 So horizontal circles continued example one. Thank you a Car is approaching a circular turn That has a radius of 60 meters if the coefficient of friction between the car and the pavement is 0.45 What's the maximum speed that the car can safely travel through the turn? Let's draw a little free body diagram What we're going to do is we're going to do a front view Here's the car We're looking straight at the car as it's coming towards us for a split second as it's going around the corner What are the forces acting on this car get the obvious ones? Is the car sinking into the ground? Yeah, Jess Yep, absolutely Now it's also going around a corner. Let's assume that the corner. It's traveling is in this direction here so If you're going around a corner, there has to be a force pushing you towards the center What force pushes you around the corner? Friction does now you'll notice SC has not appeared on my free-body diagram Centripetal force has not appeared on my free-body diagram. There's two equations here The first is that at normal force equals mg. I'm not going to write that equation I'm more interested in the going around in a circle and we're going in a circle So if I go winner minus loser no loser winner friction is The centripetal force Friction is what times what I don't know the normal force Look at another force same size is a normal force in this case. What? Okay, so this side is going to be mu mg and this is going to be m AC which AC am I going to use the one with the period or the one with the speed in it? Well, look at the question and ask either what they told me or what they want me to find that's your clue Which one am I going to use? Which one is the one with the speed in it? V squared over R. Oh, I noticed something kind of cool Turns out doesn't matter whether your VW bug or a big semi truck and Want to know the speed which I think Tyson is V. I can get that by itself. I think The safe speed to go around this corner to be mu G R Square root of it's going to be 0.45 Times 9.8 times 60 Square root of that how fast could we go around a circle? What'd you get right? I? Think it's in the 16s 16 point 16.2 or 3 which way round it off properly 16.3 meters per second about 30 miles an hour or times by three points about 50 kilometers per hour So this this would be enough You could do this as a corner a nice gentle corner on a road and people would have to slow down now You probably build a safety margin in because you'd also have to think what if it's raining But the speed limit would probably be posted as 40 35 something like that and that's where they get the speed limit numbers for the corners except In most corners, they're not flat if you're going at any speed. What do they do with the corner? They tend to bank it and we'll talk about why maybe later on example to a Test tube rotates with a period of 9.9 times 10 to negative 4 seconds What's the centripetal force on a 1.8 times 10 to negative 9 kilogram amoeba at the bottom of the test tube? The bottom of the test tube rotates in a circle with radius 0.15 meters Cool So let's suppose the test tube is right there What are the forces acting on it? Well Spencer gravity would be into the page or if we wanted to Gravity down a Normal force up. I really don't care about those even though the test tube is at an angle That's not that big an issue because the circle is flat I know I know it so that when it comes right down to it my net force is that way That's the applied force What force is pushing this test tube in a circle? Well, sorry this amoeba in a circle the test tube itself is the test tube was pushing against it. We'll call it at the fly I Don't want to call it a normal force because the test tube is at an angle and usually I like my normal force to be 90 degrees To the surface cast it who cares here's what I do know This is FC and that's what the question wants me to find I know that The applied force is what's moving it in the circle the applied force Which is the centripetal force is going to be mass times acceleration which? Acceleration equation will I use the one with the velocity? You're the one with the period in it. Oh? For pi squared r over T squared mass. Oh, there it is 1.8 times 10 to the negative 9 times 4 times pi squared times Radius is point one five I'd like all of you to try this one because some of you might not be able to go pi squared You might get an error you might need to type that in a different way But you need to find that out on your calculator now and the period is 9.9 times 10 to the negative 4 Squared so all of you try this one on your calculator just to see yo Where would yours be oh great tell you guys get that or am I wrong I? Got point zero one zero eight seven five five nine seven one one point zero nine times ten to the negative one two Newton's just because the test tube itself is at an angle who cares the amoebas going in a horizontal circle I know my overall combined net force is my FC and I can figure it out example three an Empty swing on an amusement park ride has a mass of 10 kilograms If the swing has a kinetic energy of four hundred and sixty joules What is the centripetal force on the swing they want me to find in this question and that's either gonna be Depending on whether I know the period or the speed Did they give me the mass by the way Okay, so I'm good there What else did they give me Justin? What type of energy What does that help me figure out you say V that's gonna help me figure out the speed okay So we're gonna use this one not That one and I'm gonna write down K e Equals a half M V squared and I'm gonna get the V by itself. I think V is gonna be two times the kinetic energy divided by the mass Square-rooted you know what? Let's get clever for a second Look at my original equation. What's sitting right there? Can you read it to me Andrew? No, I'm not quite what what what? But you know what I'm gonna do this and I'm just gonna put this thing. Oh boy that didn't draw right I'm just gonna put this Right in there went up I'm just curious to you can plug in the numbers and channel you be fine But I'm just curious to see what happens algebraically. I I've been on these rides before have you the craziest way I'm curious. What's going on? So I'm gonna get this the centripetal force FC is Is equal to M over R and Then V squared is 2 K E or M. What do you notice? Masses cancel I by the way I've been on the wave rider the wave swinger before and I noticed that the empty cars don't go swinging way way out or Way way in they're pretty much right in front of me They swing out a tiny bit further because I think I present bigger air resistance than the empty car and we're back in our actual physics world But for all intents and purposes if we're ignoring air resistance masses cancel I Think one of your questions in the homework asks you hey if you're on the music park ride with swings Will the swings swing out if they're empty further the same or inwards? I'm pretty sure the answer is the same Now I'm gonna crump the numbers To times the kinetic energy divided by the radius Oh in the diagram Seven what's the force that this particular empty car experiences you get a hundred and thirty one point four hundred and thirty one newtons again Chris that you didn't have to do this algebraic substitution But I suddenly saw there was a v-squared sitting there and a v-squared sitting there and I thought good gosh Why don't substitute for in for v squared because I wanted to do the square root Road safety a one thousand four hundred and fifty kilogram car can travel without slipping at a maximum speed of 32 meters per second in a circular path of radius 72 on a dry horizontal surface when it rains The coefficient of friction is reduced to one half of its original value. What's the maximum safe speed of the car under this wet condition? hmm Let's do a Dry run and then a rain run. I think the free body diagram is going to be the same for both So let's do my little free body diagram right here We're looking straight at the car. What are the forces acting on it? I'm pretty sure Gravity down Normal force up and what force pushes a car into a circle if your horizontal it's friction In fact, I think now that friction Equals FC Friction is what times what breaded? I don't know the normal force. Oh, but look look look look on another force This is a normal force. I'm gonna get mu Mg equals M friend. Do you think I want to use the equation with the v in it or with the period in it? Because they gave me a speed. So I'll use the v-squared over R Conveniently again the mass cancels. What do you think I really want to find here by the way? I think I need to find mu and then I can when it says reduces to one half of value I'll just divide it by two now and move into the rainy section now have all the values that I need now How would I get the mu by itself? How would I get this? What's the? by Bye bye G right Mu is gonna be v-squared over Grr for them to take this particular corner at this particular speed This is how much traction there has to be 32 squared Divided by 9.8 times 72 Hopefully this is a reasonable number This is what happens when I make numbers up because we're getting an answer bigger than one. Well, let's go with it for now We're getting 1.45 You need dragster tires or or spikes or studs or chains or something apparently Sorry What are the units by the way from you? Oh, yeah, there are none right? It's a unit this measure Let's go to the rain now apparently in the rain we lose half of our traction So in the rain, I'll call it mu R for rain It's gonna be 1.45 over 2 and I'll use the value on my calculator I get a point seven three Point seven two. Oh, you know what? I'll carry it to four sig figs point seven two five six Because that's not my final answer. So why round off here? What do they want me to find when it's raining though? Sorry V okay, it's gonna be the same equation friction equals circular force It's still gonna be mu Mg Equals M V square over R and that just once again. Yay masses cancel, but now we want to get the V by itself I think it's gonna be V equals mu g R Square root point seven two five six nine point eight and the radius was seventy two times nine point eight times seventy two Square root of repeat sake square root of five hundred and twelve You get twenty two point six? meters per second Sorry for that yucky you go need applications One method of generating artificial gravity in outer space Would be to build a circular space station and spin it Like that picture there If this space station has a radius of five point two meters What does the period of rotation need to be for an astronaut to experience 80% of the earth's gravitational field? Hmm, you know how I'm gonna start this one Free body diagram Now I'm going to say all right. What are the forces acting on this person, but I can't say get the obvious one Why can't I say Mg? Where are we outer space? What path is this person tracing out? So there has to be a force inwards I got that which force Alex Is this person standing on a surface? normal force How big do we want this normal force to be well normally normal forces Mg But what does this question say we don't want to quite get all of earth's gravity what percentage? So how about point eight? 80% of Mg and now I can go to my equation The normal force equals what's pushing it in a circle normal force is going to be 0.8 Mg and Circular forces mass times acceleration which acceleration will I use here the one with the V or the one with the t in it? Carissa a Period which is what I was saying check your formula sheet some you need can become familiar with this So yes, thank you for taking your formula sheet out finally Andrew Hey Will this work for a skinny astronaut as well as for a fat astronaut? cool Because the masses cancel Let's get the t by itself Okay, I think The t would move up to here and the point eight and the G would move to here. I think we're gonna get this t squared equals 4 pi square r over point a g except Rob. How do you ever squared? It's going to be for high squared 5.2 divided by point 8 g you get 5.12 seconds Don't forget square root Is it am I right? How could we make it so that the space station didn't have to spin so fast? bigger radius bigger space station or Compromise on how much gravity we want if we want to make 60% of gravity that would make this number Smaller, which would make this number Sorry, that would make this number bigger which would make this number bigger Which would mean it would take longer to go around once it's not having a sense of ask or? We'll have the middle section on a hinge Basically staying still not spinning at all This is where the spaceships could dock and this would be zero gravity which might be helpful for loading and unloading and Then as you slowly make your way down these ladder pathways as you get further and further away from the center You would feel more and more and more centripetal force which would feel to your body like gravity because you're inside the spaceship You can't tell that it's you moving you just feel something like gravity pulling you down So suppose the astronauts begins to climb a ladder that leads to the center of the space station at what radius? Will the astronaut feel 30% of his weight I? Think we're going to start out the same way Cassidy in that we're going to say 30% of mg equals my circular force 30% of mg equals m which force equation do you think I want to use now? Well, what are they asking me to find? Alice yeah, oh, there's an R in both of them What else do I know I just figured out the period I think I'll have to use the one with the period in it This one Well a little child feel 30% of his weight as well as a big grown-up at this radius. Yep What I asked you to find Alex. That's what the R by itself I think this will move to the top. These will move to the bottom. I think our ends up being 0.3 g T squared divided by four High squared point three times nine point eight times my previous answer squared divided by bracket four High squared and I get when you're one point nine five works out even that's kind of nice when you're one point nine five Meters from the center. Yes, so there's your probably most Likely way if we do ever build great big spaceships for people to live on or travel travel Interstellarly we'll almost certainly just put them into a gentle spin and it gives you a nice Ashley artificial gravity Which reduces the health issues big time because your hearts are always used to having to pump against gravity If you live in space your heart muscle you live there for about a month it atrophies It goes limp and flabby very badly. So when you get back to space a lot of blackouts a lot of issues for the first six months Or so until your heart gets back into shape. This would avoid that It brings other complications. You have your whole space station revolving which makes docking a bit more of an issue What a few other things but oh the gravatron the gravatron by the way Suppose this person Held up a ball and dropped it. What would it look like? What would happen? We go to their feet Would it fall straight down? And the answer Spencer is it depends on your point of reference I think to the astronaut it would appear to fall down and land at their feet in reality though Which way is this ball moving? What path? Its velocity is that it would fall this way However, as soon as the astronaut let it go He's also rotating this way at the same rate as this ball is falling this way And so he would see it stay the same distance next to his leg But heading straight downwards to his frame of reference, but it's actually him rotating underneath it He wouldn't be able to tell also because the air inside the space station is also rotating So he can't feel a breeze. He doesn't notice that he has no way of being able to tell He doesn't know that he's moving Are you be able to feel it? Can you guys all feel you're moving on the earth right now? Our sensation of movement comes from two things our bodies have a backwards accelerometer or if we're at a constant speed We look for wind on our face That's how we know we're moving he would not be able to tell or she would not be able to tell Kind of cool. All right The Gravitron Spins its riders around in a circle if the ride has a radius of 4.8 meters And it rotates every two and a half seconds What's the minimum coefficient of friction required between the riders and the wall of the ride you friction this? I don't know let's find out Hey, what are the forces acting on that person get the obvious ones? Gravity down are they falling down? Are they falling up? Then there must be a matching force this way. I don't know which one it is yet, but I'll drop the same link What else? What path is this person tracing out? So there has to be a net force inwards. I don't know which one yet now. By the way, I think that's it Forces pushing him inwards actually heard it over here You know what that's the surface. It's like we flip the ground on its end or this is like the Space station that we just looked at it's the normal force that's pushing him sideways inwards so then What do you think this is here and from here we get two equations Strangely we get Friction has to cancel out or equal gravity see the vertical force Horizontally there's only one force pointing towards the middle which one rub Apparently that is my centripetal force Again notice FC never appeared on here. Oh, but look look look look look. I don't know another force the same size as a normal force Hey Friction is what times what oh? What's f and the same size as I can't hear a word you're saying except I heard you giggle at the end which was great Yet this is a triple force. You know what I think I think I think I think I Can plug that into there I will get to get the mu by self because that's what this question asked me to find the minimum coefficient of friction Alex how would I get the mu by itself? Yep Minimum coefficient of friction if you're designing this ride Alex is gonna be that I don't know FC wait a minute FC is mass times circular acceleration. Let's go there This is gonna be mg all over mass times Which version of the acceleration am I gonna use here the speed or the period? Oh, do they tell you the period of the question? Every how often so this is gonna be for pi squared r over t squared. Hey, I do notice yay mass cancels That's kind of nice Real question right now some of you are going what the heck is don't write this down don't write down don't write this down Do not write this down what the heck is g? divided by four pi squared r Over t squared and I say to you easy Only yesterday How do you divide by a fraction? Yep, that's one of the multiplying fractions is the easiest top time top bottom to bottom Dividing almost as easy flip which everyone's are divided reciprocal and now multiply So this is going to be the same as G times T squared over for pi squared r apparently mu is equal to G t squared over four pi squared r 9.8 your period 2.5 all over 4 pi squared 4.8 right What did I forget squared? squared You get point three two Three two, so there's some applications of horizontal circles For your homework you can go back to lesson two and And adding to the homework or did I assign all the ones that I wanted to you can also try Number six six so one two three four five six seven. Oh Heck let's add number eight So right now it's one through eleven and thirteen one through eleven and thirteen