 Lesson two horizontal circles Imagine that you're in a car turning a corner What is the direction of your acceleration and what is the direction of the force that acts on you? Well If you're turning a corner first of all what shape are you tracing out? Circle will go simplest case So if you're moving in a circle What direction is the acceleration inwards or outwards? Yes, yes, no, no, by the way on your tests you may have noticed I often give you one of these kind of two column chart questions I always do these by using the check mark x mark method going down one whole column at a time Which way is the net force? Well forces what times what Cassidy? If your acceleration is inwards, which way do you think your force is? Because mass is a scalar and the direction for force has got to come from somewhere It comes from the acceleration. So yes, no, yes, no, and then I pull back and I say hey Which one has two check marks the correct answer is a? It says explain your answer using relevant physics principles I think I just did I said if you're moving in a circle that forces inwards and forces mass times acceleration So it gets its direction from the force. In fact, it's going to Detail the explanation. It's a little bit better here. It says this in latin the word and here's your new word of the day Centripetal That means towards the center or center seeking So when an object is moving in circular motion, it has centripetal acceleration Then it says by newton's second law actually it's by newton's First law there must be an unbalanced force in the same direction as the acceleration So an object moving in a circle has a centripetal force Some like to use the following notation. In fact, this is the notation. I'm going to use shannon AC is going to stand for inward centripetal acceleration FC is going to stand for inwards or centripetal force Where c means centripetal towards the center Oh and AC equals v squared over r or for pi squared r over t squared and FC equals m ac Most common sloppy mistake it's so that kids make because the acceleration equations, especially this one are so complicated Often Brandon. I'll see a kid do this They'll forget the mass because it looks so complicated. They forget you need it now They won't write that what they will write though is just four pi squared r over t squared forgetting to put an m in front of it force acceleration force Acceleration true statement force equals force false statement. Don't do that. Okay Now here's the second key thing that I'm going to try driving home again and again and again In our handy winner minus loser approach The centripetal force will always be the net force in other words simran My equation is going to look like this If i'm moving in a circle And you may remember rob This thing here never actually appeared on the free body diagram It was the vector sum of all the forces you will never you will never we will never write FC on our free body diagram What we will ask is hey, what is the force that's causing it to move in a circle? And then we'll go to our equation So the centripetal force will always be the net force the f net this thing It will never actually appear on a free body diagram Instead it will be the vector sum of all the forces for example a classic children's toy I need to buy another one. I bought two and both of them haven't worked so i'm going to go hunting for a good one You sometimes see them in department stores. They have a toy play plane hanging from the ceiling And the plane is moving around in a circle like this from the ceiling We call this a conical pendulum a pendulum because you have a mass on the end of a string tyson conical because the shape that it traces out is a cone So What path is this plane tracing out? What shape is its path? Justin a circle horizontal circle. Um, let's add a direction so that we're going this way So right now it's coming out of the page towards us If it's moving in a circle that means when I add up all the forces I better end up with something pointing towards the center But that's not going to appear on my free body diagram necessarily it may but we'll see What are the forces acting on this plane now before I say get the obvious ones? I'm going to talk about two the first is the propeller The propeller would be pointing right towards us right now in this picture But if it's going at a constant speed, what's happened is it's maxed out its speed The propeller and air resistance are exactly the same because it can't speed up or slow down So even though Hesham the propeller is coming towards me Away from me would be air resistance. We're going to ignore air resistance and the force from the propeller to do a simplified diagram What are the forces acting on this plane get the obvious ones? Gravity Is this in free fall? No, I heard it actually you're Tension this way back The question says which of the following is the best force diagram for the plane This one Hey, wait a minute mr. Do it. I thought you said that my force had to be towards the center. It is Doesn't look like it. Ah Watch I'm going to add these two vectors together because they're at angles to each other spencer. How do I add two vectors together? So i'm going to draw mg first And then i'm going to draw tension And do you see what the resultant is spencer? It doesn't appear on my free body diagram, but it's the combination of those two that's pushing towards the center There's my fc. Oh and for what it's worth I think that's theta robb see the zed So i'm pretty sure That's theta. We're doing a generic one here with no variables, but eventually we'll have numbers It says explain why the others are wrong. This can't be right. There can't possibly be an unbalanced force outwards That's the mythical force that we think exists because we're so used to feeling like we're getting pushed outwards in a car We said actually actually no you're getting pushed inwards, but your body wants to keep going in a straight line You have to apply the force inwards with your legs and with your torso But because our bodies are backwards accelerometers. Do you think you're getting pushed outwards? This one is wrong. It's added centripetal force actually Courtney. It doesn't need to be added It pops up when I add those two together Tipped to tail. We don't need to put fc on our free body diagram. It'll be there in a horizontal circle Vertical forces are balanced f up equals f down And you should have somewhere an unbalanced force pointing towards the middle And that unbalanced force is going to be either mv squared over r Or m four high squared r Over t squared it's going to be ma or ma Example three turn the page So example three says redraw a force diagram and then write force equations for the toy plane Note this motion is referred to as a conical pendulum since the plane traces out a cone in space So we said in terms of our free body diagram These are our two forces right But Samina when I add these together tip to tail I'll get this now whenever I'm adding two yucky vectors That aren't at right angles together Chris I always draw the easiest one first Because I know I can't watch that one that one I can get right Then I always if there was three vectors here. I always draw the toughest one next in this case. There's only two But because I know what path this plane is tracing out Then Eric I know which direction my net force supposed to be which way is my net force supposed to be if I'm moving in a circle Eric toward the Which way is my net force supposed to be if I'm moving in a circle toward the so Watch don't write this down Eric I know not to draw attention that big because there's no way that's going to be pointing towards the center That's going to be pointing up at an angle And I know not to draw attention that big because that's going to be pointing down at an angle I can actually tell you I'm going to draw attention exactly right there And that's going to give me Eric when I add those two together my lovely fc Oh and Cassidy that's theta I'm pretty sure that's theta Which means I'm pretty sure that's theta And Rob will say alternate interior. Yes. A lot of that stuff is terribly useful Which trig function relates those two together? Oh, hey, mr. Duke. Is there a right angle? Yeah Nice So which trig function relates those two together? What none of you will be doing this afternoon outside? Yeah tan Well, I don't hit tan So here's what we're going to write equals Opposite over adjacent Now this can really branch into a couple of equations. This can become tan theta equals m v squared over r over mg Or it can become tan theta equals m for pi squared r over t squared over m g I plugged in two expressions for circular or centripetal force m a m a Hey, um How many m's on top? In this first fraction here Alex How many on the bottom? And in fact because I know you guys kind of suck at fractions v squared over r over g that's actually just Don't believe me put the g over one that asks how do you divide by fractions of it multiplying the g on the bottom? Oh Alex but not Alex. How many m's on the top? How many on the bottom? Hey Usually that'll happen and in fact this tidies up into four pi squared r over g t squared two different expressions for tan of theta I Don't memorize those it's all I kind of derive it from this if I need If I have to that's the conical pendulum where you have one force at an angle Let's actually solve something with numbers. It says example four The picture below shows a frictionless tabletop in top view. We're looking down at it The mass is 4.15 kilograms It's attached to a two meter long string and it's moving at 2.5 meters per second Right the force equations and then solve for string tension. Okay. What are the forces acting on this at the obvious ones? Gravity which way since this is a top view gravity is acting into the page. Yes, which is going to be tricky to draw Oh is the mass sinking into the table No, is it flying into the air towards us off the table So there must be you know what I think what I really have is Gravity and normal force are canceling each other out That's my vertical forces Who cares what are my horizontal forces? Well Alex but not alex or alex What path is this shape tracing out? So there has to be a force this way Which force and not fc Not on my free body diagram. What force is causing this to move in a circle? Tension Is there a force in the opposite direction? No, so no lose time just winner. Who's winner? Tension equals now we used to write m a Now it's going to be m ac because we're moving in a circle By the way where this gets a bit tricky carissa T for tension and sadly t for period we need more letters in the alphabet and we don't Keep track of them and you know, what's what when you write things down Um, I'm going to use either this expression or this expression for acceleration Did they give me the speed or did they give me the period in the question speed? Oh Tension is going to be m v squared over r Do I know the mass? check Do I know the speed check Do I know the radius of the circle? I think that's the string length. Is it not? so uh 4.15 2.5 squared all over two What'd you get? Andrew would you get 12.97 Uh to three sig figs. I think 13.0. Yes Okay By the way What if I went twice as fast? What if I doubled the speed? How many times bigger is the tension not twice as big? Yeah, four times as big if I triple the speed nine times as big Yeah Yeah, how about newtons mr. Dewick? Thank you, alex Somehow i'm anticipating another question or something like that. Good gosh okay Replace the string and the mass with a car going around a corner What are the forces acting on the car get the obvious one gravity down? What else normal force up? What's pushing a car around a corner? What force would I label here for a car turning a corner? That's why it's hard to corner on ice. In fact, you could calculate since you know the radius of your corner You could probably calculate how fast you need to put the speed limit at that corner To do it safely especially if you know the coagulation friction of your tires, which let's assume icbc Or the minister your highways does know That's where the speed signs come from Oh, and i'm sure they build it a safety margin Example five. Let's write force equations for a car rounding a flat corner So you folks already told me Gravity down normal force up Now what path are we tracing out what shape? So there has to be a force towards the middle. I'm not going to call it fc Instead i'm going to say what force is pushing a car towards the middle and you guys said friction. In fact the equation would be this The normal force is mg Friction is the centripetal force Friction is what times what? I don't know the normal force. Oh, but look look look look. I don't know the force the same size as the normal force What? Turns out it's going to end up being mu mg equals either mv squared over r Or m for pi squared r over t squared. We'd be much more interested actually in this one because it has the v Can a truck And the car go around a corner at the same speed why Then that's cancelled Now there's other factors It actually has to do because trucks are higher off the ground We're assuming the center of mass is at ground level. This is our simplified physics world. Everything I teach you is wrong So trucks are tippier and that that okay that plays a role and load shift and things but first if we simplest case That's why we don't have to have separate speed limits for cars and trucks Only in certain situations So a technical comment on turning the reason that friction is inwards has to do with how wheels work When you turn Your front wheels to the left for example shannon the car wants to keep going in a straight line that that's newton's first law So The wheels because the car wants to go straight push against the road in that direction Now newton's third law kicks in forces come in pairs. The road pushes in that direction Which happens to be towards the center of the circle That's also why when you skid What do they tell you to do steer in which direction if you start to skid You steer in the direction of the skid That's because you want to turn your wheels this way to gain some Traction back and then hopefully you can null it out But if you skid you've already proven that friction isn't big enough to help you turn Turning away from the skid will only increase the force of friction required You've already proven that the force of friction wasn't big enough already and now you're asking for more. That's dumb driving example six And let's put a little Part a right here A blue ball is swung in a horizontal circle and completes a single rotation in 1.2 seconds The 0.44 meter long cord makes an angle of 35 degrees with the vertical during the ball's motion What's the centripetal acceleration of the ball? Okay Is that my computer and what was that noise about I wonder? Well Centripetal acceleration is either v squared over r or 4 Pi squared r over t squared I lifted that right from the formula sheet Which one of these two do you think I'm going to use? Did they give me the speed or the period somewhere in the question? Period okay I'm going to use This one for high oh Do I know the radius now careful? This is not the radius The radius Is that distance there? Do I know that distance there? And I'm going to say to you Not yet Not yet But I think if I draw this triangle Where that's the radius And that's 0.44 And that there is 35 degrees Come on silly. I think I see a trig function that I can pull out of my back pocket That would help me find the radius Just in which trig function Opposite you're telling me this is adjacent In fact, one could even say that you sinned when you said tan You almost signed okay Justin I'm going to say this then Sin 35 equals r over 0.44 redeem yourself Justin. How would I get the r by itself? No, we're already dividing by okay. Don't bother redeeming yourself dig yourself deeper into that hole Justin what's the 0.44 doing to the r adding subtracting multiplying or dividing So how I move it over? What's the opposite of dividing multiplying? Can I do this then folks? 0.44 Sin 35 Divide by and the period is 1.2 squared Here's my centripetal acceleration Those of you that are with me at 12 you're probably in radians make sure you're in degrees Because we're pressing the sign button It will be a sin to use radians when being degrees Mode hey, I am in degrees You get 6.9 1 8 9 9 that How about 6.92 About two-thirds of a g how many you've been on a playground merry-go-round before What's your experience with moving in a circle? Do the accelerations get fairly big fairly fast? Yeah, in fact, you don't have to get that merry-go-round moving very fast before You're having to exert a lot of force out of your arms and feet to keep moving in that circle In fact, if you don't exert enough force you'll get I'm going to say flung off You'll actually keep going in a straight line and the merry-go-round will move out from underneath you So 6.92 about Three-quarters of a g Probably pretty reasonable You didn't get the g forces up pretty fast. That's again why amusement park rides conveniently Move in a circle most of them. It's an easy way to get the thrill ride up and take up very little real estate The really cool one by the way is the crazy beach party at playland. Have you been on that one? The reason it's so that one is so nice is there's actually two types of circular motion. You're sitting in a circle And you're on the end of a pendulum that's carving out part of a circle Very complicated If you go on that ride and we do our playland field trip and you analyze it It's a very very complicated set of physics there, but nerdily cool and there you pull about I think if I recall three and a half g's Usually second or third last turn before the end. Oh b Whoops. Hey, this is part b. Mr. Newick. What's the centripetal velocity of the ball? Now I threw this on here because this is not on your formula sheet Here's how I remember this Brandon, what are the units for velocity? What a measure velocity in I think I heard you say distance over time when you said meters per second. Yes And then I remember to myself, but in a circle What is the distance once around when we call it the circumference? And I think it's actually on your formula sheet depressingly Is it just in on the back where all the equations like the quadratic equation and stuff? Is the circumference of a circle on there somewhere too? Really anyways it is 2 pi r s and what variable have we been using for once around the period? So this equation is not on your formula sheet, but hopefully eric. I just showed you a way you can remember it Uh circular velocity well velocity is meters like a distance over time in a circle a distance is a circumference All of it total time So it's going to be two times pi times 0.4 sine 35 because that's r divided by 1.2 circular or centripetal velocity 2 pi 0.44 sine 35 question 1.2 uh 1.32 is 1. Yep Yep Yep In fact, we got this equation by plugging this equation into that equation Yep I just hesitated to use a found value to find another value. I was Wherever I can I fall back on what they gave me right? By the way 1.32 meter second very fast No, but fairly big acceleration example seven example seven a nine times 10 to the negative three kilogram ball is attached to a 3.6 times 10 to negative two kilogram mass and By a string which passes through a hole in a frictionless surface the ball travels in a circular path of radius 0.42 meters. What's the speed of the ball? So, huh? Well, here's what we're talking about We have a large mass hanging underneath a small mass Hey kind of like this two masses one mass And they've threaded it through a hole in the table. I'm going to cheat and just thread it through a hole in the paper clip like that Right now If this was frictionless But if I start to spin this I can eventually Spin it fast enough. I think If I spin this fast enough Gradually instead of going straight down. I think gradually it will actually start To pull this mass up. I can spin it really faster and get higher And in fact, I can probably find an equilibrium point. I can find the magic point that you could have. I don't think this Terribly set up again. What was quite going to work. Let me try See the bottom mass raising slowly If I go faster even more and if I Pull the bottom mass down Small guy speeds up or if I slow down Low tech I'm not going to drill a hole in the table. Sorry What are they asking us to find Speed all right First thing. Let's do let's label our forces. What are the forces acting on this ball? Get the obvious ones little mg Is it sinking into the table? Must be a normal force pointing out What path is it tracing out Tension what about the big mass? What are the forces acting on it? Get the obvious one That'll be big mg since that seems to be the notation that we're using here. Is it in free fall? Tension lifting it up And this question implies and I need to retype it so that it specifically says that this is an equilibrium In other words, if I look at this guy right here Those two are the same size Otherwise it would have to be falling or raising and the question certainly doesn't mention that it's falling or raising Let's keep that in our back pocket friend And let's walk over here. What path is this tracing out? Okay Tension equals the circular force tension equals little m circular acceleration tension Equals little m which circular acceleration will I use? Well, what's the question asking me to find? Oh, let's use the one with speed in it then So let's not use the four pi squared one. Let's what's the other one Rissa Oh, I thought you were saying it. Sorry I got the m huh V squared I got the am already so just the v squared over r. Do I know r? Check Do I know a little m check? Do I know the tension? What's the tension the same as? Oh, let's rewrite this equation then one more time big m g equals little m v squared over r And now let's get the v by itself Justin What would I do with the r to move it over to this side? And what would I do with a little m to move it over to this side? And then how would I get rid of a squared? So if I hear you, I think you're saying this big m little g times r divided by little m I think you're saying 3.6 times 10 to the negative 2 9.8 That was the radius 0.42 Divided by what was the little m 9 times 10 to the negative 3 To keep this thing in equilibrium How fast must this ball be spinning around in a circle any slower and big m will start to drop down Any faster big m will start to move up You'll be okay back there You need to go get a drink go ahead Do you get 4.06? Meters per second because it's a speed I'm really not convinced that i've done enough examples or situations for you to get comfy with this I'm going to give you some homework, but i'm going to be fully expecting to take questions on this Homework i'm going to say try number one Two and three four and five number seven Oh cool amusement park ride number nine the gravatron Does Playland have a gravatron? Well, I I I think the playland one isn't part of playland. I think it's part of the p and e Okay, because I haven't seen it at playland since i've been going there for all these years spaceship So 10 and 11 9 10 11 i'm giving you lots. I know look at a couple of