on his face and a far‑away look in his eye. “Got another one. James Webb Space Telescope flies in this big circle crosswise to the Sun‑Earth line, right? But the Earth doesn’t stand still, it goes around the Sun, right? The circle keeps JWST the same distance from the Sun in maybe January, but it’ll fly towards the Sun three months later and get flung out of position.” <grabs a paper napkin> “Lemme show you. Like this and … like this.”

“Sorry, Mr Feder, that’s not how either JWST or L2 works. The satellite’s on a 6-month orbit around L2 — spiraling, not flinging. Your thinking would be correct for a solid gyroscope but it doesn’t apply to how JWST keeps station around L2. Show him, Sy.”

“Gimme a sec with Old Reliable, Cathleen.” <tapping> “OK, here’s an animation over a few months. What happens to JWST goes back to why L2 is a special point. The five Lagrange points are all about balance. Near L2 JWST will feel gravitational pulls towards the Sun and the Earth, but their combined attraction is opposed by the centrifugal force acting to move the satellite further out. L2 is where the three balance out radially. But JWST and anything else near the extended Sun‑Earth line are affected by an additional blended force pointing toward the line itself. If you’re close to it, sideways gravitational forces from the Sun and the Earth combine to attract you back towards the line where the sideways forces balance out. Doesn’t matter whether you’re north or south, spinward or widdershins, you’ll be drawn back to the line.”

Al’s on refill patrol, eavesdropping a little of course. He gets to our table, puts down the coffee pot and pulls up a chair. “You’re talking about the JWST. Can someone answer a question for me?”

“We can try.”
”What’s the question?”
Mr Feder, not being the guy asking the question, pooches out his lower lip.

“OK, how do they get it to point in the right direction and stay there? My little backyard telescope gives me fits just centering on some star. That’s while the tripod’s standing on good, solid Earth. JWST‘s out there standing on nothing.”

JWST‘s Attitude Control System has a whole set of functions to do that. It monitors JWST‘s current orientation. It accepts targeting orders for where to point the scope. It computes scope and satellite rotations to get from here to there. Then it revises as necessary in case the first‑draft rotations would swing JWST‘s cold side into the sunlight. It picks a convenient guide star from its million‑star catalog. Finally, ACS commands its attitude control motors to swing everything into the new position. Every few milliseconds it checks the guide star’s image in a separate sensor and issues tweak commands to keep the scope in proper orientation.”

“I get the sequence, Sy, but it doesn’t answer the how. They can’t use rockets for all that maneuvering or they’d run out of fuel real fast.”

“Not to mention cluttering up the view field with exhaust gases.”

“Good point, Cathleen. You’re right, Al, they don’t use rockets, they use reaction wheels, mostly.”

“Uh-oh, didn’t broken reaction wheels kill Kepler and a few other missions?”

“That sounds familiar, Mr Feder. What’s a reaction wheel, Sy, and don’t they put JWST in jeopardy?”

“A reaction wheel is a massive doughnut that can spin at high speed, like a classical gyroscope but not on gimbals.”

“Hey, Moire, what’s a gimbal?”

“It’s a rotating frame with two pivots for something else that rotates. Two or three gimbals at mutual right angles let what’s inside orient independent of what’s outside. The difference between a classical gyroscope and a reaction wheel is that the gyroscope’s pivots rotate freely but the reaction wheel’s axis is fixed to a structure. Operationally, the difference is that you use a gyroscope’s angular inertia to detect change of orientation but you push against a reaction wheel’s angular inertia to create a change of orientation.”

Kepler‘s failing wheels used metal bearings. JWST‘s are hardened ceramic.”

<whew>

~~ Rich Olcott

# Rotation, Revolution and The Answer

“Sy, I’m startin’ to think you got nothin’. Al and me, we ask what’s pushing the Moon away from us and you give us angular momentum and energy transfers. C’mon, stop dancin’ around and tell us the answer.”

“Yeah, Sy, gravity pulls things together, right, so how come the Moon doesn’t fall right onto us?”

“Not dancing, Vinnie, just laying some groundwork for you. Newton answered Al’s question — the Moon is falling towards us, but it’s going so fast it overshoots. That’s where momentum comes in, Vinnie. Newton showed that a ball shot from a cannon files further depending on how much momentum it gets from the initial kick. If you give it enough momentum, and set your cannon high enough that the ball doesn’t hit trees or mountains, the ball falls beyond the planet and keeps on falling forever in an elliptical orbit.”

“Forever until it hits the cannon.”

“hahaha, Al. Anyway, the ball achieves orbit by converting its linear momentum to angular momentum with the help of gravity. The angular momentum pretty much defines the orbit. In Newton’s gravity‑determined universe, momentum and position together let you predict everything.”

“Linear and angular momentum work the same way?”

“Mostly. There’s only one kind of linear momentum — straight ahead — but there are two kinds of angular momentum — rotation and revolution.”

“Aw geez, there’s another pair of words I can never keep straight.”

“You and lots of people, Vinnie. They’re synonyms unless you’re talking technicalese. In Physics and Astronomy, rotation with the O gyrates around an object’s own center, like a top or a planet rotating on its axis. Revolution with the E gyrates around some external location, like the planet revolving around its sun. Does that help?”

“Cool, that may come in handy. So Newton’s cannon ball got its umm, revolution angular momentum from linear momentum so where does rotation angular momentum come from?”

“Subtle question, Vinnie, but they’re actually all just momentum. Fair warning, I’m going to avoid a few issues that’d get us too far into the relativity weeds. Let’s just say that momentum is one of those conserved quantities. You can transfer momentum from one object to another and convert between forms of momentum, but you can’t create momentum in an isolated system.”

“That sounds a lot like energy, Sy.”

“You’re right, Al, the two are closely related. Newton thought that momentum was THE conserved quantity and all motion depended on it. His arch‑enemy Leibniz said THE conserved quantity was kinetic energy, which he called vis viva. That disagreement was just one battle in the Newton‑Leibniz war. It took science 200 years to understand the momentum/kinetic energy/potential energy triad.”

“Wait, Sy, I’ve seen NASA steer a rocketship and give it a whole different momentum. I don’t see no conservation.”

“You missed an important word, Vinnie — isolated. Momentum calculations apply to mechanical systems — no inputs of mass or non‑mechanical energy. Chemical or nuclear fuels break that rule and get you into a different game.”

“Ah-hahh, so if the Earth and Moon are isolated…”

“Exactly, and you’re way ahead of me. Like we said, no significant net forces coming from the Sun or Jupiter, so no change to our angular momentum.”

“Hey, wait, guys. Solar power. I know we’ve got a ton of sunlight coming in every day.”

“Not relevant, Al. Even though sunlight heats the Earth, mass and momentum aren’t affected by temperature. Anyhow, we’re finally at the point where I can answer your question.”

“Hush. OK, here’s the chain. Earth rotates beneath the Moon and gets its insides stirred up by the Moon’s gravity. The stirring is kinetic energy extracted from the energy of the Earth‑Moon system. The Moon’s revolution or the Earth’s rotation or both must slow down. Remember the M=m·r·c/t equation for angular momentum? The Earth‑Moon system is isolated so the angular momentum M can’t change but the angular velocity c/t goes down. Something’s got to compensate. The system’s mass m doesn’t change. The only thing that can increase is distance r. There’s your answer, guys — conservation of angular momentum forces the Moon to drift outward.”

“To the Moon and back.”

~~ Rich Olcott

# Here’s a Different Angle

“OK, Sy, so there’s a bulge on the Moon’s side of the Earth and the Earth rotates but the bulge doesn’t and that makes the Moon’s orbit just a little bigger and you’ve figured out that the energy it took to lift the Moon raised Earth’s temperature by a gazillionth of a degree, I got all that, but you still haven’t told Al and me how the lifting works.”

“You wouldn’t accept it if I just said, ‘The Moon lifts itself by its bootstraps,’ would you?”

“Not for a minute.”

“And you don’t like equations. <sigh> OK, Al, pass over some of those paper napkins.”

“Aw geez, Sy.”

“You guys asked the question and this’ll take diagrams, Al. Ante up. … Thanks. OK, remember the time Cathleen and I caught Vinnie here at Al’s shop playing with a top?”

“Yeah, and he was spraying paper wads all over the place.”

“I wasn’t either, Al, it was the top sending them out with centri–…, some force I can never remember whether it’s centrifugal or centripetal.”

“Centrifugal, Vinnie, –fugal– like fugitive, outward‑escaping force. It’s one of those ‘depends on how you look at itfictitious forces. From where you were sitting, the wads looked like they were flying outward perpendicular to the top’s circle. From a wad’s point of view, it flew in a straight line tangent to the circle. It’s like we have two languages, Room and Rotor. They describe the same phenomena but from different perspectives.”

“Hey, it’s frames again, ain’t it?”

“Newton’s inertial frames? Sort‑of but not quite. Newton’s First Law only holds in the Room frame — no acceleration, motion is measured by distance, objects at rest stay put. Any other object moves in a straight line unless its momentum is changed by a force. You can tackle a problem by considering momentum and force components along separate X and Y axes. Both X and Y components work the same way — push twice as hard in either direction, get twice the acceleration in that direction. Nice rules that the Rotor frame doesn’t play by.”

“I guess not. The middle’s the only place an object can stay put, right?”

“Exactly, Al. Everything else looks like it’s affected by weird, constantly‑varying forces that’re hard to describe in X‑Y terms.”

“So that breaks Newton’s physics?”

“Of course not. We just have to adapt his F=m·a equation (sorry, Vinnie!) to Rotor conditions. For small movements we wind up with two equations. In the strict radial direction it’s still F=m·a where m is mass like we know it, a is acceleration outward or inward, and F is centrifugal or centripetal, depending. Easy. Perpendicular to ‘radial‘ we’ve got ‘angular.’ Things look different there because in that direction motion’s measured by angle but Newton’s Laws are all about distances — speed is distance per time, acceleration is speed change per time and so forth.”

“So what do you do?”

“Use arc length. Distance along an arc is proportional to the angle, and it’s also proportional to the radius of the arc, so just multiply them together.”

“What, like a 45° bend around a 2-foot radius takes 90 feet? That’s just wrong!”

“No question, Al. You have to measure the angle in the right units. Remember the formula for a circle’s circumference?”

“Sure, it’s 2πr.”

“Which tells you that a full turn’s length is times the radius. We can bridge from angle to arc length using rotational units so that a full turn, 360°, is units. We’ll call that unit a radian. Half a circle is π radians. Your 45° angle in radians is π/4 or about ¾ of a radian. You’d need about (¾)×(2) or 1½ feet of whatever to get 45° along that 2-foot arc. Make sense?”

“Gimme a sec … OK, I’m with you.”

“Great. So if angular distance is radius times angle, then angular momentum which is mass times distance per time becomes mass times radius times angle per time.”

“”Hold on, Sy … so if I double the mass I double the momentum just like always, but if something’s spinning I could also double the angular momentum by doubling the radius or spinning it twice as fast?”

“Couldn’t have put it better myself, Vinnie.”

~~ Rich Olcott

# Two Against One, And It’s Not Even Close

On a brisk walk across campus when I hear Vinnie yell from Al’s coffee shop. “Hey! Sy! Me and Al got this argument going you gotta settle.”

“Happy to be a peacemaker, but it’ll cost you a mug of Al’s coffee and a strawberry scone.”

“Coffee’s no charge, Sy, but the scone goes on Vinnie’s tab. What’s your pleasure?”

“It’s morning, Al, time for black mud. What’s the argument, Vinnie?”

“Al read in one of his astronomy magazines that the Moon’s drifting away from us. Is that true, and if it is, how’s it happen? Al thinks Jupiter’s gravity’s lifting it but I think it’s because of Solar winds pushing it. So which is it?”

“Here you go, Sy, straight from the bottom of the pot.”

“Perfect, Al, thanks. Yes, it’s true. The drift rate is about 1¼ nanometers per second, 1½ inches per year. As to your argument, you’re both wrong.”

“Huh?”
”Aw, c’mon!”

“Al, let’s put some numbers to your hypothesis. <pulling out Old Reliable and screen‑tapping> I’m going to compare Jupiter’s pull on the Moon to Earth’s when the two planets are closest together. OK?”

“I suppose.”

“Alright. Newton’s Law tells us the pull is proportional to the mass. Jupiter’s mass is about 320 times Earth, which is pretty impressive, right? But the attraction drops with the square of the distance. The Moon is 1¼ lightseconds from Earth. At closest approach, Jupiter is almost 2100 lightseconds away, 1680 times further than the Moon. We need to divide the 320 mass factor by a 1680‑squared distance factor and that makes <key taps> Jupiter’s pull on the Moon is only 0.011 percent of Earth’s. It’ll be <taps> half that when Jupiter’s on the other side of the Sun. Not much competition, eh?”

“Yeah, but a little bit at a time, it adds up.”

“We’re not done yet. The Moon feels the big guy’s pull on both sides of its orbit around Earth. On the side where the Moon’s moving away from Jupiter, you’re right, Jupiter’s gravity slows the Moon down, a little. But on the moving-toward-Jupiter side, the motion’s sped up. Put it all together, Jupiter’s teeny pull cancels itself out over every month’s orbiting.”

“Gotcha, Al. So what about my theory, Sy?”

“Basically the same logic, Vinnie. The Solar wind varies, thanks to the Sun’s variable activity, but satellite measurements put its pressure somewhere around a nanopascal, a nanonewton per square meter. Multiply that by the Moon’s cross‑sectional area and we get <tap, tap> a bit less than ten thousand newtons of force on the Moon. Meanwhile, Newton’s Law says the Earth’s pull on the Moon comes to <tapping>
G×(Earth’s mass)×(Moon’s mass)/(Earth-Moon distance)²
and that comes to 2×1011 newtons. Earth wins by a 107‑fold landslide. Anyway, the pressure slows the Moon for only half of each month and speeds it up the other half so we’ve got another cancellation going on.”

“So what is it then?”
”So what is it then?”

“Tides. Not just ocean tides, rock tides in Earth’s fluid outer mantle. Earth bulges, just a bit, toward the Moon. But Earth also rotates, so the bulge circles the planet every day.”

“Reminds me of the wave in the Interstellar movie, but why don’t we see it?”

“The movie’s wave was hundreds of times higher than ours, Al. It was water, not rock, and the wave‑raiser was a huge black hole close by the planet. The Moon’s tidal pull on Earth produces only a one‑meter variation on a 6,400,000‑meter radius. Not a big deal to us. Of course, it makes a lot of difference to the material that’s being kneaded up and down. There’s a lot of friction in those layers.”

“Friction makes heat, Sy. Rock tides oughta heat up the planet, right?”

“Sure, Vinnie, the process does generate heat. Force times distance equals energy. Raising the Moon by 1¼ nanometers per second against a force of 2×1021 newtons gives us <taping furiously> an energy transfer rate of 4×10‑23 joules per second per kilogram of Earth’s 6×1024‑kilogram mass. It takes about a thousand joules to heat a kilogram of rock by one kelvin so we’re looking at a temperature rise near 10‑27 kelvins per second. Not significant.”

“No blaming climate change on the Moon, huh?”

~~ Rich Olcott

# The Top Choice

Al grabs me as I step into his coffee shop. “Sy, ya gotta stop Vinnie, he’s using up paper napkins again, and he’s making a mess!”

Sure enough, there’s Vinnie at his usual table by the door. He’s got a kid’s top, a big one, spinning on a little stand. He’s methodically dropping crumpled-up paper wads onto it and watching them fly off onto the floor. “Hey, Vinnie, what’s the project?”

“Hi, Sy. I’m trying to figure how come these paper balls are doing a circle but when they fly off they always go in a straight line, at least at first. They got going-around momentum, right, so how come they don’t make a spiral like stars in a galaxy?”

Astronomy professor Cathleen’s standing in the scone line. She never misses an opportunity to correct a misconception. “Galaxy stars don’t spray out of the center in a spiral, Vinnie. Like planets going around a star, stars generally follow elliptical orbits around the galactic center. A star that’s between spiral arms now could be buried in one ten million years from now. The spiral arms appear because of how the orbits work. One theory is that the innermost star orbits rotate their ellipse axes more quickly than the outer ones and the spirals form where the ellipses pile up. Other theories have to do with increased star formation or increased gravitational attraction within the pile-up regions. Probably all three contribute to the structures. Anyhow, spirals don’t form from the center outward.”

My cue for some physics. “What happens in a galaxy is controlled by gravity, Vinnie, and gravity doesn’t enter into what you’re doing. Except for all that paper falling onto Al’s floor. There’s no in-plane gravitational or electromagnetic attraction in play when your paper wads leave the toy. Newton would say there’s no force acting to make them follow anything other than straight lines once they break free.”

“What about momentum? They’ve got going-around momentum, right, shouldn’t that keep them moving spirally?”

I haul out Old Reliable for a diagram. “Thing is, your ‘going-around momentum,’ also known as ‘angular momentum,’ doesn’t exist. Calm down, Vinnie, I mean it’s a ‘fictitious force‘ that depends on how you look at it.”

“Is this gonna be frames again?”

“Yup. Frames are one of our most important analytical tools in Physics. Here’s your toy and just for grins I’ve got it going around counterclockwise. That little white circle is one of your paper wads. In the room’s frame that wad in its path is constantly converting linear momentum between the x-direction and the y-direction, right?”

“East-West to North-South and back, yeah, I get that.”

“Such a mess to calculate. Let’s make it easier. Switch to the perspective of a frame locked to the toy. In that frame the wad can move in two directions. It can fly away along the radial direction I’ve called r, or it can ride along sideways in the s-direction.”

“So why hasn’t it flown away?”

“Because you put some spit on it to make it stick — don’t deny it, I saw you. While it’s stuck, does it travel in the r direction?”

“Nope, only in the s direction. Which should make it spiral like I said.”

“I’m not done yet. One of Newton’s major innovations was the idea of infinitesimal changes, also known as little-bits. The s-direction is straight, not curved, but it shifts around little-bit by little-bit as the top rotates. Newton’s Laws say force is required to alter momentum. What force influences the wad’s s-momentum?”

“Umm … that line you’ve marked c.”

“Which is the your spit’s adhesive force between the paper and the top. The wad stays stuck until the spit dries out and no more adhesion so no more c-force. Then what happens?”

“It flies off.”

“In which direction?”

“Huh! In the r-direction.”

“And in a straight line, just like Newton said. What you called ‘going-around momentum’ becomes ‘radial momentum’ and there’s no spiraling, right?”

“I guess you’re right, but I miss spirals.”

Al comes over with a broom. “Now that’s settled, Vinnie, clean up!”

~~ Rich Olcott

• Thanks for the question, Jen Keeler. Stay tuned.

# 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.”

“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

# Gettin’ kinky in space

Things were simpler in the pre-Enlightenment days when we only five planets to keep track of.  But Haley realized that comets could have orbits, Herschel discovered Uranus, and Galle (with Le Verrier’s guidance) found Neptune.  Then a host of other astronomers detected Ceres and a host of other asteroids, and Tombaugh observed Pluto in 1930.

Astronomers relished the proliferation — every new-found object up there was a new test case for challenging one or another competing theory.

Here’s the currently accepted narrative…  Long ago but quite close-by, there was a cloud of dust in the Milky Way galaxy.  Random motion within it produced a swirl that grew into a vortex dozens of lightyears long.

Consider one dust particle (we’ll call it Isaac) afloat in a slice perpendicular to the vortex.  Assume for the moment that the vortex is perfectly straight, the dust is evenly spread across it, and all particles have the same mass.  Isaac is subject to two influences — gravitational and rotational.

Gravity pulls Isaac towards towards every other particle in the slice.  Except for very near the slice’s center there are generally more particles (and thus more mass) toward and beyond the center than back toward the edge behind him.  Furthermore, there will generally be as many particles to Isaac’s left as to his right.  Gravity’s net effect is to pull Isaac toward the vortex center.

But the vortex spins.  Isaac and his cohorts have angular momentum, which is like straight-line momentum except you’re rotating about a center.  Both of them are conserved quantities — you can only get rid of either kind of momentum by passing it along to something else.  Angular momentum keeps Isaac rotating within the plane of his slice.

An object’s angular momentum is its linear momentum multiplied by its distance from the center.  If Isaac drifts towards the slice’s center (radial distance decreases), either he speeds up to compensate or he transfers angular momentum to other particles by colliding with them.

But vortices are rarely perfectly straight.  Moreover, the galactic-cloud kind are generally lumpy and composed of different-sized particles.  Suppose our vortex gets kinked by passing a star or a magnetic field or even another vortex.  Between-slice gravity near the kink shifts mass kinkward and unbalances the slices to form a lump (see the diagram).  The lump’s concentrated mass in turn attracts particles from adjacent slices in a viscous cycle (pun intended).

After a while the lumpward drift depletes the whole neighborhood near the kink.  The vortex becomes host to a solar nebula, a concentrated disk of dust whirling about its center because even when you come in from a different slice, you’ve still got your angular momentum.  When gravity smacks together Isaac and a few billion other particles, the whole ball of whacks inherits the angular momentum that each of its stuck-together components had.  Any particle or planetoid that tries to make a break for it up- or down-vortex gets pulled back into the disk by gravity.

That theory does a pretty good job on the conventional Solar System — four rocky Inner Planets, four gas giant Outer Planets, plus that host of asteroids and such, all tightly held in the Plane of The Ecliptic.

How then to explain out-of-plane objects like Pluto and Eris, not to mention long-period comets with orbits at all angles?

We now know that the Solar System holds more than we used to believe.  Who’s in is still “objects whose motion is dominated by the Sun’s gravitational field,” but the Sun’s net spreads far further than we’d thought.  Astronomers now hypothesize that after its creation in the vortex, the Sun accumulated an Oort cloud — a 100-billion-mile spherical shell containing a trillion objects, pebbles to planet-sized.

At the shell’s average distance from the Sun (see how tiny Neptune’s path is in the diagram) Solar gravity is a millionth of its strength at Earth’s orbit.  The gravity of a passing star or even a conjunction of our own gas giants is enough to start an Oort-cloud object on an inward journey.

These trans-Neptunian objects are small and hard to see, but they’re revolutionizing planetary astronomy.

~~ Rich Olcott