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[Assistant] Put that baby bird right back in the tree. The baby bird was home. Just then the mother bird came back to the tree. Do you know who I am? She said to her baby.


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[Assistant] Yes, I know who you are, said the baby bird. You are not a kitten. You are not a hen. You are not a dog. You are not a cow. You are not a boat or a plane or a snort. You are a bird and you are my mother.


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[Assistant] I'm so happy that Snort was able to help the baby bird get back to his mother. Parents do so much for us.


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[Assistant] Is there something you could do to show your parents how much you appreciate them? Come back to read more stories together anytime you'd like.


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[Assistant] Or find more books to read yourself at ReadBrightly.com Bye!


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[Assistant] Welcome to Brightly Storytime. I'm Miss Linda, and today we're going to read My Friend Maggie.


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[Assistant] Written and illustrated by Hannah E. Harrison.


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[Assistant] She's great at splashing in mud puddles.


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[Assistant] She helps me reach the reddest apples.


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[Assistant] Only, Veronica doesn't think so.


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[Assistant] Veronica thinks Maggie's too big. And now that she mentions it,


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[Assistant] And her clothes are a little snuggish.


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[Assistant] I know I should stick up for Maggie.


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[Assistant] But I play with Veronica instead.


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[Assistant] And I pretend I don't see Maggie.


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[Assistant] Even though she's impossible to miss.


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[Assistant] Veronica decides that my teeth stick out too far and starts calling me Bucky.


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[Assistant] And do you know who comes charging in to stand up for me?


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[Assistant] What do you think it means to be a true friend? Maybe we could all learn a little something from Maggie.


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[Assistant] Come back again anytime to read more stories together or find more books to read yourself at ReadBrightly.com. Bye.


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[Assistant] I have no desire to stay in the place of a traitorous embarrassment. You have fallen from what little grace you had, and I know you'll pay for it.


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[Assistant] Hello, Pet. You're running the store, I see. I need some flour for my mackerel muffins. Okay, Nonna.


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[Assistant] Just a minute, Bo. Some dusters, Lord Stag?


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[Assistant] Running the general store is hard work, gull.


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[Assistant] I was thinking of asking Hachi.


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[Assistant] Yes, I was wondering if you, but perhaps if you don't mind.


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[Assistant] Hatsy, would you care to be my dancing partner at the dance party?


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[Assistant] kind of society that's quite different from ours because surely storytelling


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[Assistant] is much more diversified in the societies that we know.


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[Assistant] Uh, parents tell stories to children. Children ask, uh, for stories. We experience, uh, stories mediated through images as in television and movies. We read stories in print. Uh, I, I think that the,


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[Assistant] I think, or what's, what the challenge of storytelling is now.


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[Assistant] But John, don't you think that people make a distinction between a story that is true


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[Assistant] On the one hand, we think of story telling as a truth telling activity. Tell me the real story.


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[Assistant] Don't tell me stories, meaning don't tell me lies.


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[Assistant] And don't you think at the very center of the whole enterprise of storytelling, there is the fact that storytelling is an activity that faces in two directions. On the one hand, it's connected with an idea of truth.


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[Assistant] On the other hand, it's connected with an idea of invention, imagination, lies,


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[Assistant] One is thrilled by the story precisely because it describes something that can't happen. It's connected with fantasy.


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[Assistant] Who is telling what really happened, who is bringing exotic news and telling terrible stories generally, is only one model of storytelling.


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[Assistant] Well, of course it's true that, uh, I'm not where the story is taking place, but if you're talking about some kind of traditional storytelling which is grounded in a particular place, if I read a story of Tolstoy,


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[Assistant] I think I'm in 19th century or early 20th century Russia. I don't think it's taking place everywhere.


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[Assistant] Not necessarily. No, it might instruct my experience rather than apply to it.


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[Assistant] I mean I'm not looking to see myself projected in stories.


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[Assistant] Anymore than I'm trying to express myself when I write stories.


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[Assistant] Uhm, no I don't, I don't think stories are necessarily universal because they're read by very different kinds of people.


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[Assistant] Some, some stories, because the most powerful form of storytelling is very compact, and if it's very compact,


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[Assistant] It's likely to be very economical in its details and therefore


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[Assistant] This economy can be experienced by us with our modern ways of looking at things as something abstract. And in that sense, it may take on a certain universality.


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[Assistant] But I think it is the particularity of storytelling and the fact that we now have available to us stories from every part of the world.


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[Assistant] Maybe another way of looking at it from you because I really, uh, start not from the oral tradition, but from the literary tradition.


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[Assistant] If I think of storytelling, let's say that the parent to the child now usually in our society is reading a story rather than telling a story.


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[Assistant] We have so many stories available to us. In fact, sometimes people say that we have a surfeit of stories, which is why storytelling is often thought to be in a state of crisis.


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[Assistant] Uh, and so my model is originally a literary model and I'm a city dweller.


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[Assistant] And you're someone who has lived a large part of your life though you grew up, as I understand, in London.


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[Assistant] Uhm, in a peasant community in the south of France. And therefore you feel closer by choice because you've chosen to live there.


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[Assistant] Uh, to another, uh, what many people would think of as an older form of storytelling, which is, which is oral. And you have the stories of these people's lives whom you come to know.


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[Assistant] And you want to tell their stories to other people because there's, I think, an ethical assumption behind your work that there are stories which have not been told.


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[Assistant] Uh, certain kinds of people and certain kinds of experience which have not found voice.


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[Assistant] And you want to give voice to experiences which are neglected or regenerate or revive certain


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[Assistant] Forms of storytelling which are neglected now in terms, in favor of perhaps a more sophisticated, uh, city-oriented, uh,


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[Assistant] Print oriented kind of storytelling. Isn't that what you're saying? When you told this fascinating story about the shepherd and his unrequited love and his kind of suicide.


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[Assistant] The reason you wanted to tell this story is that you were moved by him.


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[Assistant] And he was a kind of person that perhaps you don't find in, in our literature as


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[Assistant] That's a point I think that's worth talking about, why death seems the natural end.


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[Assistant] of a story and in the most traditional form of storytelling I think that it is. Is it because a death is the natural end of a life and what stories are about are lives?


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[Assistant] See, I think they're about much more than that, and uhm,


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[Assistant] But that is certainly one model of storytelling, to tell the story of a life, to, to, to give the value of a life in the form of a story which has its natural end as a death.


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[Assistant] But if I think, for instance, of the first stories that I remember reading, which really marked me as a...


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[Assistant] As a person, and undoubtedly as a writer too, uh, were the stories of Edgar Allan Poe. And those are not stories of a life.


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[Assistant] Uh, but they are real stories, absolutely unforgettable stories.


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[Assistant] Which may or may not end in death, it often does. But they are terrifying stories. They are fabulous stories. And on the whole, they are invented stories.


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[Assistant] And the, the storyteller is the person who invents. We have examples of that now in writers like Calvino and Borges, who certainly think of themselves as storytellers, but they're not telling stories of lives. They're telling stories of dilemmas, let's say, which often are catastrophes.


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[Assistant] What about that kind of storytelling, as opposed to rendering a life?


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[Assistant] I think people want to travel outside themselves. I think there's a need for a vicarious experience that is not necessarily identical with the need for truth. I think there's a need, uh, uh, for fantasy.


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[Assistant] I think there is a longing to see taboos broken, violated. Uhm, there is a, well, the imagination runs riot.


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[Assistant] Uh, or the life of someone I know. But I feel as if I'm lending that to, to, to something that is basically invented and therefore basically fantastic.


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[Assistant] And I think of fiction, but again, I'm just speaking for myself, uh, and thinking of examples of fiction that I like very much. I think of fiction as, uhm, a kind of, uh, speculativeness, a kind of fantasy. I'm interested in, in, in moral fantasies, moral science fiction even.


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[Assistant] That's very strange to me what you're saying because I can't feel the subjectivity of the listener or the reader.


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[Assistant] Of course I think if I write a story it will be read. I wouldn't write if I didn't think that I could be read.


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[Assistant] But I can't imagine how that could actually be part of the story.


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[Assistant] Maybe that's because I think the difference between hearing stories and reading them is, is greater than, than you do. We have to stop now and we'll be back in a moment.


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[Assistant] We have an old musical instrument. It is called a clavichord.


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[Assistant] The instrument was bought by my grandfather many years ago.


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[Assistant] She tried to play jazz on it. She struck the keys too hard and two of the strings were broken. My father was shocked. Now we are not allowed to touch it.


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[Assistant] It is being repaired by a friend of my father's.


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[Assistant] Eventually, I did rebuild my identity, and the why for me was about competing again, because pursuing success and risking failure was simply how I felt normal.


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[Assistant] And I finished the rebuild on the 10th anniversary of losing my sight. I took part in a 43-day expedition race in the coldest, most remote, most challenging place on earth.


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[Assistant] It was the first race to the South Pole since Shackleton, Scott and Amundsen set foot in Antarctica a hundred years before.


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[Assistant] And putting the demons of blindness behind me with every step towards the pole, it offered me a long lasting sense of contentment.


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[Assistant] As it turned out, I would need that in reserve, because one year after my return, in arguably the safest place on earth, a bedroom at a friend's house, I fell from a third-story window onto the concrete below. I don't know how it happened. I think I must have got up to go to the bathroom,


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[Assistant] And because I'm blind, I used to run my hand along the wall to find my way.


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[Assistant] That night, my hand found an open space where the closed window should have been. And I cartwheeled out.


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[Assistant] My friends who found me thought I was dead. When I got to hospital, the doctors thought I was going to die and when I realized what was happening to me, I


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[Assistant] Thought that dying might have been, uh, might have been the best outcome. And lying in intensive care, facing the prospect of being blind and paralyzed, high on morphine, I was trying to make sense of what was going on.


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[Assistant] Trying to explain how I should respond. It was called Optimist Realist.


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[Assistant] Or something else and it drew on the experiences of Admiral Stockdale who was a POW in the Vietnam War.


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[Assistant] He was incarcerated, tortured for over seven years. Circumstances were bleak, but he survived. The ones who didn't survive were the optimists.


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[Assistant] They said we'll be out by Christmas and Christmas would come and Christmas would go and then it would be


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[Assistant] Christmas again and when they didn't get out, they became


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[Assistant] Disappointed, demoralized and many of them died in their cells. Stockdale.


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[Assistant] Was a realist. He was inspired by the stoic philosophers and


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[Assistant] He confronted the brutal facts of his circumstances while maintaining a faith that he would prevail in the end.


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[Assistant] And in that blog I was trying to apply his thinking as a realist to my increasingly bleak circumstances.


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[Assistant] During the many months of heart infections and kidney infections after my fall, at the very edge of survival,


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[Assistant] How do you resolve the tension between acceptance and hope? And it's that that we want to explore with you now.


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[Assistant] Spinal cord injury strikes at the very heart of what it means to be human. And it had turned me from my upright, standing, running form into a seated compromise of myself.


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[Assistant] And it's not just the lack of feeling and movement. Paralysis also interferes with the body's internal systems which are designed to keep us alive.


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[Assistant] Multiple infections, nerve pain, spasms, shortened life spans are common. And these are the things that exhaust even the most determined of the 60 million people around the world who are paralysed.


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[Assistant] Over 16 months in hospital, Simon and I were presented with the expert view that hoping for a cure had proven to be psychologically damaging.


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[Assistant] It was like the formal medical system was canceling hope in favor of acceptance alone. But ...


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[Assistant] Canceling hope ran contrary to everything that we believed in.


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[Assistant] Yes, up to this point in history, it had proven to be impossible to find a cure for paralysis, but history is filled with accounts of the impossible made possible through human endeavor.


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[Assistant] The kind of human endeavor that took explorers to the South Pole at the start of the last century. And the kind of human endeavor


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[Assistant] Why can't that same human endeavor cure paralysis in our lifetime?


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[Assistant] Wasn't a normal feeling, but with the stimulator turned on upright in my exo, my legs felt substantial. I could feel the meat of my muscles on the bones of my legs. And as I walked,


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[Assistant] Because of the stimulation, I was able to voluntarily move my paralysed legs and as I did more,


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[Assistant] The robot intelligently did less. My heart rate got to a normal running training zone of 140 to 160 beats per minute and my muscles which had almost entirely


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[Assistant] Disappeared. Started to come back. And during some standard testing throughout the process, flattened my back. Twelve weeks.


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[Assistant] Six months and three whole years after I fell out that window and became paralyzed, the scientists turned the stimulator on


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[Assistant] And clearly we're not finished, because when we left that pilot study and went back to Dublin, I rolled home in my wheelchair, and I'm still paralyzed, and I'm still blind, and we're primarily focusing on the paralysis at the moment, but being at this conference, we're kind of interested. If anyone does have a cure for blindness, we'll take that as well.


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[Assistant] But if you remember the blog that I mentioned, it posed the question of how we should respond, optimist, realist or something else. And I think we've come to understand that the optimists


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[Assistant] rely on hope alone, and they risk being disappointed and demoralized. The realists, on the other hand,


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[Assistant] They accept the brutal facts and they keep hope alive as well. The realists have managed to resolve the tension between acceptance and hope by running them in parallel. And that's what Simon and I have been trying to do over the last number of years.


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[Assistant] I accept the wheelchair. It's almost impossible not to. And we're sad sometimes for what we've lost.


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[Assistant] I accept that I and other wheelchair users can and do live fulfilling lives despite the nerve pain and the spasms and the infections and the shortened lifespans.


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[Assistant] And I accept that it is way more difficult for people who are paralysed from the neck down, for those who rely on ventilators to breathe and for those who don't have access to adequate free healthcare.


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[Assistant] So that is why we also hope for another life, a life where we have created a cure through collaboration, a cure


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[Assistant] That we are actively working to release from university labs around the world and share with everyone who needs it.


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[Assistant] I have just received a letter from my brother, Tim. He is in Australia. He has been there for six months. Tim is an engineer. He is working for a big firm.


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[Assistant] A small town in the centre of Australia. He will soon visit Darwin. From there, he will fly to Perth. My brother has never been abroad before, so he is finding this trip very exciting.


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[Assistant] I don't know, Lily. It'll snow when it's cold enough.


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[Assistant] No chance you can conjure us a couple of those, can ya? Sadly, no. I'm afraid without my grimoire, my powers are just a tad limited in the human world. What, you can't memorize your fucking spells? Oh, your memory's so great, what's his phone number? Fuck you. Exactly.


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[Assistant] Thank you for coming on and taking the time to talk to us today. There was something you mentioned earlier when you were giving an explanation of the problem of first principles that Sherman explores in this book.


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[Assistant] Sort of buried history of a materialism which is sort of...


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[Assistant] Almost in principle disorder that starts from disorder rather than order. Um, and I was wondering if you think there's any relation between what perhaps Althazair is trying to do there, what we're reading in Sherman and perhaps in your own work.


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[Assistant] In so far as it's still, it still remains a kind of systematic and metaphysical understanding of the world. It's precisely what Sherman thinks is, it's almost already dead, right? Exactly. Okay. That's great. Thank you.


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[Assistant] Because I, I'm working on the idea of radical democracy, and that's, democracy, of course, is quite an important element of Rancière's thought.


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[Assistant] This is more of a comment than a question, but uhm, you're right, I sort of, I agree with you, I noticed in Rancière's thought, it's that he comes so close to wanting to say something like anarchism.


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[Assistant] Yeah, and politics and police in general. Politics, of course, as you said, he will grant it a certain place within our lives, but it's always a kind of...


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[Assistant] So I was, I was thinking back to the, uhm, the Sherman text and this idea of the closure of being, uhm, sort of the end of, uh, metaphysics and systematicity and, and so on. Uhm, and I suppose I was wondering,


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[Assistant] For Nietzsche, at least on my reading of Nietzsche, there's this worry about the modern age following the death of metaphysics in a certain way, of a kind of nihilism. And in a book by Bernard Regenster, he talks about this and he says one of the kinds of nihilism that Nietzsche worries about


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[Assistant] Is uhm, a kind of groundlessness. The, the lack of any.


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[Assistant] Presumably Sherman's conclusion is that this is precisely the situation that we have.


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[Assistant] Uhm, and so I was wondering when it comes to, uhm, to, to praxis, to, to anarchism in, in particular, uhm, how, how should we think about, uhm,


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[Assistant] Uh, this, this lack of principles, is, is it, uhm, is there some way of, uhm, orienting ourselves without them, or, yeaah, I don't know any better way to say it, how, how, how do we get by without, uhm, uh, grounding for these principles, or without principles at all?


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[Assistant] Yeah, no, that's great. Uhm, I wanted to put it, the question to you and then you get it, you know, kind of the answer I have to admit I was somewhat looking for in a way. Uhm, it validates quite a bit of what I've been researching, but uhm, it's a great answer. Thank you.


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[Assistant] Hey guys. So in part two of the series, I made a mistake with the code that manages our player animation.


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[Assistant] So the number right here is just the XFrame variable of the player, which if you remember is basically controlling what frame the player is on. So you can see here that the last frame,


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[Assistant] Frame 8, which is actually the 9th frame, remember, because we're starting at 0. So frame 8 isn't playing properly. It's only staying here for a fraction of a second before looping back to the start. And that's not what we want, we want the frame to play fully before we loop.


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[Assistant] Now, when we're animating, it doesn't actually just snap from one integer to the other. I've just done that for simplicity right here, but our XFrame is actually changing by fractions of a number.


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[Assistant] Remember it was our NMSP, which was 12 over 60, so that's gonna be a decimal number.


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[Assistant] So, and as an aside, this is also how the player's image index is gonna be working, even if you're not using a sort of custom animation system, like the one that we've set up. What this means is that, so for example, frame 0, the first frame,


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[Assistant] Frame 0 is going to be playing while XFrame is equal to a number between 0 and 1. Then once it hits frame 1, frame 1 is going to start playing while XFrame is a number between 1 and 2, and it goes on like that.


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[Assistant] So our frame 8 should be playing until it hits 9, but that's not what's happening here, as you can see. It's getting to 8 and then immediately coming back to the start to 1. So let's look at our code.


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[Assistant] So this right here is saying if X frame is less than anim length minus one, then increment it. Otherwise, loop back to the beginning.


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[Assistant] So if you recall, we had nine character frames for our walk animation. Our anim length was nine. So if we substitute that in, we're going to get, if X frame is less than nine minus one, then increment it, but otherwise loop back to the beginning. So you can see our problem.


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[Assistant] It's only allowing us to increment if we're under 8. And as soon as we get over 8, it's gonna loop back to the beginning. It's not gonna allow anything between 8 and 9.


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[Assistant] So we're gonna have to change this. So firstly, let's drag this code up here. I think it makes more sense to be dealing with what frame we're on before we draw the character. Right now, because it's down here, it's sort of preparing the X frame for the next step, which to me doesn't really make much sense, so let's move it up. Okay, now we have to change this.


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[Assistant] So we're gonna change the logic a little bit. What we're gonna say instead is only increment the X frame if it doesn't put us over the maximum number of frames we have.


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[Assistant] If this is less than our maximum anim length, then go ahead and increment it. So if incrementing it would put us over the max anim length, then loop back to the start. Okay, and if we have a look in game now,


[User] Please transcribe this text

[Assistant] We can see that it's working properly. It gets to 8 and it plays through the whole animation before looping back to the start. So that's it. I'm sorry for that mistake guys and a big thank you to Sam for pointing that out to me. I hope it's helped you get a better understanding of how our animations work. So bye guys and I'll see you next time.


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[Assistant] Middle Earth is the Midgard that we've encountered in Old Norse, or Midanyerd in Anglo-Saxon. And it simply, in those contexts, means the Earth in the middle between the sky and hell, surrounded by the ocean.


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[Assistant] The One Ring actually has a malevolent spirit, part of Sauron's spirit, living inside of it. And so this malevolent spirit works on people to change them, to manipulate them, to do evil things.


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[Assistant] And the ring acts as an addiction. The longer you have it, the more you desire it. It's like a bottomless pit.


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[Assistant] Originally. One day, he and his friend Deagal went fishing. And Deagal sees something glinting in the bottom of the river. He pulls it out. It's the ring. And it's so beautiful.


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[Assistant] But Gollum, whose name was Smagol at that time, wants the ring.


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[Assistant] Who is an interesting name because it means wise in Old Norse and Anglo-Saxon. And Frodo is the one who gets stuck with the ring.


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[Assistant] Hobbits are little people, probably four feet or shorter. They don't wear shoes because they have very thick soles on the bottom of their feet and lots of fur on the top of their feet. They're sort of homebodies, they don't ever really go in for adventures.


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[Assistant] Frodo is not a typical hobbit because he's learned.


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[Assistant] And he cares about the outside world enough to sacrifice everything he actually loves.


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[Assistant] Huge pile of papers, poems written in elvish, and english, and histories, and the publishers were like, ughh, we have no idea what to do with this.


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[Assistant] Tolkien was an extremely devout Roman Catholic for reasons of personal faith and also for family history. His mother converted to Catholicism and when she did, her family sort of disowned her.


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[Assistant] She raised her two children Catholic and then she died of diabetes when Tolkien was very young. He was adopted by a Roman Catholic priest who took care of him and his brother. So the whole work is informed by Catholic thought.


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[Assistant] The famous stories he was grading exams, a student had left a page blank, and Tolkien wrote on it, in a hole in the ground there lived a hobbit.


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[Assistant] Tolkien actually wanted to make the runes be representative of a real language.


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[Assistant] It was an idea of secret writing, of magic writing, but also it was connected with his invented languages.


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[Assistant] It's got very close connections with Dante's description of hell. In that there's the burning plane in hell, the dry desert with the flakes of fire falling from the sky.


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[Assistant] In line 512 of Beowulf, there's a description of all the evil creatures that have been descended from Cain after Cain killed his brother Abel. And those are Eotanas and Ylpha and Orcneas. And that is Etans and Elves and Orcneas.


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[Assistant] Gandalf has also been compared by some people to Jesus. He sacrifices himself, is dead, and comes back clothed in white.


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[Assistant] It sucks in everything good about you and about your personality, just like any other kind of addiction, until all you can think about is the ring.


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[Assistant] The whole work is informed by Catholic thoughts. The very end Tolkien said was illustrating the last two petitions of the Lord's Prayer. He says,


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[Assistant] Frodo has made it to the very crack of doom. The edge of the chasm in the volcano where the ring was forged. And he has the ring on its chain. But he can't destroy it. It's become too much of his personality.


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[Assistant] As evil as Gollum is, it's Gollum who saves Middle Earth by doing something evil.


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[Assistant] If Gollum hadn't done that, the world wouldn't have been saved. So it's a nice little twist about how this all works together.


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[Assistant] This is a relatively simple topic. In fact, any of you took any kind of


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[Assistant] High school physics, you would have done this thing with fluids.


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[Assistant] So this is the subject for today, fluid dynamics.


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[Assistant] And static, so I'll start with a simple static problem.


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[Assistant] We are going to take, whenever I say fluid, uh, you are free to imagine water or oil. That's it. That's a good enough example of a fluid. One important property of the fluid, the density denoted by rho.


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[Assistant] Density of water, I would probably denote by a subscript w, and you know that's mass per unit volume. For water, that happens to be 1000 kilograms per cubic meter.


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[Assistant] So let's not linger there, I think that's a fairly simple concept. This motorcycle concept is the one of pressure.


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[Assistant] So you have the notion of a pressure. If you go into a swimming pool and you dive down to the bottom, you know the pressure is going up. So what's the formal definition of pressure? Is it a vector? Is it a scalar? Does it have a magnitude? Does it have a direction? That's what I want to explain to you.


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[Assistant] So you can pick a point on the fluid there, and say the pressure there is such and such. What do we mean by that? We mean by that the following. If you get into that fluid,


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[Assistant] And you want to carve out a little space for yourself. You want to make a little cube, maybe a glass cube, and you want to live inside that cube. So I'm going to blow up the cube, like this.


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[Assistant] The water is trying to push you in from all sides and compress this cube.


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[Assistant] You therefore have to push out on the two, on all the walls.


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[Assistant] If the force you exert on this wall is some F,


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[Assistant] And the area of that wall is A. That ratio is called a pressure.


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[Assistant] Pressure is an intensive measure of how hard the water is trying to push in.


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[Assistant] That pressure is still there, but one way to measure the pressure is to try to go in there and push the fluid out and ask, how hard does it push you?


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[Assistant] But meter square, and we use another name for that. Call it Pascal.


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[Assistant] 1 Pascal is 1 Newton per meter squared. Here's another example of pressure. You have a gas.


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[Assistant] We'll do a lot of this after the break. There is some gas inside the cylinder. There's the piston.


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[Assistant] Now if the pressure of the gas and the pressure of the outside world are the same, there's nothing you have to do, assuming the piston is maskless.


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[Assistant] But if you want to increase the pressure in the gas, you put some extra weight.


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[Assistant] Is the atmospheric pressure plus the extra weight you apply divided by the area of this piston. Because atmospheric pressure is everywhere.


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[Assistant] So when you push down on the piston here to compress the gas further, you're adding to the atmospheric pressure this extra force divided by area.


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[Assistant] Sometimes, I mean, this is called the absolute pressure, and that's called the atmospheric pressure, and that's called the gauge pressure. The gauge pressure is the pressure on top of atmospheric pressure. For example, when your car has a flat,


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[Assistant] The bright side of it is, the pressure inside the tire is in fact equal to atmospheric pressure.


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[Assistant] Doesn't help you because there's a pressure inside and there's a pressure outside. And if you want to keep your car moving, you really have to increase the pressure in the tire. And when you stick this gauge in, you measure something, 32 pounds per square inch, that's the gauge pressure. That's in excess of the atmospheric pressure.


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[Assistant] So you should understand that the pressure is a condition in a fluid.


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[Assistant] And one of the important properties of pressure is that if you took a fluid and you went to a certain height, all points of that height have the same pressure.


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[Assistant] And we understand that as follows. So I go to this fluid. I imagine in my mind a little cylindrical section of the same fluid.


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[Assistant] Just draw a dotted line around that region and focus on that little chunk of fluid and say,


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[Assistant] Can the pressure on the two sides, this face and that face, be different?


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[Assistant] The answer is no, because the pressure on the left is bigger than the pressure on the right. I take a cylinder of some area A, pressure times area on the left will exceed pressure times area on the right, and therefore the fluid should move to the right. But it's not doing anything. It's in equilibrium, and the only way that can happen is if it's pushed equally from both sides. So notice that the pressure


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[Assistant] On this chunk, from the left, point this way, and then that chunk points that way. They're trying to push it in. So pressure cannot change at a given depth.


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[Assistant] Remember, this is not a real cylinder. This is the same water. We mentally have isolated a part of the water that looks like a cylinder. Maybe you want to draw it in dotted lines. It's just a fluid of cylindrical shape, base area A and height H. I can ask the following question.


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[Assistant] Let's say h1 is the measure of that one from the surface, and h2


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[Assistant] This is the depth from the surface to the lower phase of the cylinder.


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[Assistant] Pushing up and pushing down will cancel. No net force on the cylinder. You can ask them, what's keeping the cylinder of water from falling down?


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[Assistant] Well, there has to be a force to equal the weight of that cylinder. Therefore, there has to be a net upward force. That means the pressure down here, pushing up, better be higher than the pressure on the top. We're now going to calculate what the pressure difference is. So, let's call the pressure downstairs P2 and upstairs P1. Then, the upward force is P1 times A.


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[Assistant] That's the net upward force. That's got to be equal to the weight of that amount of water.


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[Assistant] The weight is found by first the mass, which is area times


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[Assistant] How you keep track of science is a little subtle. I'm balancing two magnitudes. I'm balancing net upward force.


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[Assistant] Because this area, see the pressure between two points.


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[Assistant] Should not depend on the area of this fictitious cylinder I took. In fact, I find it is equal to ρG times h2 minus h1, which is the difference in the depth of these two points. Yes?


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[Assistant] In fact, it really happens to be normal, our perpendicular area, because the pressure will push straight up and straight down. How about forces on the side of the cylinder?


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[Assistant] They cancel at every height, because at every height, the push from the left and right are equal. I already showed you that. So for the mechanical stability of this chunk,


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[Assistant] I need all forces to add up to zero. Horizontally, there is no gravity and the pressure at every height is equal. So that cancels out. Vertically, that is the force of gravity and is cancelled by the difference in pressure. So we can see that P2 is equal to P1 plus ρg times the height difference.


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[Assistant] And people write this formula as follows. It's very standard to take P1 to be the point right at the surface.


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[Assistant] And P2 to be any point inside, and to call the depth of that simply as h.


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[Assistant] Then the pressure at the point P in the fluid is the pressure at the top.


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[Assistant] So I don't have an H1 and an H2 because H1 I've chosen to be 0 and H2 I'm simply calling H.


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[Assistant] This says something very simple. If you go to a lake, the surface of the lake, the pressure is due to the atmosphere. You take a dive, you go down some depth H, the pressure goes up by this amount.


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[Assistant] Now if you go to the bottom of the ocean, it can be an incredible amount of pressure. That's why you and I cannot survive in the bottom of the ocean.


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[Assistant] Because the outside pressure, inside pressure is usually atmospheric pressure. You're breathing the air in your lungs, you go down with that. Outside is atmospheric plus this, and that can kill you. That's why when you build a submarine, you gotta make sure it can withstand the pressure.


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[Assistant] But fish don't have the problem because fish are breathing the water. The water is going into the system and outside the system. So that's one way for you to live.


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[Assistant] What's the origin of that? The origin of the atmospheric pressure is that we are ourselves living in the bottom of a pool, but it's filled with air. The air above our heads goes on for maybe a hundred miles, but the density decreases.


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[Assistant] Eventually vanishes, so I cannot tell you precisely where the atmosphere ends, but I can say the following.


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[Assistant] If I go far above where the absolute pressure is zero in free space, that is just vacuum. The pressure here is the atmospheric pressure. Atmospheric pressure at the bottom of the earth


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[Assistant] So, rho is equal to 0, plus rho gh, where rho is the density of air, g is g, and h is the height of the atmosphere.


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[Assistant] And that happens to be 10 to the 5 pascals.


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[Assistant] So we are living in the bottom of a pool where the pressure is 10 to the 5 pascals relative to empty space, interstellar space.


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[Assistant] But as I said, that pressure doesn't kill us because the pressure can be felt both from the outside pushing in and inside going through your nostrils and everything else pushing out.


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[Assistant] But you have all seen the dramatic experiment where you take a can of something and you heat it up so the air goes out. Then you seal it.


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[Assistant] Then the air that's been driven out reduces the pressure. So when you seal it and you cool it, then the pressure drops.


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[Assistant] And even the drop in pressure doesn't even drop down to zero, but it's big enough for the whole can to implode.


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[Assistant] What you get here, let's do the following.


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[Assistant] This is rho gh, and you can ask yourself the following question. If this was a swimming pool, how high would the water be?


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[Assistant] In other words, what height of water above the earth will produce the same pressure as our atmosphere does? Then I say, if I want 10 to the 5 Pascals,


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[Assistant] That the density of water is 10 to the 3, g, let's pretend, is 10, and h is what I'm looking for. Cancel all the powers of 10, h is like 10 meters, and it turns out is, to the best of my knowledge,


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[Assistant] So we are at the bottom of a pool. If it is filled with water it will be 32 feet of water.


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[Assistant] We are going to take this formula P equals P0 plus ρgh and put it to work. Get some mileage out of that. So what are the things we can do with that formula? First thing you can do is to build yourself a barometer.


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[Assistant] Now the atmospheric pressure, when I said it's 10 to the 5, that's a typical pressure. It doesn't really stay locked in the value. Each day there are fluctuations. That's why the weather person tells you pressure's going up, pressure's going down. So, let's find a way to measure pressure, and here is one way to do that.


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[Assistant] A can of something, fill it with some liquid, take a test tube,


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[Assistant] Evacuate it completely. Suck all the air out of it and stick it into this.


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[Assistant] When you do, there is complete vacuum here and the atmosphere is pushing down, so the fluid will rise up to some height h.


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[Assistant] Zero pressure here, plus this ρgh, which is the pressure there, will be the same as the pressure here, because they are two points of the same height. Pressure here is the atmospheric pressure.


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[Assistant] So pressure here, atmospheric pressure, is 0 at the top of the tube, plus ρgh.


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[Assistant] So if you build this gadget, this barometer, out of water, the water column will rise to a height of 32 feet.


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[Assistant] But now nobody wants a gadget 32 feet high, so what you use instead is mercury.


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[Assistant] Because it's very dense. So you want to get the same atmospheric pressure, but you want to have a bigger rho and get a smaller H.


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[Assistant] If you go look up a book and find the density of mercury, you're going to find the height is something like, I don't know, 750, 780 millimeters.


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[Assistant] That's why the weather guy says the pressure today is so many millimeters. And the mercury is dropping.


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[Assistant] Now I'm not sure why they bothered to give the numbers, because for most of us, including me, those numbers don't mean anything.


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[Assistant] Here's the number, 746 millimeters. Does it speak to you? Not to me. So it, it speaks to one of you guys.


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[Assistant] But when I hear the mercury, I think it's just a waste of time. Anyway, they are telling you.


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[Assistant] How much the mercury is, okay? You can use any fluid you like, but you gotta agree on using mercury because if it's 760 millimeters and it's water the person's talking about, of course you are in serious trouble. Understood? We are talking about mercury.


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[Assistant] Also, Mercury is used in thermometers. That's the subject of after the break. So there the Mercury falling could stand for falling temperatures and rising temperatures, but here it's for falling pressure and rising pressure. So this is the first gadget you can build with what I've taught you.


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[Assistant] Next gadget you can build is you can imagine, uh, this is you trying to drink, uh, this is your ears and nose and what not. You've got a straw and you're trying to drink something.


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[Assistant] So how do you do that? You know, when you drink from a straw, you create a partial vacuum in your mouth. So the pressure here, so in the case of the fluid that you want to drink, this is your head. The pressure here is less than the atmosphere. If it's less than the atmosphere by some amount, then the fluid can start climbing up to some height.


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[Assistant] And climb to such a high so that the lower pressure plus that ρgh will equal atmospheric pressure.


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[Assistant] If you just, if you want to just make it to your mouth, it depends on the length of the straw, the height of the straw. If you want to suck it up a certain height, then that height, ρgh,


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[Assistant] Plus the pressure in your mouth is the pressure here, and that's the atmospheric pressure.


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[Assistant] So if you want to drink water from a well that is more than 32 feet deep, you're out of luck. Even if your face and your head are a complete vacuum, you cannot get the water to climb more than 32 feet.


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[Assistant] So that's another illustration of this. So, one more example of this P equals P0 plus ρgh is, if I give you another fluid,


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[Assistant] And this is the other fluid. So this is oil, and this is water.


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[Assistant] If the two heights were equal, then you know we're talking about the same fluid, but this is supposed to tell you that oil is less dense than water, and we can check that by comparing two points in this fluid at the same height, and saying the pressure must be the same at those points.


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[Assistant] You understand how I get that? That pressure and that pressure are equal, because you can draw a cylinder there horizontally, which cannot be pushed.


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[Assistant] Uh, sideways. From here to this pressure at that ρgh and that ρgh, and you come to that point, you conclude that pressure is also equal.


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[Assistant] You cannot jump into the new fluid here because the rho for here and the rho for this are different.


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[Assistant] But these two points have the same pressure. They are in the same fluid.


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[Assistant] So let me write that statement that this pressure and this pressure are equal by saying atmospheric pressure is on the top for both of them.


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[Assistant] Height goes to H2. First fluid, the height is equal to H1. The density is ρ1 and ρ2. Atmospheric pressures cancel. Then you find H1 over H2 is equal to ρ2 over ρ1.


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[Assistant] So by comparing the heights, you can find the relative density.


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[Assistant] Whatever it is, if you know the density of one fluid, you can find the density of the other.


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[Assistant] Okay, yet another application of this law that the pressure is equal at a given height is the famous hydraulic press.


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[Assistant] Here are two pistons of different radii. This has got cross-section area A1, this has got cross-section area A2. And here, I want to put some incompressible fluid.


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[Assistant] Is something whose volume cannot be changed no matter how much you press it. The water is pretty close to incompressible.


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[Assistant] It does have a compressibility, but it's not going to change very much for our purposes. So now here, I have a piston, and I have a piston here, and I push down here with the force F1, and I ask, what force will I get at the other side?


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[Assistant] You might think F1 equals F2, but since these fluids are at the same height, we only know P1 equals P2. That means F1 over A1 is equal to F2 over A2. So this fluid here,


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[Assistant] But I want to amplify the force. So what do you really want to do? Well, it's a matter of who you want to emphasize. So what do you want to emphasize here? Maybe if I drew a picture, I would know exactly what I'm trying to do. This picture is fine. In practice, this is not what you do. You want to push down here and raise something there.


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[Assistant] Because what you really imagine is some elephants. Some elephants standing here, and you want to lift the elephant by applying a force here.


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[Assistant] So let's still call your force F2. The force on the other side is the force you apply times A1 over A2. So A1 over A2 could be 100.


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[Assistant] What that means is, if you apply 1 Newton here, you'll get 100 Newtons on the other side.


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[Assistant] That's the way to max, to convert a small force into a large force.


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[Assistant] If you have a support like this, you can put a weight here.


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[Assistant] And let me see, that's right. So tiny weight here can lift a big weight here because that mg times that distance and this mg times this distance will be equal. But you must know, even from that example, that you don't get something for nothing. In other words, if you lift the elephant here, by pushing down here, the fact that the forces don't match is perfectly okay.


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[Assistant] Must be the work delivered at the other side, long conservation of energy. The work you do on this side is the force multiplied by the distance.


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[Assistant] Force multiplied by distance, I'm going to write as the pressure times the cross-section area times distance.


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[Assistant] But what is A times A2 times dx2? A2 times dx2, if you push the liquid here, if you moved it a distance, Δx2,


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[Assistant] Area times Δx2 is the volume of fluid you push down here. That's the volume that will come up on the other side.


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[Assistant] So you're free to write that as P2 times A1, Δx1.


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[Assistant] So he just says that the force times distance on one side is the force times distance on the other side.


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[Assistant] That means the work you actually do is not amplified by the process. You cannot get more jewels out of one side by any device. What you put in is what you'll get out. But still it's useful because in practice,


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[Assistant] You may have to move a whole meter here to lift the elephant by one centimeter.


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[Assistant] But the point is you can lift elephants this way. That's the important thing. That's why it's worth doing.


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[Assistant] And this is how you, the brake in your car works. You know, you pump the brake pedal, there's a little cylinder there, and there's a fluid there, and the fluid is pushed by your feet. And you push it quite a bit, several centimeters. At the other end, there's another cylinder,


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[Assistant] Whose piston is right next to the drum that's rotating and pushes on the drum. And it exerts an enormous amount of force.


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[Assistant] But it moves a very tiny amount. The shoes that grab your rotating drum move a very tiny amount, whereas your feet move a large amount.


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[Assistant] So a lot of hydraulics is based on this simple amplification of force.


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[Assistant] So, we all know the conditions under which this was discovered, so I will not go into that, other than to say that Mr. Archimedes noticed that if you immerse something in a fluid, it seems to weigh less.


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[Assistant] What I mean by that is that if you attach it to some kind of a spring balance and you weight it,


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[Assistant] So that the kx of the spring was the mg of the object, and if you did the same thing now, you will find it seems to weigh less.


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[Assistant] And the question was, how much less? And our community's answer is very simple. The amount by which you lose the weight, or the weight loss,


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[Assistant] Because we don't, right now, so many years after Archimedes, we will not accept this unfaith. We want to be able to show this is the case.


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[Assistant] There are several ways to prove this. One way which I like is to say, if you, if the thing you were hanging here was itself a chunk of water shaped like that,


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[Assistant] You don't have to do anything, because that chunk of water can float at that height for free.


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[Assistant] But now, if you took the chunk of water and put a stone here of the same shape,


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[Assistant] The rest of the fluid doesn't know what you're doing. It applies the same force that it would to its own colleagues. Namely, if this is water, the rest of the water is in a configuration ready to support that amount of water.


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[Assistant] So if you took that water out and put something in, the water will apply the same amount of force, and the rest of it is your problem. You apply the remaining force. So the water is ready to support its own kind of any volume.


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[Assistant] That's the meaning of being in equilibrium. Any chunk of water is in equilibrium, therefore it's getting an upward force equal to the weight.


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[Assistant] What we're saying is, if without disturbing the environment, you take the water out and put something in its place, the rest of the guys will apply the same force. Now one formal way to prove that


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[Assistant] You can take various geometries, but I prefer a cylindrical geometry.


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[Assistant] So this is, in this example, not a piece of water, but a new material you've introduced inside the water, and you want to find the buoyancy force. What is the net force of buoyancy? Well, it's the pressure at the bottom times the area, minus pressure at the top times the area.


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[Assistant] We've already seen the difference in the pressure is ρghA, but h is now this height.


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[Assistant] Well, h times a is the volume of the water, rho times that is the weight of the water, the mass of the water, that times g is the weight of the liquid displaced.


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[Assistant] You can prove this for a cylinder, you can prove this for oddball shapes by thinking a little harder. But this is good enough.


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[Assistant] Then the density of iron is less than, is more than the density of water.


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[Assistant] So the weight of that chunk of iron will be more than the weight of the water displaced. So you will have to provide a net force, and you can support it with the cable. But suppose this was not made of iron, but made of cork.


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[Assistant] It won't want to be there, right, because then the applied force by the water is more than the weight it takes to support it. The cork will then bob up to the surface.


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[Assistant] If you want to keep it down, it's like rubber ducky, you want to keep the rubber ducky inside the water level, you got to pull it down.


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[Assistant] Or you want to tie it to the, I think one of the problems they gave you is somebody tied this piece of whatever to the floor. Then it'll stay.


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[Assistant] But things will bob up to the surface. Now the question is, how far up will it go?


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[Assistant] We know part of it is going to be inside and part of it is going to be outside and you can ask how much will be outside and how much will be inside.


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[Assistant] You can already guess the answer, but let's prove it. Let F be the fraction immersed.


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[Assistant] Then we can say the weight of the liquid displaced is equal to the fractional volume.


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[Assistant] You understand? This is the, this is the weight of water equal to the full volume. This is the fraction of water displaced. So, this is simply the weight of this shaded region here.


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[Assistant] And that is going to be equal to the weight of the thing that's floating, that is ρg times full volume.


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[Assistant] If you cancel the G and you cancel the volume, you find that the fraction that is immersed is the density of that material divided by the density of water.


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[Assistant] In other words, if this material has 90% of the density of water, it will be immersed by 90%.


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[Assistant] And that's exactly what happens with ice. As you all know, ice has a smaller density than water. One of the great mysteries. Normally when you cool something,


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[Assistant] Condenses and reduces in volume and the density will go up. But ice actually expands when you cool it. That's why density of ice is less than density of water. That's why ice floats on water.


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[Assistant] That's a very peculiar property of ice. That's why you have these movies like the Titanic where you got a huge ice thing and it's floating. First of all, if density of ice is more than density of water, these two actors would still be alive. Okay? But what happens is,


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[Assistant] It is floating, and not only that, what you see is maybe a small fraction of the whole thing. That's why the big ship went down. Okay, but it's all thanks to this accident of nature that here is one substance which when cold, increases its, decreases its density.


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[Assistant] Alright, now Archimedes principle got hundreds of applications. If you want to build a boat, here's how you build a boat. Here is the steel boat.


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[Assistant] Now, you cannot come to me and say, how do you make a steel boat float in water? Okay?


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[Assistant] This is a thing made out of steel, but it's completely hollow. So what we claim is,


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[Assistant] So you can easily calculate how deep this one should sink.


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[Assistant] The balance is weight, right? That I assume you know how to do that. Take this to be a rectangular boat. Cross-section is rectangular. It's gone down to a depth row, to a depth H.


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[Assistant] Therefore, the volume of water displaced is h times the area of the boat, cross sectional area of the floor of the boat. That's the volume of water. That's the mass of water. That's the weight of water. That's the weight of the boat.


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[Assistant] Then of course you can load more and more cargo in this. You know, if you have a top here, start putting more and more cargo, this will go down until the boat looks like this.


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[Assistant] How much cargo can that boat take? Well, that's very simple. Weight of the boat plus weight of the cargo.


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[Assistant] Is maximal in this critical situation, when it's just about to go under.


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[Assistant] Therefore, the total volume of water displaced times the density times G will be the weight of your boat plus cargo.


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[Assistant] These are all elementary applications and they come from two principles. One is the pressure increases with depth. The second is that the weight of the liquid displaced is the weight of the, is the reduction in weight.


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[Assistant] So you can imagine three cases, one where the density of the material is more than the density of water, in which case it will start going down, and you will have to hold it up with a cable, but you won't have to apply the same force as you would outside.


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[Assistant] With a certain fraction of it immersed and a certain fraction of it outside.


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[Assistant] The fraction immersed is simply the same fraction as what you get by dividing its density by the density of water.


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[Assistant] Also density of water changes. In the Dead Sea, because of the salt concentration, density is a lot higher, so it's a lot easier for people to float.


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[Assistant] This is the first time I'm going to consider fluids in motion. So far, my fluids were at rest. And it's really, very simple. But notice one more time, all I ever invoked was Newton's Law of Motion.


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[Assistant] You realize that, I just took this fluid and took that fluid, balanced the forces, and said there should be no acceleration.


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[Assistant] If I can convince you of the one thing I have accomplished something, to realize that all the mechanics we have done


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[Assistant] Does not appeal to any other law than F equals MA. In all these problems of equilibrium, A is zero, F is zero.


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[Assistant] Just from that fact, and clever applications of F equals ma, for example, it's very clever to think of a piece of the water and demand that it be in equilibrium. That's how we find how the pressure varies with depth.


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[Assistant] But there's no new principle so far. In fact, there's going to be no new principles at all this term. I mean, relativity was different. The whole space-time was modified, but non-relativistic mechanics is all coming from Newton's law, as is this problem. So, we're going to describe for the first time fluid motion. We're going to take the most general case of fluid motion, where there's water in some pipe. This is the most famous picture in all the textbooks.


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[Assistant] We have not thought of a better picture now. We're all working on it. This is all we can come up with after 300 years. Water flowing in a pipe. This point is going to be called 1. This point is going to be called 2.


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[Assistant] Everything here will have a subscript 1. That means, measured from some ground, this at a height h1, that's at a height h2.


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[Assistant] The velocity of the fluid here is some v1, the velocity of the fluid there is v2.


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[Assistant] So imagine now a steady flow of some fluid through a pipe.


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[Assistant] But we're not considering pipes that break up into four or five pipes. This is just a single pipe. So here is the first law that you have to satisfy. If the fluid is incompressible, that's a certain law.


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[Assistant] And the velocity here to the area here and the velocity here. And the relationship should not surprise you.


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[Assistant] The basic premise is going to be, you're shoving in water from the left.


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[Assistant] And the water cannot pile up between here and here because it's incompressible. That means in that volume, you can only pack in so much water. So what comes in has to go out.


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[Assistant] Can you visualize in your mind that in one second a certain amount of fluid comes in, maybe until that point?


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[Assistant] And the cross section of that fluid is A, the distance it travels is V.


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[Assistant] So the volume of the fluid coming in from the left is really A1.


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[Assistant] So in a incompressible fluid, if I tell you the flow rate at one point, I've told you the flow rate everywhere. Think of cars going down a freeway and the freeway is getting narrow. But unlike in real life, we don't allow the cars to pile up.


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[Assistant] We want the density of cars to be the same. That means if there's a narrow road, they gotta go faster to maintain the traffic.


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[Assistant] That follows, then it follows that if I go to one checkpoint and see how many cars cross me per second here, same number will cross anywhere else.


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[Assistant] Now, let me show this to you in another way. It's going to be helpful. Let me wait a small time delta t. In a small time delta t,


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[Assistant] This front that was here would advance there. It'll go a distance b1 times delta t.


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[Assistant] On the other end, this front will advance the distance V2 times delta t.


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[Assistant] Now the volume that got pushed in in the time delta t is A1, B1, delta t, and that's the volume that came out on the other side, and you can cancel the delta t's and come up with the result I gave you.


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[Assistant] So wait a short time, see what comes in to the left face, and see what goes out of the right face, and equate them.


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[Assistant] Okay, now we are going to find a constraint between the state of the fluid here and the state of the fluid there.


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[Assistant] What I'm going to do, look, think about what's going to happen before you derive any formula. Suppose you're going uphill.


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[Assistant] Does it make sense to you that when the fluid climbs uphill, it's going to slow down?


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[Assistant] It will slow down on the way to the top because it's got to work against gravity. Even if it threw a rock up there, it's going to go up and slow down.


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[Assistant] Therefore, there's going to be some connection between the gain and height and the velocity of the fluid. That's from the law of conservation of energy. That's all I'm going to do. Here's what I'm going to do. You remember, if there are no external forces on a system, kinetic plus potential is kinetic plus potential.


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[Assistant] If there are external forces on a system, kinetic plus potential after


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[Assistant] Minus kinetic plus potential before is the work done by external forces.


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[Assistant] What are the external forces and what's the energy is what I'm going to think about. So here's what I'm going to focus on. Take this region of fluid.


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[Assistant] It advances to the new region there. Maybe I should draw a better picture here.


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[Assistant] Advance of the rear guard, and in the front, the fluid that used to terminate here has gone up to there.


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[Assistant] In other words, if I colored this fluid, just this part of the fluid between here and here, different color from the rest of the fluid,


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[Assistant] Short time later, that colored liquid will be occupying this new volume. It's the same numerical volume, but it's a slightly different volume.


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[Assistant] I'm going to compare the energy of this chunk of fluid before and after.


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[Assistant] This region in between the shaded regions is common to both the chunks of fluid.


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[Assistant] Point wise, the fluid is the same location going at the same velocity, so I don't have to worry about that. That's common to before and after.


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[Assistant] The only difference between final minus initial is the energy contained in that region minus the energy contained in this region.


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[Assistant] The rest of it, like here, it was part of the old fluid. It's part of the new fluid. It's at the same height, going at the same speed. So you don't have to worry about it. So what's the volume? What's the energy of this region?


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[Assistant] Delta x2 is the distance it advances here. That's the volume of the fluid. That's the mass of the fluid. Sorry, that times g is the mass of the fluid.


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[Assistant] And the kinetic energy will be mass times V squared over 2.


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[Assistant] Potentially g will be, I'm sorry, am I doing something right?


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[Assistant] Let me do it more slowly. What's the mass of this shaded region? It's got a base A2. It's got a height, delta x2. That's the volume. Time density is the mass. mv squared over 2 is the kinetic energy.


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[Assistant] Potential energy would be the same mass times gh. But I should call it gh2.


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[Assistant] Oh, you're worried about the height varying over the tube? Yeah, we are neglecting that aspect. That's an important point. You can ask, when you say H2, are you talking about the middle or the end and so on? We are going to be dealing with problems where the height differences are much bigger than the cross section of the pipe, so we don't worry about that.


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[Assistant] That's the final energy minus initial energy. That's got to be equal to the work done.


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[Assistant] Ah, now gravity, you don't count as a force when you have a potential energy for gravity. Gravity is included. The minute you write rho gh or mgh, you don't count gravity again.


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[Assistant] In question, who is acting on that fluid? Not what it's doing to others. Who are the others doing something to this fluid? Yep.


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[Assistant] The walls of the pipe will apply normal force, and that will not do any work. Yes?


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[Assistant] Yeah, but that person is somewhere here. But remember in mechanics, you don't have to go to people in remote places. I've told you, when you apply Newton's law,


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[Assistant] The only kinds of forces are what? Remember from day one?


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[Assistant] Contact forces. Except for gravity, which reaches out and grabs everything. We've included gravity. Who is this body of fluid in contact with?


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[Assistant] This fluid in between, but I'm talking about this fluid. I'm talking about all of this fluid.


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[Assistant] My taking the difference of that shaded and that shaded region was a convenience, but I'm looking at the energy change in all of this fluid.


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[Assistant] Yeah, but the point is the fluid to the left here is pushing it, and it is pushing the fluid to the right.


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[Assistant] Okay, that's the only way. It is true there's a pump somewhere pushing the fluid in the beginning, but you don't have to go all the way to the pump. In the end, you only ask, who is in contact with me? Well, it's the guys to the left. So they will exert a force P1 times A1 times delta x1.


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[Assistant] Because that's the force, that's the distance, that's the work done on the fluid.


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[Assistant] And the work done by the fluid is P2A2Δx2, because if it moves to the right, it is doing work on the left. It has work done on it.


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[Assistant] Now you can see that A1 delta x1 and A2 delta x2 are all equal, because that is just the volume of the fluid displaced in the short time. That is just the continuity equation. If you divide everything by delta t, if you like,


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[Assistant] Then you will find that delta x2 over delta t is v2. I'm just using the fact a1 v1 is a2 v2.


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[Assistant] So if you cancel those factors, what do I get?


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[Assistant] This is, this is a derivation that you can go home and study at leisure in any book you like. They're all the same, the story is the same.


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[Assistant] I can only tell you if you'd like to know what the trick behind the derivation was. I don't mind going over that.


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[Assistant] Because maybe there are different volumes in question, you may have gotten confused. But first let me write down the result.


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[Assistant] By writing everything with 1 on the one side and 2 on the other side, I get this result P1 plus ρV1² over 2 plus ρGH1.


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[Assistant] This is nothing other than the law of conservation of energy applied to unit volume. If you take 1 meter cubed of the fluid, its mass is rho times 1.


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[Assistant] So this is really a mass of one cubic meter of the fluid. This is his kinetic energy, that's his potential energy, that's the kinetic, that's the potential.


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[Assistant] You might say, why aren't we just equating kinetic plus potential to kinetic plus potential? It's because this chunk of fluid that I focused on is not an isolated system.


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[Assistant] So you gotta take the difference between the work done on it and the work done by it.


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[Assistant] Then, you will find they don't quite cancel because the pressures are not equal.


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[Assistant] Pressures are unequal, you get an extra contribution. But this is simply the law of conservation of energy. I got it by writing the law of conservation of energy.


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[Assistant] But it takes a while for you guys to get used to applying the law of conservation of energy. You may be always used to saying, kinetic plus potential is kinetic plus potential. You got used to that. Or you also remember, kinetic plus potential may be less at the end than the beginning because there was friction. This is yet another thing where there's no friction, but there are external forces acting on a body.


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[Assistant] That's the walls too. I think some of you brought up a very good point. The walls are exerting a force, but that's perpendicular to the fluid. In a real fluid.


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[Assistant] The walls will in fact exert a force parallel to the fluid, and it's called viscosity. So in a real fluid there will be a drag on the fluid, because the fluid


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[Assistant] Really doesn't like to move right up against the wall. It doesn't mind moving in the middle of the tube. So as you go near the edges, the speed of the fluid will relax to zero. And that's a region where there's a lot of dissipation. So we're ignoring what's called viscosity. We're ignoring all other losses. Then this is simply law of conservation of energy.


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[Assistant] And as to what I did with those two volumes, maybe I'll repeat this so you can all follow this. I'm saying, take all that fluid, now.


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[Assistant] Find the total energy in it and ask what happened to the fluid a little later.


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[Assistant] Okay, where is this fluid a short time later? I want you to think about it. This end has moved here, this end has moved there. So the fluid at the end of the day is sitting there.


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[Assistant] I want the energy of all of these guys minus the energy of all of this. Then you can see in the comparison, it's only this part that is gained.


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[Assistant] In this part that is lost, this part is common. When I say common, not only is the mass common, the location at every location, the height and velocity are the same.


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[Assistant] So in the subtraction, this will cancel that, that will cancel that. This will have nothing to cancel, and this will have nothing to cancel. So just focus on the relative increments, and that difference is what gives you the net change in energy, and you equate that to the work done. Okay, so now we are going to learn how to use this Bernoulli.


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[Assistant] The main thing to notice about Bernoulli is for a minute if you focus on fluid at the same height. Just think about the same height.


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[Assistant] He tells you whenever the fluid picks up speed, it's going to lose pressure.


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[Assistant] Trust a Bernoulli at a qualitative level. I'll give you some quantitative examples. So one example is a baseball. Here's a baseball coursing through air.


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[Assistant] If you like, you can sit and ride with the baseball and say the ad is going backwards.


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[Assistant] Now suppose you spin the baseball like this.


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[Assistant] If you spin the baseball, it carries some of its own air due to friction between the leather and the air. So that velocity is countered to the velocity of the drift you have on the top and additive on the bottom.


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[Assistant] So the actual air velocity in the bottom will be more in the bottom and less in the top because you're subtracting this vector from the top and adding this vector in the bottom.


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[Assistant] So if the velocity is different, this is the higher velocity region, this is the lower velocity region, so pressure here will be less than the pressure there, so the ball will sink down.


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[Assistant] If you spun it the opposite way, the ball will rise up. If you spun the ball side to side, the ball will curve from left to right.


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[Assistant] You gotta spin the ball and release it, then it produces extra forces.


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[Assistant] This rising and falling is on top of the falling due to gravity. Even in a planet without gravity, you will have this extra force.


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[Assistant] It's coming simply because the velocity at the top and bottom have been modified. Here's another example. Here's an airplane wing. The plane is going like this.


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[Assistant] As if the wing were not there. But near the wing, this is the flow of air past the wing. Notice that these guys


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[Assistant] Above the wing have to travel further than the particles below the wing so they can catch up here where everything is the same.


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[Assistant] So the velocity on top of the airfoil is faster, therefore pressure is lower.


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[Assistant] Pressure is lower than in the bottom, then the difference in pressure times the area of the wing will push the wing up. That's the lift you get when you start a plane. When you're on the runway, this is why the plane goes up. Once you're airborne, you can tilt your wing. When you tilt your wing to this angle,


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[Assistant] Now what if you made a wing that looks like this?


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[Assistant] Okay, so you're all laughing. But this has to use too. Have you seen it anywhere? Yes? NASCAR.


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[Assistant] On the race track, you want the opposite effect because you want to push down on the car, and therefore,


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[Assistant] You manufacture wings, you go to the factory where by mistake they built a wing that looks like this.


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[Assistant] Then you can bring it to your race car and attach it, then it will keep the race car down.


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[Assistant] Of course you also have the other option of going to the wing and tipping it over and applying it to the airplane, but if it did not occur to you.


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[Assistant] If you thought once the wing is made like this, I'm stuck with this wing, well go take it to your race car and attach it. That'll push it down. Because you want to get traction. So you can get the upward lift or the downward lift.


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[Assistant] Now it turns out this lift theory is actually somewhat naive, and I believed it for a long time. Now I know the story is more complicated. There is a lot of truth to this, but it's not the whole story. So if you go into aeronautic engineering one day, you will find out that it's a little more complicated reason, way to calculate it precisely.


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[Assistant] But this general notion that when the air moves fast it loses pressure is true. So, uh, here's another example.


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[Assistant] If you have an atomizer, you know, you have a perfume here, and you got a pump, and then you have a tube here. When you squeeze that pump,


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[Assistant] The instant you squeeze the pump, you're driving a lot of air here at high velocity. Whereas the air here is at rest.


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[Assistant] So high velocity air has a lower pressure than low velocity air. Therefore, it'll suck the perfume and spray it right on your face.


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[Assistant] Alright, so now I'm going to do two problems where I really start putting numbers in. This is very qualitative. I'm going to do quantitative problems.


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[Assistant] And there are only two kinds, so when we are done, we're done. Here's a tank of water. And I poke a hole in this, somewhere here. At a depth, uh, let's call it H.


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[Assistant] Question is, how fast will the water come out of here?


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[Assistant] Turns out you can use this Bernoulli's, by the way, this is called Bernoulli's equation. You can use Bernoulli's to derive this.


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[Assistant] You just have to figure the points 1 and 2 cleverly. So this is going to be point 1. This is going to be point 2. Point 2, in fact, is just outside that hole.


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[Assistant] So what does Bernoulli say? T1, which is atmospheric pressure,


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[Assistant] Let's modify this as follows. Let's pick H2 to be 0, then H1 is just this. I mean, you can pick this 0 any way you like. So, where the hole is, I'm calling 0. So, I don't have to worry about this one.


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[Assistant] What about P1 and P2? P1 is atmospheric pressure there.


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[Assistant] And right outside the hole, it's also the atmospheric pressure. So p1 and p2 cancel because they're both equal atmospheric pressure.


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[Assistant] You know that if you punch a hole in the tank and you drain it, it's going to start moving down.


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[Assistant] But we're going to imagine that's a huge tank, 10 meters diameter, and this is a tiny pinprick. Then this velocity is negligible.


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[Assistant] Rb2 squared is 2gh. Now you remember this formula from day one. This is the velocity a droplet would have if it, say, spilled over the top and fell straight down.


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[Assistant] Because that is just the mgh being converted to 1 half mv squared. Because


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[Assistant] By the law of conservation of energy, the minute the fluid starts coming out of here and draining out of the top, I have traded droplets of water at the top for droplets of water at the bottom.


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[Assistant] Therefore, since they have potential energy at the top, they must have kinetic energy at the bottom.


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[Assistant] But be clear about one thing, the drop coming here is not the same drop moving at the top.


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[Assistant] If I push it over the top, that would be the same drop, and it's very clear to all of us. The beauty of this is, this water's pushing down and something's coming out here, but it's coming out with the right speed, so that drop by drop,


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[Assistant] The kinetic energy of what comes out here, the potential energy of what's draining at the top.


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[Assistant] So you can punch the hole at various places, you know, once you punch a hole here, it's going to land there, and you can imagine the fun you can have with this kind of problem.


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[Assistant] Where do I punch the hole so that if my dog is standing here, I'm going to feed the dog?


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[Assistant] Or don't feed the dog, go in front of the dog, go past the dog, get the biggest range, smallest range, whole bunch of problems. They will combine.


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[Assistant] This Bernoulli who will tell you how fast it's coming out with Chapter 1 of how to do trajectories once the fluid is coming out this way.


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[Assistant] Okay, last topic. It's again, it's not even a new topic. Last application of Bernoulli has to do with the, what's called a Venturi meter.


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[Assistant] By the way, you can imagine this is just a cross section of possible applications of fluid dynamics. You can take a whole course on fluid dynamics. So let's take the following problem. So you're going in an airplane.


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[Assistant] How do you think you find out? I will show you now one device people use to find the flow rate, to find the speed of the plane through the atmosphere. So you go to the plane and you attach the following device to the underside of the plane.


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[Assistant] I'm imagining in my mind a symmetric thing. This is a pipe with a constriction. Here the air is coming in at the speed of the plane itself.


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[Assistant] See, in real life, the plane is going through the air, but go sit with the plane, which gives the air is going backwards at the speed of the plane. So that's the cross section A1 here.


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[Assistant] V1 is what we are trying to find. I hope you understand. That's our goal.


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[Assistant] Then it comes to this region, where it has to speed up, because a1b1 is a2b2.


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[Assistant] Here's another example. You're 5000 feet above the ground. Don't worry about altitude till here or there or there.


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[Assistant] 5,000 is the big height, and that ρgh cancels on both sides. We don't worry about the variation in height over this gadget. It's a very, very tiny thing you attach to the underside of the plane. So then I can say p1 plus ½ ρv1


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[Assistant] Plus ρG h1, I'm not going to write, because h1 and h2 are going to be equal.


[User] Please transcribe this text

[Assistant] Now V2 is not the speed of the plane, but we know what V2 is, because V2 times A2 is equal to V1 times A1.


[User] Please transcribe this text

[Assistant] So we can write here 1 half ρ B1 square minus B2 square is B1 square times A1


[User] Please transcribe this text

[Assistant] In this problem, a1 is bigger than a2, so you may worry that 1 minus a1 over a2 squared is negative, but so is p2 minus p1, because p2 here is certainly going to be lower than p1 here.


[User] Please transcribe this text

[Assistant] You see that? You can flip it over, provided you flip it over here. But ask yourself, what do I need to know to find the speed of the plane? The speed of the plane is here.


[User] Please transcribe this text

[Assistant] Rho is the density of what? Can you tell me? Rho is density of what?


[User] Please transcribe this text

[Assistant] Rho is the density of air. A1 and A2 are known because you designed the tube.


[User] Please transcribe this text

[Assistant] So if you can find the pressure difference somehow, you can redrop the speed of the plane.


[User] Please transcribe this text

[Assistant] So what people do to find the pressure difference is they go and they take, they punch a hole there and they put here.


[User] Please transcribe this text

[Assistant] Like oil. If the plane is not moving, you can tell these two heights must be equal.


[User] Please transcribe this text

[Assistant] Because that's the condition for hydrostatic equilibrium. Because these two pressures are equal.


[User] Please transcribe this text

[Assistant] But if the plane starts moving, and the pressure here is lower than the pressure here, imagine this is a high pressure, that's low pressure, it'll push the fluid up, and it'll start looking like this.


[User] Please transcribe this text

[Assistant] And how much extra is it? Well, let's think about it.


[User] Please transcribe this text

[Assistant] The pressure at these two points is the same. The pressure here is what I call P1. The pressure here is P2 plus ρg times height of this fluid. This ρ is not the density of air. This ρ is the density of oil, let's say.


[User] Please transcribe this text

[Assistant] So the minute the plane begins to move, the pressure here will be higher than the pressure here, the fluid will be pushed up, and the difference in these two heights


[User] Please transcribe this text

[Assistant] Directly is a measure of the pressure difference, that in turn is a measure of the velocity.


[User] Please transcribe this text

[Assistant] So what you will try to do is then forget all about the height difference, and if you're clever enough, you can put markings so that by counting the difference in the markings, you can translate to the speed of the plane.


[User] Please transcribe this text

[Assistant] You can also use this to find the rate at which oil is flowing. Because oil is flowing in a pipe. And you want to know the rate at which oil is flowing. Again, create a constriction in the flow. Then put some tubes.


[User] Please transcribe this text

[Assistant] These cannot be oil, these got to be some other fluid. Once again, there'll be pressure difference, and there'll be height difference, and from the height difference, you can find the rate at which the fluid is flowing here.


[User] Please transcribe this text

[Assistant] Well, I don't, I think all I'm, as far as I can tell, there's going to be a pressure difference that's going to translate into height difference. And if the two do not mix,


[User] Please transcribe this text

[Assistant] I don't know what the danger, I think one danger you may have is, if the fluid comes in here, and if it can penetrate, it will start mixing.


[User] Please transcribe this text

[Assistant] Right, so for that purpose it may be better if this one is a higher density than the one on the top.


[User] Please transcribe this text

[Assistant] Now in practice, I'm not really sure what kind of fluids people use.