Ep. 356: Rotational Inertia

An object at rest stays at rest, and object in motion tends to stay in motion. This is inertia, defined famously by Isaac Newton in his First Law of Motion.

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Show Notes


Transcription services provided by: GMR Transcription

Fraser Cain: Astronomy Cast episode 356, inertia. Welcome to Astronomy Cast, our weekly facts-based journey through the cosmos. We hope you understand what we know but how we know what we know. My name is Fraser Cain. I’m the publisher of the university and with me is Dr. Pamela Gay, professor at Southern Illinois University, Edwardsville and the director of CosmoQuest. Hello, how are you doing?

Pamela Gay: I’m doing well. How are you doing, Fraser?

Fraser Cain: Good. So we’re recording this episode on the Monday after the spaceship to disaster and one week for space flight.

Pamela Gay: Yeah, it was the Antari’s incident with org three and then spaceship two all in one week. That was a tough week but it all comes down to, they say it’s not rocket science for so many different things because rocket science is so damn difficult and we have to figure this out and we’re still at that learning to explore stage, the analogy I used with several people last week was we are essentially trying to get to the new world where the new world is getting colonies above earth’s atmosphere and we’re still at that stage of sending out the explorers but we haven’t yet sent out the colonists and we’re gonna lose ships as we go and this is why they’re test pilots.

It still sucks every single time and our hearts are not just with the people who lost spacecraft but the people who also lost their instruments and everything else between those two different spacecraft last week.

Fraser Cain: This is it. We’ll be talking quite a bit about this on the Weekly Space Hangout. We’ll be unfolding that week after week after week so we record that on Fridays at noon so if you wanna join us there, that would be great. I wanna give a shout out as well. We got some great communities that are around everything that we do but one community has been really great and fairly new is the folks who’ve kind of come together for the weekly space hangout. It’s the WSH Crew. They’ve created their own Google Plus community over on Google Plus, which is user name WSH Crew and a lot of familiar names that we always talk about.

So if you’re a fan of Astronomy Cast, I think you’d probably fit right at home with those folks as well so I wanna recommend that. All right, let’s move on.

Fraser Cain: Isaac Newton defined inertia as his first law, the vis insita or innate force of matter is a power of resisting by which everybody as much as it lies, endeavors to preserve its present state, whether it be of rest or of moving uniformly forward in a straight line. So when you’re in giving your introduction to physics, how do you describe inertia to your students?

Pamela Gay: Well, I break it down and I’m like okay, so we’ve got force which is what you apply to an object to accelerate that object to get it moving through space and what is resisting your ability to get it moving with that force is the amount of mass that it has.

Fraser Cain: A lot of math is required to do the calculation.

Pamela Gay: Right. It’s the amount of mass that matters. Well, when it comes to getting something rotating, the amount of mass plays in but it plays in two different ways. It plays in because you have to get that mass moving but the distribution of that mass varies with how, makes it vary on how hard or easy it is to rotate something so when you’re looking at something’s inertia, you’re looking at what axis are you trying to rotate it around, where’s the force relative to that axis and how is the mass distributed.

So this is the point where I usually pick the burliest looking human in the classroom and I say okay, walk over to that door and try and push it closed by pushing three or four inches away from the hinges so your rotational axis is the hinges essentially. You’re applying your force a few inches in from those hinges. You have those little tiny distance and your mass is spread out over this large sheet that we simplify as pretending it’s a rod usually. It’s a really weird-shaped rod. You’re not gonna succeed very well.

Instead and here I grab the most diminutive looking human in the class and I say come over here. Now just use the tip of your finger and push as far away from that hinge as possible. Well here, you’ve increased your moment arm that lever that you’re using to rotate the door, way easier. So that’s one part of it. Now the other part of it is how that mass is distributed so it’s how far you’re pushing away from your rotational axis. The other thing is how your mass is distributed so this is where I now find the person in the classroom with the largest wing span, the biggest arms in the classroom and if I can find the person with the biggest arms and the skinniest body works even better.

We have because we’re evil, these small platforms that you can stand on and they rotate and so I’ll get a pair of like 5-lb. hand weights, handed them to this long-armed skinny person and have them start with their arms all the way out and give them a gentle shove on their hands because that requires at least force to get them rotating and we’ll start rotating and our platform doesn’t have that much friction in it so they’re going on a fairly constant velocity or in this case they have a constant rotational speed. It’s not really velocity because they’re constantly changing.

So now their mass is spread out. They’re essentially a rod in the center and then kind of the lamest shell of mass ever. I have them bring in their arms and because they’re tall, skinny individuals, they’ve now suddenly increased their central mass by a large percent as in like 5 or 10 percent and they start rotating much faster. That’s because we’ve changed their moment of inertia. So the moment of inertia defines how quickly some things going to rotate because of the way its mass is spread out as a function of the force that you’ve applied to it.

Fraser Cain: So this is obviously, this is this idea like skaters who will hold their arms out and then as they pull their arms in, they rotate faster and faster and faster. Go ahead.

Pamela Gay: No, keep going. You’re good.

Fraser Cain: I’m just gonna say and we see this as well in things like some of the rotational like the planetary formation, how the in the earliest days of the solar system, how the, as the gas and dust collected together, it had to maintain its total inertia and would spin up and flatten those disks.

Pamela Gay: We also see it when a star bloats out to being the giant star. Its rotation rate slows down. When it collapses down to being a neutron star, it seriously speeds up and this is also like your mom’s way of checking or your dad’s way of checking if an egg has been hardboiled or not after Easter. I don’t know about you but we always had the great post Easter mystery of which eggs were the ones that got hardboiled and not painted and which are the ones that are still raw.

Well, if you put a raw egg and a hardboiled egg side by side on the counter and you rotate them, the one that’s solid is going to rotate more readily than the one that’s still liquid on the inside.

Fraser Cain: Right because it will actually push out to the outside edges of the shell, right? Change the blob around inside the shell and change the inertia. I guess we already sort of started to move towards it which is that this is Astronomy Cast and of course everything used to have its astronomical connotations. So where in the universe and I’d love to go back to that idea of these neutron stars, where in the universe do we see inertia really playing out on the grandest scales?

Pamela Gay: Absolutely everywhere.

Fraser Cain: All right.

Pamela Gay: It’s one of those things that’s absolutely fundamental. It drives students crazy because it requires lots of evil calculus. This is one of those places where calculus comes into play. You have to use all of your advanced maps. There’s just certain places that require calculus and where it starts to become difficult is things like the planet earth, we don’t have a continuous distribution of mass and in fact if you look at how we wobble because the sun is trying to pull is over, it’s exerting a torque on us.

When you look at our wobble due to the sun trying to pull us back up straight, it turns out that our moment of inertia is this combination of a solid mass of single density and a point source in the center and this is because we have this really high density core and then we have the crust and water on top and we run into neat things like our rotation rate will change after earthquakes because the mass gets redistributed just enough.

Fraser Cain: Yeah and people always run the calculations for how much our day has changed after the earthquakes. It’s not very much.

Pamela Gay: Earthquakes and another one that really affected us is the dam that they’re building in China as that continues to fill up, it may already be completely filled up but the big dam in China actually changed the moment of inertia of the entire planet.

Fraser Cain: Right and I guess in the worst case scenario, this is where the tidal forces start to, the interactions that eventually given enough time, the sun would tidally lock the earth to the sun to finally bounce out that inertia, slow it down to the point that it’s sort of in a perfect spot.

Pamela Gay: Well, with our moon, I’m not sure it can actually get that far but what the sun’s trying really hard to do is currently our earth is tipped over on its side 20 some odd degrees and that how many 20 some odd degrees it’s tilted over actually wobbles somewhat because it’s between roughly 21, roughly 25 and as it wobbles between these two different values, that wobbling which is tied to the procession of the poles, that comes from the sun trying to get us to be completely upright and luckily we have this moon orbiting around that’s locking in our rotation between those two different bodies.

So the sun’s never gonna succeed in tidally locking us or bringing us upright or anything like that but it’s that interplay between our axis of rotation and the procession of that axis that comes from our planet’s moment of inertia.

Fraser Cain: All right so we’ve got the I guess the interactions between the sun and the earth. Where else would we see this?

Pamela Gay: So as you already pointed out, we see it with the collapsing of the former solar system and then sticking with our solar system for a minute, we see it in the asteroids so if you look at how the asteroids rotate around different axes, you can actually start to slowly get out what their distribution of densities are, what is all of this weirdness in terms of do they have the same stuff on the inside in different places by looking at the way they rotate around different axes and the way these different axes rotate.

Fraser Cain: That’s really amazing. I mean one of the things that I always find kind of almost magical is how that if you have like a planet and a moon, the moon orbiting, but once you have those two masses together, you can get at the mass that I guess the combined mass, the individual mass of the objects themselves but if you’ve only got one body, it’s really tough to get at the mass but watching it rotate –

Pamela Gay: You still can’t get at the amount of mass but you can get at the distribution of that.

Fraser Cain: Distribution, the density, the you know, and then what it, maybe it’s even formed out of which is just mind bending.

Pamela Gay: Certain approximations are required because what you can get at with all of these wobbles and stuff, you can get at the variation. You can get at the ratio. What we know it’s probably not gonna have anything more complicated than an iron core so if you know what is the ratio of its core to its outer shell, you can start doing all sorts of neat approximations to get at what that density probably is. It’s amazing what you can do if you do a whole lot of math.

Fraser Cain: Right and that’s it, you’re gonna have to crack out some really tough math on this one.

Pamela Gay: This is where we’re also starting to be able to get estimates of where there’s probably underground oceans because underground oceans will change the moments of inertia of moons. We often but not always get lucky enough to see these geysers, these ice volcanoes but borrowing that, we also can measure over time if we have the right imagery, the wobbling of their axes and these wobbling of the axes can be due in part to the moment of inertia and a variety of different gravitational interplays.

So if you have that tilt and you have that procession of the orbit, it allows you to get at the moment of inertia which might tell you hey, there’s liquid here. It’s rotating more like a can of soup than a can of pumpkin filling.

Fraser Cain: It’s like spin but again it’s like spinning those eggs. You spin Enceladus or you spin Europa and you can watch those movements and that’ll tell you whether there’s something swashing right there.

Pamela Gay: This is a really neat one to get to teach to kids to try to get them to understand it because you can literally take a can of soup and a same sized can of something more solid like pumpkin pie filling and just roll them down a ramp and they won’t go down the ramp at the same rate and it gets you at the moment of inertia and how it affects rolling bodies.

Fraser Cain: Because it’s solid it rotates more evenly but because with the liquid one, it’s more sloshing around and more resistant to the turn.

Pamela Gay: Now if you wanna make it really complicated, you can do empty can, can filled with a different can in the center and suspend it so like if you do can styrofoam, rod in the center of the styrofoam, and then do your pumpkin pie filling, now you’re starting to get at even more complicated stuff and you can start to see how all of these different maths start to combine and that combination of can, styrofoam, rod in the center is sort of the cylindrical version of what a lot of worlds are more like.

Fraser Cain: Well, this is great so keep going. I wanna hear some more examples of where this inertia comes up.

Pamela Gay: So as you get out of our solar system and start looking at the stars, stars as they go through their various stages of evolution, their rotation rate, unless they’re eating mass of of something else or losing mass to something else, their rotation rate is pretty much set at the moment of formation. You start out with this big old blob of gas and dust. It’s kind of hanging out big and blobby and then something triggers it to collapse. During that collapse, whatever it was that triggered that collapse, probably instigated some sort of a force that set this whole thing rotating.

As this whole thing starts rotating, initially it’s really big. As it collapses, it’s like that ice skater bringing her arms in and as it collapses, it gets faster and faster and faster and the center star ends up with the majority of that momentum. Now, as the star ages, it’s gonna change size. It’s eventually gonna bloat up to be a giant star and like the ice skater putting her arms out and in this case grabbing sticks and moving the mass out further using sticks, that’s going to slow the rotation rate.

Then when the star collapses down, it’s going to speed back up so you have this change in the rotation rate of stars as they change in their radius and that’s all due to the moment of inertia.

Fraser Cain: The part that’s really amazing to me is this idea that every single atom, every particle in the entire dust cloud has its own individual inertia but the final, essentially orientation and speed of rotation of that final gas disk is when you’ve added up all of the directions and all of the inertia from all those particles you come up with this average because now they’re all essentially connected as one big body and that final orientation is just what you get if you average out the motion and inertia of all of those individual particles.

Pamela Gay: This is one of those quantities that gets conserved in systems. So if you’re going to stop that ice skater from rotating, you’d have to exert a torque somewhere. You’d have to exert a force at a distance from the rotational axes that acts to stop the rotation. Now if you’re a particle orbiting in a solar system, you have some sort of an angular momentum. Now at the same time, if you’re a particle in a galaxy trying really hard to fall into that central super mass of black hole, you’re gonna have an angular momentum as well.

So it’s possible for a particle to have an exact, its center of mass is lined up with the center of mass of the black hole and it just falls straight on in. That’s possible. What happens most of the time is you have a particle that’s on an elliptical orbit where a perfect circle is still a form of an ellipse. That elliptical orbit pulls it down towards the black hole and it has to somehow shed momentum somehow if it’s going to fall all the way into the black hole and that gets shed usually through interactions with other particles so there’s a force enacted upon the particle, that energy, it gets conserved so it gets turned into things like heat.

This is where you get these really hot accretion disks in the centers of galaxies. That’s all the stuff that angular momentum prevented from falling straight into the black hole. Instead it got piled up into a desk. As it gets piled up into the desk, the momentum has to go somewhere, it goes into heat and stuff that sheds enough momentum is able to fall all the way in.

Fraser Cain: Now I wanna let the record show that you’re the one who brought up black holes in this conversation, not me but now you’ve opened up the door, let’s talk about black holes.

Pamela Gay: Okay. So black –

Fraser Cain: Where our comprehension of the laws of physics you know, start to break down and inertia, I’m sure is one of these where you push these ideas to their very limits.

Pamela Gay: What’s kind if fun is when they first teach you about black holes, all of the equations are for non-rotating black holes and then kind of at the end of the class in grad school, they’re like anything actually rotates and if you do research in that field, you’d have to solve it in a rotating frame and all of us who are observational folks were like oh and the theorists who like to chew on math, they get excited and then they go off and chew on math and when you compare the rotation rates for this as far down to the event horizon as we can get with the predictions that factor in time dragging and everything else, the predicted rotation rates for central super mass of black holes over the masses observed, it all seems to line up so far.

So folks who are a whole lot more loving of doing math than I am, I’m a computer girl, people who do that and have figured out predicted rotation rates, the physics matches the observations and we really can’t ask for better than that.

Fraser Cain: Yeah, we’ve done a bunch of stories on this. The fact that the most rapidly rotating super massive black holes are rotating at the speed predicted by Einstein and the scary thing here is that it’s a significant portion of the speed of light that these things are rotating so quickly that they can’t rotate any faster which is mind bending.

Pamela Gay: It’s really cool to look at all of the things that are nominally happening down at the surface of the event horizon which is different than saying it’s happening at the surface of the black hole and I know there’s a whole lot of debate on whether or not anything can collapse down to a singularity but we do know there is mass confined inside of an event horizon defined as that surface where you have to be going faster than the speed of light to escape.

I stopped talking when we hit the event horizon but, so whether or not you believe it’s fully collapsed, up to you. There’s a horizon at which we can’t observe beneath and at that horizon, the physics matches the observations.

Fraser Cain: One of the other interesting kind of more speculative used for black holes then is that you can actually extract this rotational inertia out of the black hole that you can drop objects in as it kicks it back out. It gives up a little bit of its inertia so you can slow down the rotation of the black hole and extract the energy out until you’ve stopped it and then you’re gonna have to feed it again to speed it back up again.

Pamela Gay: I do wanna clarify that you’re not dropping an object into the black hole. You’re dropping an object into the gravitational well of the black hole so you’re essentially gravitationally slingshotting the object and so some of the angular momentum from the black hole gets transferred to this object that’s on a highly elliptical orbit and so by essentially throwing things towards the black hole that whip around it, gaining momentum through the gravitational slingshot. The black hole has to give up some of its momentum and essentially what this also means is that every time we use Jupiter, Mars, or Venus or any of the other planets to give a gravitational boost to the velocity of a spaceship, every time we do that, we’re actually stealing angular momentum from these planets.

But the mass of the spacecraft is so small that it’s not perceptible so we don’t actually worry about slowing down Mars every time we slingshot a spacecraft around it.

Fraser Cain: Yeah, if we did it enough, we could eventually cause Jupiter to crash into the sun because we’ve stolen all of its orbital –

Pamela Gay: No, wrong momentum.

Fraser Cain: Right because what’s slowing, you know, we’re lowering Jupiter’s orbit every time one of our spacecraft uses Jupiter’s orbital speed to kick into a higher orbit, it’s having to take a little bit away from Jupiter so it’s lowering Jupiter’s orbit a fraction.

Pamela Gay: Okay, that’ll give you, yeah.

Fraser Cain: So I’m saying if you had enough spacecraft buzz past Jupiter and use them for those gravitational slingshots, eventually Jupiter would run out of, it would spiral in and eventually crash into the sun. It’s all the same.

Pamela Gay: Yeah, that I’ll agree with.

Fraser Cain: But so one last question, you’ve been using sort of the terms angular momentum and inertia and momentum fairly interchangeably so for people who aren’t clear on the distinction between inertia and momentum because they are very similar, how would you sort of describe the difference?

Pamela Gay: So momentum is what you deal with when you have a whole bunch of things that collide together, something that is already in motion will share its motion with another object through a collision. They can either bounce off of each other in which case when you sum up all of the motions so mass times velocity, that will sum up to what you had going into the system. So mass times velocity for all of the things that are interacting stays a constant. Rotational inertia, momentum, has to do with now you’re looking at a rotating system so when you’re dealing with linear things colliding, you’re dealing with force.

When you’re dealing with angular, you’re dealing with torque, which is force at a distance away. So momentum says things that are in motion stay in motion and all the motions across the ensemble are going to be conserved. Inertia says I’m going to resist this force you’re imparting upon me and I’m going to refuse to rotate by this amount because of how my mass is distributed whereas with strictly linear stuff it’s just the mass sitting, they’re going yeah, I don’t wanna move.

Fraser Cain: So I’ve got one last question for you. It’s kind of philosophical but why? Why is there inertia? I think this is – well, maybe this is one of those things that I think, the investigation into the Higgs Boson has been helping science understand there is this theory about why mass tends to resist movement.

Pamela Gay: So what you’re trying to get at is the fact that all of these different particles are coupled to a quiescent field that permeates all of space and time but at the end of the day it’s the ensemble of particles going yeah, I don’t wanna move and I’m locked into this ensemble and you’re gonna have to get all of us moving and the ability to get that entire ensemble moving depends on how tightly clustered there are. The analogies that often get used are if you have a crowd of friends who are kind of distributed all over the place and you send a famous person through the room, the famous person is going to have a huge pull on that crowd of people and that’s the analogy for the Higgs Boson is the famous person has a huge mass and attracts all the rest of the people around the famous person.

Well, with inertia, the way to think about it is how hard is it to get all your friends moving? All your friends are spread out all over the room and they’re in their individual conversations, they’re gonna be a pain in the insert whatever you want to get moving but if they’re all tightly focused and all together, a point source of people then they’re really easy to get moving. So the moment of inertia reflects on how hard it is to get that ensemble moving and the more spread out the ensemble, the harder it is to get moving.

Fraser Cain: That’s really cool. All right, well, thank you so much, Pamela.

Pamela Gay: You’re welcome.

Fraser Cain: Thanks for listening to Astronomy Cast, a nonprofit resource provided by Astrosphere New Media Association, Fraser Cain, and Dr. Pamela Gay. You can find show notes and transcripts for every episode at You can email us at Tweet us at Astronomy Cast. Like us on Facebook or circle us on Google Plus. We record our show live on Google Plus every Monday at 12:00p.m. PST, 3:00p.m. EST or 20:00 GMT. If you miss the live event, you can always catch up over at

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