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check it out so I've got this spout it's like a sprinkler it's got these tubes and then water shoots out the end so I want you to think about what direction the water is going to go when it comes up okay let's turn it on okay so as we suspected the water comes straight out but now I'm going to spin the pipes and I want you to think about what the path of the water is going to do is it going to stay going straight or is it going to curve and trail behind or maybe curve and go ahead check it out so if you guessed that the water curves and trails behind you are right but now I'm going to replace these pipes with something a little bit different okay so these pipes curve back inward so when we turn on the water it's going to squirt
toward the middle so yeah so now that water's shooting straight back Inward and I'm going to spit it again clockwise and I want you to think about this time which direction the water is going to curve when I spin it is the water going to trail behind the tube or is it going to curve a head or is it going to this time keep going straight let's check it out who that is so weird it is so unintuitive and if you guessed it would jump ahead I don't know how you knew but hopefully by the end of this lesson we're going to understand why this happens hello I'm Diana Cowan and welcome to lesson 15 of Diana's intro physics class also known as AP Physics 1 review also known as physics by Diana today's lesson is about
the property that you feel inside of your head every time you spin around for a while and then you get dizzy and fall down why do you get dizzy because the fluid in your head keeps spinning even after you stop spinning basically objects that are spinning will just keep spinning this is a property of objects it's like the momentum of a fast car but it's for spinning momentum it's angular momentum and that is today's lesson our theme for today is the apocalypse because if the Earth stops spinning that would be the end of our days here's a fun question if everyone on Earth gathered at the equator and then started running in the same direction could we change how fast the Earth is spinning in today's video
you're going to learn the tools to answer that question okay we are done being humorous as you know from the last lesson the key to understanding rotation is to link it to linear motion Concepts that we've used before so for example a linear Force makes this spherical cow accelerate and the rotational analog is that a torque makes it change its rotational speed you also remember the definition of torque which is it's a force times the distance between where the force is applied and the point about which it will spin which could be either a fixed axle like a door hinge or the center of mass of something like the center of a frisbee when you spin it so we did all of that in the last lesson but we didn't talk about the feeling of
spinning or spinning something it feels much easier to spin a top that it is to spin a bowling ball and it would be even harder to spin a real cow and the moon well I personally don't have the strength to spin the moon some things are just hard to spin so why is that it turns out you already know what we're about to do right now is find a concept like f equals m a but for rotation so an equation with torque instead of force angular acceleration instead of acceleration and then we're going to figure out what this new term is that's the rotational analog of our linear mass and it's called the moment of inertia but we're first going to derive this equation then understand what it means and then understand what
that moment of inertia term is and then from last time rotational acceleration Alpha is related to linear acceleration like this acceleration equals Alpha linear acceleration rotational acceleration times R and then we have torque equals fourth times R now I'm going to combine all these equations and simplify I'm going to put M A into here so my torque equals m a r and then put my a here with the Alpha and the R so torque equals m alpha R times R so torque equals m r squared times Alpha and now I have a new model torque equals Mr squared Alpha and if you're just doing these lessons a reminder that one of the main things that we do in physics is to come up with mathematical models to describe reality as closely as we can so this looks a lot
like f equals m a especially if I put parentheses around this part and it is this is the rotational f equals m a force torque acceleration angular acceleration but this part here well we had mass and f equals m a and so over here we've got Mr squared what is that is the moment of inertia yeah I said it because that's what it's called and remember Mass tells us how much force we need to apply to accelerate something linearly so similarly moment of inertia tells us how much torque we need to apply to get something to spin to get something to accelerate angularly and we're going to give it a new label we're going to call this I for a moment of inertia so I is that Mr squared in this case and then check it out torque equals I Alpha now we've got f equals m a
and Tau equals I Alpha but hold up if I spin say a record what is the m in this equation I equals m r squared and what is the r and what I mean by that question is if I swung an egg on the string it's way more obvious the m is the mass of the egg and the r is the length of a string and it turns out that this eye we just found I equals m r squared is exactly the right moment of inertia for a particle or an object that has all of its mass in one spot so like the egg on the string but what about a spinning record the mass is all spread out and the record is really a collection of a gazillion tiny little pieces little record molecules so when I spin the record all the tiny pieces well they all have the same angular velocity
and this confused me at first because the pieces on the outside are going faster linearly than the pieces in the middle but recall angular velocity refers to revolutions per time and all the pieces make one round trip or two Pi radians around the circle in the same amount of time but since the outer pieces have a larger r a larger Radian yes then they have to travel farther to make the revolution which means their acceleration is what bigger smaller the same it's bigger so the outside pieces need more torque to spin up to angularly accelerate so to get the moment of inertia for a specific object like this we actually have to add up all the little moments of inertia of all the little pieces using
their mass times the radius away from the center squared and it's unfortunately much more complicated than just the total mass of This Record Times its radius because think about if the record were some weird shape what would you use for radius but fortunately people have done these calculations to find moment of inertia for common shapes like spherous cylinders donut like rings and you can too if you take calculus based physics here's some moments of inertia for common shapes and some intuition for you first what shape do you think gives you the biggest moment of inertia meaning that it's the hardest to spin it is a hoop because all of the mass is as far from the axis as you can get it that is if you're spinning the hoop like this and
the moment of inertia for this is Mr squared now what's the smallest moment of inertia you can get it's a sphere that's why planets oh can you pass me Jupiter thank you that's why planets are all spherical because nature is minimizing the energy to spin or pretty close to spherical so the moment of inertia for a solid sphere of uniform density is two-fifths Mr squared the moment of inertia for a solid cylinder is one half Mr squared so it's half of this one but that is if you spin the cylinder about the middle I'm going to grab another cylinder nice the moment of inertia if you spin the cylinder instead of about the middle about this middle is different that moment of inertia becomes 1 4 Mr squared plus 1 12 ml squared
where L is the length of the cylinder oh confusing all right so many rotational analogs to our linear motion Why Stop Now so we did forces we did acceleration we did f equals m a let's do energy we know the energy of a cow Ball moving linearly that's just one-half MV squared what about a cow ball rotating oh this is so easy using analogs the rotational World equivalent of mass is moment of inertia and equivalent of linear velocity is angular velocity Omega so the energy of an object rotating the rotational energy is one half I Omega squared done I know it's ridiculous how easy that was but don't worry it's going to get complicated again right now I just gave you all the tools to answer this next question which is a very tricky
quiz question so imagine that I have two cows one spherical cow one box cow now I'm gonna Slide the Box cow down a ramp and for Simplicity that's let's imagine that we turned off friction so imagine this is sliding down with no friction and it just continues to speed up and then with friction the spherical cow is going to roll down the ramp foreign now which cow is going to reach the bottom first the rolling spherical cow or the Box cow sliding with no friction now a lot of people think that the rolling cow reaches the bottom first because rolling things get going really fast right but let's dig into it and we're going to think about it using energy if the spherical cow and the Box cow have the same mass and they're
starting at the same height they're going to both have the same amount of gravitational potential energy at the top of the hill before they start sliding or rolling down so they're both going to start out with the same potential energy MGH now when the Box cow reaches the bottom of the hill it is just sliding it's not rotating so it has converted all of its potential energy into linear kinetic energy but when the spherical cow reaches the bottom of the hill it is moving linearly and it is rotating and we just saw rotation requires energy by one half I Omega squared so some of the spherical cow's initial potential energy has to go into that rotational energy and some of it will go into the linear motion of the
spherical count so my energy here is going to be one half I Omega squared plus 1 one half MV squared box and Sphere for box cow and spherical cow so some of the spherical cow's initial potential energy is going to go into rotational energy so the bottom of the hill the spherical cow will have less linear kinetic energy than the Box cow so the Box cow reaches the bottom of the hill first simply because it doesn't have to put energy into spinning this is one of the unintuitive quirks of physics so now let's actually do some math because on a test you would get this problem with some numbers so we're going to assume that the spherical cow has a mass m and they're both at a height h up on the hill now the potential energy
of the ball cow is MGH we already wrote that down and at the base of the Hill I'm going to assume that all their energy gets converted into this into this type of energy so this is their total one half I Omega squared plus one half MV squared now I'm going to plug in more stuff that I know the moment of inertia of a solid sphere we have it's two-fifths Mr squared so I'm going to plug that in here and we know that Omega we can write that in terms of this velocity so Omega the angular velocity is always linear velocity and I'm going to write vs because it's the spherical Cal over R so this is going to be VR squared now if we look we've got an m in every single one of these terms so I can cancel those M's
and we can rewrite now I've got GH equals one-half times two-fifths r squared times v s squared over r squared plus one half v s squared now this r squared cancels with this r squared 2 cancels the two and I'm getting one-fifth vs squared plus one half v s squared and I can simplify all of that to g h equals seven tenths v s squared now if I wanted to I could solve for vs this seems like it's good enough but if I wanted to solve for BS I could I would get vs equals the square root of the seven tenths would flip to 10 7's GH Ada so we get that the velocity at the bottom of the hill only depends on the gravitational acceleration of Earth and the height of the hill so the mass and radius of the ball are totally irrelevant to the situation so long as
the radius is not so big that it compares to the height of the Hill but that means that any solid sphere as long as it has a uniform density will reach the bottom of the hill at the same time as any other sphere this surprised me the first time I heard about it but if you think about it's really the rotational equivalent of Galileo's big idea about the feather and the hammer reaching the ground at the same time the mass and the size don't matter here but the shape does now seems like a good moment as any to introduce the big idea for today think about that spherical cow rolling down the hill when it gets to the bottom of the hill it's going to just keep rolling if it fell off a cliff it would keep spinning in
the air if it fell into a vacuum it would keep spinning there things that are spinning will keep spinning with the same angular momentum until something exerts a torque on it the way that we would say that in physics terms is the conservation of angular momentum and as always angular momentum has a linear analog which is just momentum so linear momentum is mass times velocity I remember that equation P equals MV so angular momentum is moment of inertia we use l for angular momentum I don't know why I moment of inertia times angular velocity Omega and I'm actually going to write this out I'm going to use Mr squared for I so that we can think about this a sec this is Thea Ulrich she is an aerialist
and a maker but most importantly for us she's a fantastic demonstrator of physics Thea let me pass her some weights and then give her a bit of a spin and then showed what happens when she brings the weights in toward her body she speeds up a lot enough to make even an aerialist dizzy and this is the same concept as our crazy little water demo you've got these pipes spinning around all going the same angular velocity the same number of revolutions per second but the parts further away from the center where the water comes out have a higher linear speed sideways than the parts of the pipe in the middle which are going slower linearly because they're going around a smaller Circle and then just as the water droplets
leave the end of the pipe where they were moving with that higher sideways linear velocity they're now sprayed toward the part of the pipe I'm has the smaller radius the part moving with the lower linear velocity so the fast water is going to pass those parts closer into the center and you can see now exactly why you have to spin faster when you decrease your radius in order to conserve angular momentum when you move inward your initial linear velocity closer into the center corresponds to a higher angular velocity so actually I don't see this demo as necessarily being explained by conservation of angular momentum I kind of think it explains conservation of angular momentum that's
cool so Thea the areolas spinning in there doing amazing tricks is a great example of conservation of angular momentum because there's no motor to make her spin faster or slower she is controlling how fast she spins by changing the position of her arms and legs so if you watch closely for a while you'll notice that even without the weights when her limbs are closer into her body she spins faster when they're further out she spins slower that is conservation of angular momentum because when she brings in her arms her R goes down so her moment of inertia goes down now let's tackle the question that we asked at the beginning of the video if we got everyone on the planet to start running in the same direction
along the Equator could we change how fast the Earth is spinning we're going to answer this question using conservation of angular momentum now the Earth already has some angular momentum because it's rotating but we want to figure out how much angular momentum we can transfer to the Earth by all of us running together along the Equator and how big of a change in Earth's angular velocity that would translate to so ultimately we want to find change in angular velocity or also change in so first let's estimate the angular momentum of everyone running in a big old circle around earth's equator so angular momentum is moment of inertia times Omega times angular velocity so what's the amount of inertia here angular
momentum is moment inertia times angular velocity so I have to figure out the moment of inertia of all those people running well if we treat every single person running on the equator as a point particle then the total moment of inertia of all of the people running around will be the total mass of people on Earth times the radius of Earth squared because it's kind of like a ring of people moving around so we're going to use Mr squared and actually this moment of inertia is the same for a ring which is kind of what we have this ring of people going around let's pick an estimated average mass let's say 65 kilograms so my total moment of inertia is going to be the sum of all my
mass times r squared which is going to be the radius of Earth so we're going to say this is 65 kilograms times the number of people on Earth which is about 7.8 billion or some 0.8 times 10 to the 9 people so that's going to be our Mass and then we need the radius of Earth which is roughly 6400 kilometers so we want that in meters that's 6.4 times n to the 6 meters that's the radius we're going to use and now what's the angular velocity of all those people running in a circle well that's their linear velocity divided by the radius of the Earth because we know that equation Omega equals V over R we'll say that everyone's running at 5 meters per second look this whole thing is far-fetched five meters per second it's about 10
miles an hour five meters per second around the radius of Earth was 6.4 times into six meters plug those numbers in to L this equation and solve for l so L equals my I here we go 65 kilograms times 7.8 times 10 to the nine all of my people all their mass times 6.4 times 10 to the 6 meters squared then we're going to multiply it by this Omega so times 5 meters per second and then divide all of that 6.4 times 10 to the 6 meters that actually cancels with one of these so I'm going to cancel this squared and then if we do all this math out I don't know in your head and paper on the calculator whatever L our angular momentum equals 1.6 times 10 to the 19 and then our units are kilograms times we've got a meter squared here per
second ta-da so that's how much angular momentum we would be able to transfer to Earth but would we be able to change how fast Earth spins yes but by how much we can use this equation for angularwentum l equals I Omega and use Earth's moment of inertia I to see how much of a change in angular momentum this would correspond to so l equals I Omega well I will say this is our change equals L Earth something like that so that equals 1.6 times 10 to the 19 kilogram meters squared per second equals and then I need to know the moment of inertia of Earth which I looked up because it's not easy to calculate it since Earth is not a sphere of uniform density it's more dense in the middle so I looked it up
Earth moment of inertia is 8 times 10 to the 37 kilogram meters squared and then we've got Omega because we're doing is angular momentum for Earth so that was Earth's moment of inertia times Omega and I can solve for Omega and when I do that when I solve for Omega I get Omega equals 2 times 10 to the minus 19 radians per second so that's a small number so our answer is yes we can change how fast the Earth is spinning but only by an amount less than.0003 percent or minus 2.7 times 10 to the minus 15 of Earth's rotation speed so not much fun fact humans running may not be able to change how fast Earth spins by much but big earthquakes actually can that magnitude 9.0 earthquake that hit Japan in 2011 made earth spin faster and
therefore shortened the day by almost two millionths of a second according to measurements by NASA scientists so that's our lesson on angular momentum for today when they ask what you learned on YouTube today here are your two key takeaways one the rotational equivalent of mass is the moment of inertia and two rotational energy and momentum act just like their linear friends they are conserved and as always don't forget to go solve the problems yourself that we did so here are all the problems from today's lesson if you want to review which you definitely should because that's the only way to really learn physics and almost lastly here's an out of this world example of angular momentum so when some massive stars explode as
supernova super No Lie their cores collapse into Super dense compact objects called neutron stars that are only a few kilometers across usually the massive star was spinning before it collapsed so it had some angular momentum and angular momentum has to be conserved so the much smaller neutron star that's left after the Stars collapse typically ends up spinning super duper fast to make up for its now much tinier size sometimes they spin hundreds of times per second and also some of these neutron stars shoot out intense beams of light that sweep around like a lighthouse as they spin these are called pulsars because from our point of view on Earth we see these far away Cosmic lighthouses as regularly timed
pulses of light these pulses are so strangely regular that astronomers nicknamed the first ever Pulsar observed lgm1 for Little Green Men because they thought it might be a message from Aliens and if you keep studying physics you might get to learn more about pulsars and other really cool examples of angular momentum in a class on classical mechanics or on astrophysics and I do hope you continue on with physics and now let's hear a message from a special guest hi there everybody my name is Carrie Byron former mythbuster current crash test girl and big fan of the study of physics maybe it was all my time blowing things up and dropping things off cliffs but I love physics you get to uncover the secrets of the universe well I hope you
stick with this course and I hope you love everything that you're doing and good luck and find yourself in a world of science
