 Number seven first thing I ask is are we back here now First thing I ask is are we in orbit? Well if I read this it looks like we're launching straight up from the earth because it says before stopping and then falling back down Okay Then I ask myself is this question asking me how much work So anyways, since I'm not an orbit. I know I'm not gonna do this Okay, not an orbit Then I ask myself fret is this question asking how much work was involved It's meant to be a really easy question to answer. It's meant to be really obvious Is this question asking how much work is involved? No, it's talking about work. So this is my checklist orbit FG Goes FC. No work No, then I fall back on conservation of energy the amount of kinetic initial and the amount of potential Potential initial has to equal the amount of kinetic final and the amount of potential final That's how I'm gonna handle this one Here's the other way. I was able to come to that conclusion Emily Do you notice we have an initial launch velocity? Oh, we have been initial kinetic Stopping. Oh my final kinetic Brett zero kinetic energy is a Half m v initial squared Plus now I can't use MGH anymore. We're doing a cosmic question here with big distances So I'm gonna have to use negative big G big M little M over R Initial which I think is going to be the Earth's radius because we're starting out on the surface of the earth equals negative big G big M little M all over our Final and our final is what they're asking me to find That would be the distance from the center of the earth. I Don't think I would try and get the our final by itself because this is way too complicated an expression I think I would plug in the numbers plug in the numbers get an answer and Then recognize the our final moves up and the answer moves down. It's gonna be negative G Mass of the earth mass of the satellite divided by whatever the heck this I'd worked out to Is that okay? Oh, you try it the rest of the way yourself, but that's how I would approach that one Is that okay for anybody else would read Okay, then find the distance the rocket travel in kilometers will divide by thousand Oh, it wants the height from the surface. Well, I wouldn't ask that but you can figure that out Is that all right? We're good any others um, I Have liked that question. I have liked that question. I have liked that question I got a couple of different versions of tests. I'm pretty sure it's Although I think to make it funner I move it to the moon or to Mars or something like that Because you know why not? Any others? Let's finish off the unit then. All right, so a little reminder your test is gonna be on Friday, February 24th and Let's begin roller coasters This was actually Caitlyn the bonus question that I threw at you on quiz one Says this example one at the top off should say at the top of the 12 centimeter radius loop the 350 gram toy car experiences a force from the track of 200 of two Newtons Assume the car is released from rest and that the track is frictionless a figure out its release height and Be figure out the range are that it travels after leaving the ramp I don't know if any of you when you're growing up had the little matchbox cars and the little Plastic stick together race tracks that I didn't growing up give you're nodding your heads most of the guys But some of the girls too Okay, it always came with a little loop-to-loop which was way cool. I thought And I don't know if you were a nerd enough to try and actually experiment a little bit with What's the minimum height I could get it so that if my car would just barely make it around the loop We're actually going to come back to this question in a second so Let's go to example two Orbits a satellite in orbit around the earth is at radius R and speed V So it's in a stable orbit But it's hit by a small rock and that small rock causes it to slow down it loses some kinetic energy What happens to the radius of the orbit assume the satellite begins and ends up in a circular stable orbit Okay, so the satellite Collision is the new orbit the same radius as before is The new orbit at a higher radius than before is The new orbit at a let lower radius than before and convince me Hmm Any suggestions? What would happen if it slowed down? Why? Well the original V was exactly enough to maintain Stable orbit and what was orbit Kara orbit was actually free falling to the earth, but free falling to the earth at Just the right sideways speed so that when you fell to the earth Even though you were falling you were matching the curvature of the earth. You're moving sideways So if we slowed you down you wouldn't be falling as fat You wouldn't be moving sideways as fast you'd be falling, but you're not moving sideways as fast Your radius is slowly going to decrease. It's going to decrease. It's going to decrease till eventually That's how the shuttle leaves orbit It fires its reverse engines and it just decreases its speed a tiny bit and it will gradually Gradually gradually free fall closer and closer and closer to the earth. Let's mention that here as V decreases the orbit will fancy word decay since sideways V no longer free falls the Curvature of the earth Little technical for what it's worth actually the final orbit is actually any lips that passes through the collision point Then moves closer to the earth on the far side of the orbit It's not quite circular anymore. Oh, sir. Look up. You want to read that? Okay example three This is why I skipped example one example three is actually is Actually a nicer question an introduction to a roller coaster loop a roller coaster barely makes it around a vertical loop of radius four meters Find the normal force of a 50 kilogram passenger When the coaster returns to the bottom of the loop, okay? I think we're going to do a bunch of stuff before that We're actually going to add in part we're going to add a part a to this part a the Minimum height H of the ramp First question I want to ask myself is What's the minimum height required here? This is going to be an energy question, but what I really need to look at is location a and Location b the top of the loop The kinetic energy at a Plus the potential energy at a has to equal the kinetic energy at b Plus the potential energy at b Let's assume we're starting from rest and I now have this m g h at a equals a half m v at b squared plus m g h at the and we're on the earth so I can use the shortcut potential energy I don't need to go cosmic here. I could be more work than necessary Say what do you notice about the m's? You have to step on a scale before you get on a roller coaster. Yeah How fast am I traveling right here? Well if I barely complete the loop Here's my free-body diagram right there. What are the forces acting on this get the obvious ones if you barely complete the loop So you're just about ready to fall off the track What's the normal force? How heavy would you feel if you just barely made it over if you were just about to fall the track Jacob? Zero so and since normal force is what we interpret as how heavy we feel. What's the normal force? Zero in fact, there's my free-body diagram right there Okay, no normal force What path is this coaster tracing out? Ah, so I can say this Gravity is what at the top Pulling me into a circle The reason I walked over here Emily is I want some kind of an expression for the velocity at the top So I think I'm gonna plug in if that's okay Hey Hey once again whoo-hoo and I get an expression for V V is gonna be the square root of actually I was gonna get an expression for V Brianna and say it was the square root of Gerr, but I don't actually want a V over here. What do I really want over here a V? What oh? heck why don't I just do this and It turns out at the top That's an expression for V I get this G height at a equals I'll replace the one half with a point five V squared is actually Gerr Plus and now something really bizarre happens Zay What do you notice about the gravitational field in this question? G what happens? They also cancel Say what it means if you could somehow like Star Trek beam this roller coaster to Jupiter It would work just fine there, too The gravitational field plays no role in this whatever height that we find works on any planet on the moon on the earth Doesn't matter it works actually independent of gravity, which is kind of nerdly cool fact I get this The minimum height is equal to Point five R Plus the height at B And now I'm going to get a tiny bit clever one more time Kara because this ends up with such a lovely equation It makes my little math nerd heart go pitter-patter. What do I call this distance right here? What I'm called it traditionally Begins a letter R at radius. So How far from here to here? Just from the middle here Radius Jacob how far from here to here also a single radius and now what did you say? What's the total distance? You know what I can replace the height with Two Rs. Can I not the height at the top of the loop? And look what that gives me this is kind of nerdly cool The minimum height is point five R Plus two are Katie call me silly, but those are like terms What is point five R plus two are squared really? Don't think so. We're not multiplying You know what the minimum height of any roller coaster hill to hit a Circular loop has to be has to be two and a half times the radius of a loop Otherwise, you're not going to make it and that's minimum. They would build a huge safety margin in But I'm expecting roller coaster designers know this equation like the back of their hand In our case, what was the radius? I've scrolled down. What's the radius of this loop? What does it say? for I don't need the calculator Two and a half times four and think is ten I have to say I do kind of like that equation I'm not saying I like this question like this question. I'm just saying as a nerd I love the fact Mitchell that it becomes so clean How high does your hill need to be two and a half times the radius that's also ignoring friction I'm willing to bet if you tried it that with your old Hot Wheels track two and a half times exactly Wouldn't quite make it you'd lose some energy due to friction You probably have to go three times or three and a half times or something like that Now let's see if we can answer the question. Let's put a little B right there. I did a find the minimum height H of the ramp first B Find the normal force of a 50 kilogram passenger when the coaster returns to the bottom of the loop Okay, what are the forces acting right here get the obvious ones? Gravity what else got to be just at the end of this loop Andrew what path is this thing still tracing out So who's winning Where does your net force always have to be pointing if you're moving in a circle toward the So you know what I forgot to put a normal force on there. I think now my equation is going to be Normal force minus mg Equals m v squared over r the normal force equals m v squared over r plus mg Which Mitchell tells me I'll feel heavier at the bottom of the loop because normally I feel mg and now I'm adding something Unless my velocity is zero in which case I'm standing still and it's kind of a pointless question. Is that okay so far kiddo? Yeah, did they give me the mass 50 kilograms Do I know G? Yeah, do I know the radius? Yep. Do I know how fast I'm traveling at the bottom? Nope No, oh So let's make a little note here you need v Kinetic energy initial plus potential energy initial equals kinetic energy final plus potential energy final Look up for one second Let's call Right here Location C That okay is a and what I'm really going to say then is instead of initial initial and final and final At B at the top of the loop at C at the bottom of the loop The amount of kinetic energy you have in the top and the amount of potential energy you have at the top is equal to the Mental kinetic energy at the bottom. Oh, how much potential energy do I have at the bottom of the loop? kinetic energy is a half m v at b squared plus Mgh at b and that equals a half m v at C squared Zay look look look look look and I had an expression for v squared at the top of the loop I've scrolled down, but we wrote it somewhere. What was v squared at the top of the loop as it turned out equal to? It's kind of a nice little expression. What was it? Grr, so if I hear you and I can go point five Grr Plus oh, and I had an expression for the height at the top of the loop to did I not? to grr right G and then to R equals point five VC squared, you know what I think I can I got like like terms just about here Let's times by two times by two and times by two to get rid of that one half and get rid of that one half yo Wouldn't what be two point five Here, what's my height at B? How far from here to here are how far from here to here are? Okay, now this by the way at a I didn't want to deal with that because I Don't know if the question didn't tell me if I was standing still or not here I am I said minimum height I assumed I was but I know they told me that I barely made it through the loop So I use this height here because I knew more information about it. Is that all right make make make it good good good good In fact, I get this Connor Grr plus I guess for grr. What is V squared at the bottom? I think five grr Isn't it four and one? Yeah, Emily or no. Yeah, let's plug that into here the normal force at the bottom is going to be m Five grr. I don't need to square it because I solved for V squared over r Plus mg question Brett Good. I have a minimum strike Again something kind of need happens turns out Connor what what cancels in this fraction and you get this Five mg's plus one mg. You know how much heavier you feel at the bottom of a roller coaster loop if you've just barely made it Six mg's you feel six times heavier That's why you feel pressed down into your seat at the bottom of the loop Yeah Here it's MV squared over r. Is that not the circular motion equation? Not we're not doing gravity here right gravity had an R squared on the bottom But circular motion had an R right here So v squared is what we found an expression for right So I can replace the v squared with a 5 gr and there was already an R sitting there on the bottom I just dropped it down because I replaced v squared That out is that right? I'm running out of room here, which is part of the problem, too. I didn't use good spacing here Oh, so final answer will be six times 50 times 9.8 Next time you're on a roller coaster with the loop When you come out of the loop at the bottom Notice how heavy you feel or I think I mentioned this if you're on a good roller coaster The California screaming at Disney is a great one as you hit the loop Lean forward and then try straightening up while you're in the loop or try holding your arms straight out While you're in the loop and you'll find at the bottom you feel a force pushing them down. It's actually inertia, but You feel six times heavier. Is that all right? Let's go back to example one then here's our toy car more roller coaster loops Except now we're on a smaller scale But this time it doesn't just barely make it Tells me that at the top of the loop it experiences a normal force a force from the track of two newtons Says find its release height. Okay We'll still start out by going kinetic energy initial potential energy initial equals kinetic energy final and potential energy final Where this right here is initial and this right here is final are any of these zero Well, it says released. So I think my initial kinetic I think we're starting from rest because release suggests you're holding it let it go. I get this Mgh initial equals a half MV final squared plus Mgh final Zay once again. Yeah, the masses cancel Now unfortunately this time velocity squared is not going to be gr. Let's draw our Free body diagram. There's the roller coaster while it's at the top What are the forces acting on it Connor get the obvious one and then it says the track is also pushing with Two newtons a normal force Two newtons what path am I tracing out at the top of this track Connor? So where must my net force be toward the you know what both of these are towards the middle They're both winners. My equation is gonna look like this Mg plus two equals m v squared over r Sadly not as nice You know what I think I'm gonna go mg plus two really quickly here the mass is 350 grams, which is point three five zero kilograms point three five times nine point eight plus two I Get five point four three Equals MV squared over r now. I can get the V by itself a bit easier. I think V is gonna be five point four three times r divided by M Square-routed gonna be five point four three times. What's the radius is it twelve centimeters? so point one two divided by point three five Square-routed that the velocity at the top is one point three six four four. I can now plug that into here. Oh and Nicole how would I get the H because they want the initial height how would I get the H by itself how to move the G over? Yeah, you know what? I think my equation is gonna look like this the initial height is equal to point five times one point three six four four squared plus nine times Point two four the final height is twenty four centimeters because the radius is twelve centimeters all divided by nine point eight point five times this answer squared Plus nine point eight times point two four all divided by nine point eight you get thirty three point five seven point three three four five someone else double-check me. Yeah point three three five meters and Connor the minimum would have been two point five times the twelve centimeters if you started thirty centimeters high you would just barely make it over the track ignoring friction here We had some extra force which meant we had some extra kinetic energy to do we started a little bit higher So that also jives in with my observation. Is that okay? It would be thirty three point five centimeters, but I changed it to meters I did Right, I didn't put a twenty four there. I put a point two four there, right? Yeah, yeah Okay We're done the unit Circular motion and gravitation the biggest unit of the year we spent a lot of time on this That's why you have two unit reviews You only have to hand in one I would do both of them the five review both answer keys are online Some of the questions get a bit repetitive. What's your homework a little bit more practice? Number one is good Number four is good Number five is good Six is good Seven is good eight is good nine B and then work on the review