(host) So here I've got a fun little challenge. You tie a string between two chairs, and then you hang two bottles off that string. Now here's the challenge-- you can only touch the bottle on the right-hand side. So how do you get the bottle on the left to start swinging? Check it out. Eventually the bottle on the right stops swinging and the bottle on the left starts swinging. So we're going to figure out why that works.

[off-screen] It worked.

Hello, I'm Dianna Cowern, and welcome to lesson 10-- we've actually made it so far-- of Dianna's Intro Physics class, also known as AP Physics 1 Review, also known as Physics by Dianna. Today's lesson is conservation of energy. And today's theme is nature.

(Describer) She and another hand put small potted plants on the table. She sniffs a cactus flower.

[birds chirping]

These are fake, which is not really relevant at all today, except for the word "conservation" is a word that you usually hear related to nature, and we're not talking about that kind of conservation, but we needed a theme. So in the last lesson, we introduced energy.

[bell rings]

Like, not for the first time ever because philosophers started doing that in the fourth century. But we learned a lot about energy and the different kinds of energy--energy of moving things called kinetic energy, and energy stored in fields, like gravitational potential energy. And energy can be transferred from one form to another, like potential to kinetic. We learned that. And of course, the big idea is that during this transfer of energy, energy is never lost. It is conserved.

[bell rings]

I don't know whether we actually even stated that last time. Did we? If we didn't, shame on us, because, girl, this is a huge idea. Energy is conserved. As they say, energy is neither created nor destroyed. It just changes forms. And here's a really cool chart showing where the many different kinds of energy on Earth come from and what they turn into. But today, right here on the internet, we're going to show how framing questions in terms of energy and its conservation can give extra useful insight for you and make problems way easier to solve, like way. For example, at the end of this lesson, we're going to revisit that problem about the bottle flipping. Does it go faster on the way up or the way down? And I promised you that we'd come back to it and do it in an easier way. And we're going to answer the age-old question--whoops-- we're going to answer the age-old question: Is it better to sit in the front or the back of a roller coaster? You could start debating now.

[music playing]

(Describer) Viewed from a back car, a roller coaster reaches the top of a hill, then speeds down.

[people screaming]

Amusement parks are awesome for physicists because there's physics everywhere. There's this viral video of a carnival ride that's a cage usually with two people inside attached to some stretchy cords. And then they pull the cage all the way down, and they let it go and it flies up like nobody's business. So in the video, there's this young couple on the ride. So the operator straps them in and tells them that their seat belts are way too loose right before he presses the button to launch the ride. And it's evil and hilarious. And you can look at it because I can't show it because of copyright. So in this ride, there's clearly conservation of energy happening. But the first question: Where does the energy come from? Older versions of this ride used huge bungee cords. But it turned out that that was kind of dangerous. You can actually see other viral videos of the bungees breaking right before the ride starts. I don't necessarily recommend looking those up, although no one gets hurt. So modern versions of this ride use steel cables hooked up to hundreds of smaller springs. The energy comes from the springs. This is interesting. It turns out, springs come in a lot of physics problems, including places that you wouldn't suspect, like in molecular bonds and quantum mechanics. So we should probably check out some springs.

[springing]

So what do springs do?

(Describer) They move the plants off the table and she brings in a large spring.

So when you stretch a spring, it wants to pull back to its original state. And when you push it or compress it, it wants to push back to its original state. The more that you push it or pull it, the more that it wants to push or pull back. So what's going on here? And I just want to complain a second about that very question, because physics classes often ask, "What's going on here?" But they don't really give you context. They don't explain what they mean by "What's going on here?" And what they usually mean is, Can we find an equation? Can we find an equation that's good enough to describe the behavior of a spring and one that we can use to predict how springs will behave, how much they'll push back, how much energy they store? So that's we're going to look for, that equation. A typical experimental setup with a spring trying to find this equation might look like this.

(Describer) Setting the spring aside, she pulls a big sheet of paper onto the table and catches a thrown marker.

There's a spring. You're going to have a spring with one end attached to the wall.

(Describer) She draws.

And the other end is attached to a force meter. Force. Like one of those scales that has a hook and a dial that tells you what the force is when you stretch or compress it. So let's graph what we would see for the force if we actually did this and stretched it to different distances. So on my graph...

(Describer) She draws positive and negative x and y axes.

I'll draw it over here. On my graph, I've got force on the y-axis in newtons. And I've got displacement on the x-axis in meters. And by "displacement," I mean how much the spring is stretched or compressed from its rest position. And we'll call the rest position right here x equals 0. So I can either go forward-- positive x is stretching-- or backward--negative x is compressing. So when I stretch the spring with a positive displacement in the x direction, the spring is trying to pull back toward the zero position. So the x is positive over here, but it's exerting a force in the negative direction, a minus F. So I'm going to have a point here, and positive x but minus F. And then if I stretch it even further, it has a greater position, positive x, but it's pulling with a bigger force, minus F. So I'm going to have a minus point over here.

(Describer) She plots a couple more points down and to the right of the first.

And maybe just a couple more points like that. So if you actually do this with a spring in real life, your dots should fall on a graph that's pretty much a straight line. So now let's try compressing the spring or pushing it in. And when I squeeze the spring, the end here undergoes a negative displacement, minus x. And the spring is now trying to push back in the opposite direction. So that's a positive force, a plus F. So I'm going to have a minus x and a positive F-- somewhere here; I'd have maybe another point here.

(Describer) Up and left.

And obviously if I was just at 0, nothing would be happening with the force. Now all these points should fall in approximately a straight line, too, because if I actually measure this spring's position in meters and the force exerted in newtons, then I would find that this relationship is linear. Ahhh! This linear relationship was discovered in the 1600s by an English scientist dude named Robert Hooke. So the relationship is called Hooke's law, or the spring force equation. And the equation that describes this graph is the equation that we were looking for. And it's F equals minus kx. The k here is the slope. And that's called the spring constant. And it tells you how stiff the spring is. So let's check the units of k. We got newtons on the left, and I've got meters on the right. So then k is going to have to be newtons per meter. Different springs have different spring constants depending on the size, composition, construction. So I measured this spring, I measured this spring earlier

(Describer) She gets a small spring and a small scale.

from a clicky pen--

(Describer) She compresses it.

me, me, me. And I measured on a scale. So I press it down until it maxed out at two ounces, which is 0.55 newtons. And I divided that by the displacement, and it only displaced about 2 millimeters, which is 0.002 meters. OK, I'm using this equation right now to figure out the spring constant. The force was two ounces, which is about 0.55 newtons. I'm looking for k. And then the x-- the x was really hard to measure. But it was about 2 millimeters-- or I need that in meters. It was negative, so negative 0.002 meters. And that works out to give me a spring constant of 275 newtons per meter, which is a pretty typical spring constant for a spring of this size. But a spring like this would definitely

(Describer) The large one.

have a bigger spring constant because it takes more newtons, more force, to go a certain number of meters. And the coils attached to your car wheels might have a spring constant of something like 50,000 newtons per meter. That's what the spring constant tells you. Fun fact, when you hear "spring," you probably imagine this or this.

(Describer) The small pen spring. The large one.

But some unexpected types of springs are other things, like maybe an archer's bow string or a pair of tweezers, because they spring. And some researchers have even studied the math of how a chameleon's tongue acts like a bunch of connected springs when it shoots out to catch bugs.

(Describer) A chameleon eats. Title: Yummy.

OK, cool. We have a model now for the force a spring exerts. But what about finding a model for the energy it stores? That sounds fun. I know there is energy there because when I let a spring go...

(Describer) She puts a small pillow on top of the large spring, which she compresses and lets go.

there is obviously a kinetic energy that appears. So you'll be happy to hear that you already know how to think about the stored energy in a spring, because we know how to think about work.

(Describer) She puts the spring and pillow aside. She moves the paper and writes.

OK, so work is... w equals F delta x, right? In the last lesson when we worked with work, we were using cases with a constant force. But here, though, we have a force that's not constant. As we stretch the spring, we change the force. What a headache. But don't despair, we can actually find and use the average force trick, which is very similar to the earlier lessons when we used the average velocity trick. So let's say we stretch our spring some distance, x. I'm going to mark it on this graph.

(Describer) On the x-axis.

So we stretch our spring some distance, x.

(Describer) She draws down to the diagonal.

Then the force of the spring is minus kx.

(Describer) She draws across to the y-axis. Zero.

And here, when the spring was relaxed before we stretched it, the force was 0. So then the average force it took to stretch this spring to this point is halfway between 0 and kx, or 1/2 kx.

(Describer) She marks that.

We don't actually have to worry about the minus sign right now because we don't care about the direction of the force, just the size of it. But now we can figure out the total work done to stretch the spring to this point, x. The total work is going to be the average force, which we just figured out, 1/2 kx, times our delta x which is the distance that I stretched it, x, which ends up being work equals 1/2 kx squared. Check it out. This is just the area under the curve here.

(Describer) ...between x and the diagonal.

Beautiful. That's the work that it takes to stretch or squeeze a spring by some amount, x. And that's also--surprise-- how much energy gets stored in the spring when it's stretched or compressed by some amount, x. So I can just write this as energy. This is a little confusing, because e as a symbol of energy is not super clear. What kind of energy we're talking about? So some people write this as U, and I'm going to do that. But you could also write it is PE, potential energy, with a little s, so you know you're talking about spring energy, or you could write it as energy with a little s, as long as everyone knows what you're talking about. So now I have a simple model for the energy stored in a compressed or stretched spring. So congratulations, we have made it past the part of the video where we say, What's going on here? And we come up with new models, which we did-- new equations to describe what is going on. So now we can think about the bungee-cord ride we talked about, which honestly terrifies me. So looking at this ride, I was really curious about the springs powering the ride. So I did some research and I found that the modern, safer version of this ride can have up to 720 springs

(Describer) She moves the paper.

all attached in parallel. So we'll draw it like this, springs all attached

(Describer) She draws a few springs inside a square frame.

in parallel. Imagine there's 700 of these. This thing is moving up and down and then pulling on one end of the steel cable. And then the cable's attached to a bunch of pulleys, like this.

(Describer) ...curving up and down in a connected frame.

And then the springs can compress and stretch a length of about 1.25 meters.

(Describer) She writes that by the springs.

So the steel cables attach to a bunch of pulleys so the springs don't have to stretch all the way to the ground; the springs get stretched only about 1.25 meters; and the humans then get launched to a typical altitude of 80 meters. So here they are starting at the ground. Here's my people. "Aah!"

(Describer) She draws people in a circle on the ground and in the air.

They get launched up to this height of 80 meters.

(Describer) It's shown in real life between two posts.

Whoo. So given all of this, I can figure out the spring constants of the springs. Exciting. So let's do the math. So the problem we're trying to solve is to find the spring constants. We have everything we need. A typical mass for the humans and the steel ball together could be around 400 kilograms. And we're going to need that mass. And if they're launched to 80 meters, then we can find the gravitational potential energy, which is mgh. P, potential energy, is mgh. Then we've got 400 kilograms, g is about 10 meters per second squared. If you're just joining these lessons, we typically use 10 for the acceleration of gravity because it's 9.81 meters per second squared, but we round it to 10, because it makes things easier. And h is 80 meters. Do all this math out and you get about 320,000 joules of gravitational potential energy up at the top of the ride. That's a lot. And where did all of that energy come from? From the stretched springs. Let's divide up 320,000 by around 700 springs to get 400. If you divide 320,000 by 700, you get 460 joules. And what does that? Well, that's the energy per spring. This is the energy stored in each spring when it's fully stretched. I'm looking for k. So I'll go back to my spring constant equation. I know the potential energy of the spring is 1/2 half kx squared. For each spring, it's 460 joules. So that's 1/2 k. And my x is not the height. It's this distance here, the 1.25 meters that the springs were stretched, squared. 1.25 meters squared. If I do all this math, I get that k is approximately 580 newtons per meter. Ta-dah! That's the spring constant of all these little springs. These are big springs. Now I want you to take a minute and look at our math model and convince yourself that it's OK what we did, to just divide up the energy between all the springs. This is often overlooked. Well, all the springs have that 1/2 in common. And they all have the x in common. So the only thing that might be different is the k. But let's assume that they're all the same spring. So it's the same spring constant. So, yeah, you can just divide it. Now ask yourself, Could I just add up all of my 700 spring constants and model my system as one big spring in a line? Do you get the same result? No, you don't. If you had a huge long spring and you just stretched that 1.25 meters, you would feel barely any force at all. So here's the problem for you. If I take a spring and I cut it in half, what happens to the spring constant? Does it stay the same or does it get bigger or smaller? I'll answer that in the next lesson. That's about it for conservation of energy and springs. But before we go, there's a couple more things I promised. Let's revisit a question we asked way back in lesson 2. When you throw an object like a bottle up into the air,

(Describer) She gets one.

does it take longer on the trip up or on the trip down, like this? Imagine it's not spinning; imagine it's solid, whatever. Conservation of energy makes this tricky problem so simple. When I throw the bottle up, it has some kinetic energy from that motion. It's moving up. And then when it gets to the top of the trajectory, most of the kinetic energy will become potential energy, gravitational potential energy. Notice that I said "most." It doesn't all go to potential energy because some of the bottle's energy will have leaked out into the surrounding air through air resistance. Energy wasn't lost here; it just leaked into the surrounding air as heat. It's still conserved overall. So when the bottle falls back down, its gravitational potential energy converts back into kinetic energy, and it loses some more energy to air resistance, again, on the way down. So the end of the bottle's trip, it has less kinetic energy than it did at the start of the trip. And since kinetic energy is 1/2 mv squared, that means energy is proportional to v squared. That means the bottle's velocity then is slower right before I catch it than it was when it left my hands. That means the average velocity on the trip down was slower than the average velocity on the trip up. And I'm done. The trip down took longer. No calculations, no kinematics. Just sweet, simple, conservation of energy. And now we can easily explain the challenge at the beginning of this lesson. Since energy is conserved, it can be transferred from one bottle to the other and back. And finally, the question that I promised I would answer: Is it better to sit on the front or the back of a roller coaster? Think about it.

(Describer) She puts the bottle down.

Let's draw it. So they crank you up that first hill. Whoo. Here's all the people.

(Describer) She draws a block with people near the top of the hill.

Whoo. They don't have arms. Doesn't matter. They crank you up the first hill and then they let you go, and whoosh--the entire ride is about converting gravitational potential energy into the sweet thrill of kinetic energy. It's 100% a conservation- of-energy joy ride. But if you want to maximize the thrill, should you sit at the front or the back? Does it matter? Is one actually faster? Let's think about that first hill. The roller coaster has a lot of potential energy up here. And then it goes, whoosh, down the hill. And when the first car reaches the bottom...

(Describer) She draws it there with people at the front and back.

we lost a few people. Whoops. So when the first car reaches the bottom, has all of the potential energy of the car been converted into kinetic energy? No. Because there's still some height of this back car. So there's still some potential energy that can be converted to kinetic energy. So there's still some roller coaster left to fall. So it hasn't reached its maximum speed yet. Once the last car has reached the bottom, then it's all kinetic energy, and you're going as fast as possible. So any time the roller coaster is falling, the back car gets to go down the hill or loop faster than the front car did, like a whiplash. And on the flip side, when the roller coaster starts going up a hill or a loop-- let's draw this is a loop, whoo-- going up a hill or a loop, the front car is going to enter that loop faster than the back car will because it's going to be going into the climb with the maximum kinetic energy. But as soon as it starts climbing the loop, then the kinetic energy starts changing into potential energy, and the back person ends up going a little slower. So there is a difference. But which one is better is up to you. Do you want to be heading into loops faster than anyone on the roller coaster? Sit up front. Are you a big fan of the whiplash and higher Gs leaving loops? Then sit in the back. Your call. But come on, it's totally the front. So there--with some careful application of physics, amusement parks can become even more fun, except that they just tell you where to sit and you don't get to choose. So that's today's lesson on conservation of energy. When they ask you what you learned on YouTube today, here are your two key takeaways. One, if you can, use conservation of energy to solve physics problems, and, two, the energy stored in a spring is 1/2 kx squared, where k is your spring constant. And here are all the problems we did in today's lessons, because the best way to make sure that you understand these concepts is to practice applying them yourselves. So do go, redo these problems. I promise, it'll help. Go practice the problems.

(Describer) Problem 1: You compress a spring by 0.2 centimeters. You measure a restorative force of 0.55 Newtons. What is the spring constant of that spring? Problem 2: A scary amusement park ride shoots you and your friend 80 meters into the air. The total mass of the people and car in the ride is 400 kilograms. What work was done by gravity on the trip up? How much potential energy is now stored in the gravitational field? Problem 3: All of that energy in the previous problem came from around 700 springs, in parallel, each compressed by 1.25 meters. What is the spring constant of each of those springs? Problem 4: Can you use the conservation of energy arguments to show that Dianna is wrong and the *back* of the roller coaster is by far the best place to sit? Edit: Heyyy! Signed, Dianna.

(describer) Problem one: You compress a spring by 0.2 centimeters. You measure a restorative force of 0.55 newtons. What is the spring constant of that spring? Problem two: A scary amusement park ride shoots you and your friend 80 meters into the air. The total mass of the people and car in the ride is 400 kilograms. What work was done by gravity on the trip up? How much potential energy is now stored in the gravitational field? Problem three: All of that energy

(in the previous problem) came from around 700 springs, in parallel, each compressed by 1.25 meters. What is the spring constant of each of those springs? Problem four: Can you use conservation of energy arguments to show that Dianna is wrong and the back of the roller coaster is by far the best place to sit? Edit: Heyyyy! Signed, Dianna. And now I'm going to tell you about something really cool that's just random and unrelated to this lesson at all, except for the fact that I used this little thing

(Describer) She gets the pillow.

a couple of times. You're probably like, what is that? although some of you may know. So a couple guys were doing a physics experiment a couple decades back. And they were getting this noise, this background signal, of a bunch of microwave radiation, like the microwave waves that are in your microwave oven. But they were hearing the microwaves with their antenna. They couldn't figure out what it was coming from. It turns out that there are microwaves coming from all over in the entire sky, all over the entire universe. This is a map of the microwaves of the little fluctuations, and more or less microwaves, across the universe. So what was the signal? What was this coming from? Well, this is a signal from the Big Bang. It turns out the microwaves are the radiation, the leftover image of the Big Bang from about 400,000 years right after the Big Bang. And we now have a map of that. This is called the cosmic microwave background. It doesn't have two eyes and a mouth, but the rest of this is correct.

(Describer) Printed on the pillow.

And what this shows us is a picture of what the universe looked like 400,000 years after the Big Bang. That's amazing. And actually, the waves weren't initially microwaves. They were more like orange-yellow light. But since the universe is expanding and stretching those waves over time, they've turned from small, orange-yellow waves into longer microwaves. And eventually they'll stretch into radio waves. And who knows, because we don't know whether the universe is going to continue expanding or collapse. Something for future physicists to figure out. Anyways, that's my fun physics of today. And now here's a message from a special guest. Hello there. My name's Sophia Chen. And I'm a writer for Physics Girl. In fact, I helped write the video that you just watched. I'm super excited for you to be taking this course, not just because I'm helping to write it, but also because physics is a huge part of my life. I majored in physics in college and I got a master's degree in it, and now I'm a science writer and science journalist who writes about physics. I've written for magazines like Wired and Science Magazine and Gizmodo. And of course, I get to put words in Dianna's mouth. And she wanted me to tell you a little bit about my favorite part about my job. And one of my favorite moments, I think, involved me traveling. And I actually got to go to Paris in 2018 and watch as the world redefined the metric system. I went to Versailles. I went into this hall with all these diplomats who voted to get rid of this hunk of metal that was the kilogram and replace it with Planck's constant. And so that was a really amazing moment. I guess what I wanted to say is just that physics is this really powerful and really insightful way of looking at the world. And I hope you stick with it because you never know where it'll take you. With that, happy physics-ing. I've always wanted to say that. Accessibility provided by the U.S. Department of Education.

(Describer) Titles: Executive Producer/Host/Writer: Dianna Cowern Lead Writer/Course Designer: Jeff Brock Producer: Laura Chernikoff Video Editor: Rachel Watson Researcher/Editor: Sophia Chen Researcher/Editor: Erika K. Carlson Videographer: Levi Butner Curriculum Consultants: Lucy Brock and Samantha Ward Accessibility provided by the US Department of Education.