 Here we go first before we before we begin on The weekend I came across a great video and I think it works into forces here I'll show it to you and then we're gonna try and talk about what's going on with actual forces in this video And I'm not quite sure in fact I may even end up posting this on to the physics teachers News group for some feedback Did I put it here? I think I put it I think I put it here mr. Duke Let's see arrange by Date modified there it is A company claims that they have invented a machine that can clean up spills in a very remarkable manner Watch this is from Japan This this is bordering to me on witchcraft almost. I'm like, okay This is the machine That's some mustard. Oh, it gets better Let's start smooching some of this around Let's make sure it's actually liquid So there's your standard spatula, which just smears it I've tried to figure out what's going on there because it's clearly applying a force in one direction and There should be friction in another direction me. I'm really not quite sure what's going on here I got a sit and really think about this free body diagram because what should that catch up be doing? It should be smearing in the direction that that thing is pushing should it not It's what ketchup is always or it spills have always done for me yet somehow this isn't good morning, sir How are you doing sir? Did we just see a pretty cool video? Was it actually almost blowing your mind cool? Oh, it would be sure needed someone else had seen that video Feel lucky. What we got? Let's see. I won't even ask What are we at for okay, yes, pick it up first. Yeah, there you go. Good pass. Don't hit the objective. Thank you And Here you go Dylan. Oh, how's that happen? I don't get the that video I don't get the free body diagram because again, I'm going if it if that machine is pushing to the To the right or to the left there has to be a force going to the but friction that huh? Because it almost looks like that was frictionless it almost looked like it slid smoothly right underneath it and that that Liquid that smear whatever it was the ketchup almost looked like it had no longer had any friction I'm wondering maybe if that platform that little machine is vibrating really really really really fast So that the ketchup almost stays like on a cushion of air almost first. I don't know But it is pretty cool Meanwhile to today's lesson Many body problems more than one mass in These problems we have several objects connected together together by strings and The easy approach is to realize that we can treat the entire system as a single object Since all the parts are accelerating at the same rate what I'm going to say to you is if objects are connected with strings There's no way that I can make these accelerate at different rates the bottom mass and the top mass The string means unless I put slack in it and Devon we're not going to put slack in it Yep, you map them as long as I'm moving nice and smoothly the direction of the acceleration is different But if this is accelerating at two meters per second to your right, this is accelerating meters per second squared Sorry, this is accelerating up at two meters per second squared. I can't have them accelerate at different rates That's the first logical conclusion. We're going to reach So example one says find the acceleration and Then also find the tension between the strings right there And we're going to start out with no friction our frictionless surface What's the solution we're going to treat this system as a single object and write a force equation What we're going to do is we're going to walk along this string and we're going to list all of the forces I'm going to start out with this two kilogram mass, and I'm going to say what are the forces acting on this two kilogram mass get the obvious ones gravity What else? Normal force up. What else? Well, I'm just going to put an arrow here reminding me that there's a 20 newtons I'm not doing a separate free body diagram I'm actually just labeling the forces on this lovely diagram that they gave me I haven't represented the forces as a dot. I'm doing kind of a easier sloppier free body diagram for convenience What else well there is one more as it turns out There is tension in that direction because it's having to pull this mass And so it feels resistance from this mass on this string What are the forces acting on this mask at the obvious ones? Now because I have two masses I'm going to call this m1g and I'm going to go over here And I'm going to call this m2g and my standard Alexis as I always label the masses from left to right The first one is mass one the second one is mass two the third one is mass three Starting from left to right. You don't have to go in that order I'm just going to say that's the way I'll do it if you want your work to look like mine But we are going to have to label the masses as separate What else? Sorry, I heard it. Normal force because it can't it's not sinking into the ground Oh, and you know what since I have two normal forces I'll call this normal force number one and normal force number two because they're different What else? Tension the string is pulling it forward not the 20 newtons It doesn't know about that 20 newtons applied force. All it knows is this string is causing me to accelerate forwards Here's the nice thing Use your imagination think a little bit If I have a piece of string Is it possible for me to have a different tension over here than I have over here? I don't think it is I think Sean that if I'm pulling on the string I think both those tensions have to be identical I can't put somehow magically more tension from one end than the other and this forces come in pairs The tension is going to be the same in both So there's our free body I'll call the free body diagram. There's our labeled forces To find the acceleration We're going to first of all ask ourselves who's winning who is winning if anyone Which way do you think this thing is accelerating? To the right so who's the winning force? Okay, we're going to go winner minus loser, but we're going to walk down the entire string the entire rope starting here Winner That means anything to the right kodi is going to be winner Anything to the left is going to be losers. So these are vertical. I don't care about those guys. I'm walking down the rope Winner or loser Loser negative winner or loser positive Hey, alex. What's going to happen to the tensions? See it alexis. I said alexis alexis. What's going to happen to the tensions? They're going to cancel. Oh, I love it when I lose tension. It's so relaxing Uh We've come to the end of the rope Now the only difference here jacob is instead of writing ma You know what? We're moving two masses. It's the mass of both of them times a That's the only difference Now remember what I said we're going to treat the system as a single object. We're treating it as one big mass tension cancels So we have this 20 equals And m both I think that's m one plus m two times a Find the acceleration. How would I get the a by itself? Going to move that whole bracket to the bottom of the 20 the acceleration is going to be 20 divided by m one plus m two it's going to be 20 divided by five Don't reach don't reach for your calculator Dylan in your head. What is 20 divided by five? Four you don't need your calculator for this one. Uh, how many sig figs? It looks like uh, two sig figs here Because the 20 has a decimal two sig figs two sig figs. So i'm going to go 4.0 Meters per second squared. That's the only twist we're going to add if we have more than one mass We're going to label all of the forces and they're going to walk down the entire chain Listing all the masses winners and losers depending on what the winner is But it's going to be the mass of both times a Or if there's three masses the mass of all three times a or if there's four what you're never going to get But if there was four, you know what I could say the mass of all of them times a That's it now. I'd like to find the tension Because it said Find the acceleration check Find the tension To find the tension We look at a single mass I'm either going to look at this mass Or i'm going to look at this mass and I think i'm going to choose this mass because it's got less stuff on it Both are fine. You'll get the same the right answer for both But this one here Dylan a little cleaner So i'm going to scroll down. You guys still have this diagram in front of you. Yes So i'm going to scroll down and i'm going to say, okay Just looking at this mass only Who's winning Cassidy tension Who's losing on that mass? It's a trick question Nothing What about that 20 at 20 is not touching that mass that 20 is not touching this mass so tension That's the winner no loser Equals Are we looking at both masses or just one mass this time? So it's going to be m one a Now they're asking me to find the tension. Do I know mass one? Did I just figure out the acceleration by looking at both? In fact, that's going to be our Strategy or plan of attack we find the acceleration by looking at the whole system And then we can find any individual force by looking at any mass that has that individual force What was mass one three? What was a four dillon in your head showing no work? Woohoo, is that two sick figs? Yep units we just found a force because tension is a force newtons Hey, mr. Dewick, what if I hadn't used the three kilograms? What if instead I'd use this one? The two kilogram no problem. You all still have the diagram On the two kilogram mass Who was winning 20 Who was losing? There is a force on the two kilogram mass pointing to the left. Which one? Minus tension equals and now we're looking at mass two. So it would be m two a Let's get the tension by itself I think to do that devon I would plus the tension over here and I would minus this over there Is that okay? Could I do that in one step? Oh Thank goodness. You guys are so much better at equation solving than you used to be So if I hear you correctly devon you're saying this 20 minus m two a equals tension Tension is going to be 20 minus m two was two a was four and holy smokes. What do we get as an answer? 12 We've just proved the tension is the same But now that we know that we don't need to prove it again We'll always just try and pick whichever object has the least number of forces because it'll be an easier equation It doesn't matter which one you pick though. In fact, honestly ashley on a test I'd probably do both and if I get the same answer for both. I know I'm right I wouldn't do both right away. It's finish the test go back check your answers. Oh, I'll find tension using the other mass Hey, I got the same answer. I know I'm right Take the same diagram, but we're going to add a little pulley And hang it over a cliff Conor you're gonna make it Because I don't know where you are, but you aren't here you're gonna make it Yeah, I'm not okay So if we have a hanging mass and no external force Then we need to understand that it's the weight the mg of the hanging mass that accelerates the system. So What are the forces acting here? Let's do the hanging one first get the obvious ones Gravity and I'm gonna call this m two g. I'll call it mass two because I always go from left to right Is this in free fall? No, so something's slowing it down What the string? Oh, what do we call that force? Tension Is there a normal force? No, because it's not touching a surface normal forces only occur when you're touching a surface I think that's it for this part What are the forces acting over here get the obvious ones And one g what else normal force what else? The rope Tension and devin this is where it gets when you're finished you on it kind of weird a little bit Because the nice this is vertical. These are horizontal, but it's really the forces along the rope So I'm going to say first of all, let's clue in Is there any way that this can be moving to the left at all? If this is moving it's because this thing's pulling it down into the right So my winner Right there So what I'm going to do is I'm going to walk down the entire length of rope And any force that would end up pointing down is going to be winner Any force that ends up pointing up is going to be loser Winner. Oh, I ran into a force on the rope winner or loser Loser minus Oh, I ran into a force on the rope now if I follow it follow it follow it follow it follow it follow it follow it Which way will it end up pointing up or down? Winner. Have I done all of the forces that are parallel to the rope? Yep Equals Justin how many masses? Two and we're looking at both of them because we've done force We've done a force from this mass and this mass in this equation Then we better have m1 plus m2 times a or mass of both Times a I don't care which one you write some kids like m both because it's less writing Some kids like m1 plus m2 because they can see what's going to be going there. Whatever I can feel the tension just leaning what happens to the tension Ah, it's like going to my happy place Beach and Maui Tension cancels To develop a tension always cancel. Yep Can we just not write it? Nope I'll be fussy on that because I do want you to realize it's not that there's no tension I used to let kids not write it and then I found when I said find the tension We're no tension. No, there there is So we'll for the minus t plus t crossing out the one second that it takes I think is worth it Alex how would I get the a by itself? Yep A is going to be m2 g divided by m1 plus m2 It's going to be m2 4 times 9.8 divided by I think I can add m1 plus m2 in my head 6 You are allowed to use your calculator this time Although really you shouldn't need to that's 2 3rds 9.8 times 2 is 19.6 divided by 3 is going to be 6 point She's trapped the three carry I give up. I'm tired. What do you get? If you wanted you to do it in your head, mr. Duke Sorry fighting the cold Brain doesn't want to work 4 times 9.8 divided by 6 6.53 I should have been able to get that Oh, how many sig figs are we going to here? Let's see How many sig figs 3 how many sig figs 3 now this 9.8 is kind of a weird one It's sort of an arbitrary number of sig figs because it's really 9.8 If you went 2 or 3 I wouldn't freak out But I usually go from the data that they give me I don't include the constants in my sig fig calculation So I'm going to say 3 we're going to say the answer is 6.53 meters per second squared So there Is the acceleration what else did they want me to find? In this question sorry No, they didn't want me to find said identify the net force. We did there it is We wrote our equation Then find what my friend Acceleration check what else did they want me to find? of tension To find the tension we don't look at both masses because if you look at both masses the tension is always going to cancel Which means we're not going to be able to find it. We're going to look at one mass It doesn't matter which one although one will probably be easier than the other one. Which mass would you like me to use? Mass one or mass two. I don't care pick Alexis You say mass two this one Okay, now I am going to be saying to you in a second I would have chosen this one you'll see why in a second, but I want to show you you can get the same answer for both So here is using mass two. We're only looking at these guys now looking at mass two looking only at the forces here Who's winning? Okay, so it's going to be winner loser Equals and I'm only looking at one mass. Which mass did you say alexis m2? a There's our force equation There's our Winner minus loser tug of war Once again to get the t by itself alexis. Do you mind if I go plus t over here minus that over there? Can I do that one step? Yeah, okay. I'm going to go tension equals. I think it's m2g minus m2a Tension is going to be m2 4 times 9.8 minus 4 times 5 mr. Do it times 6.53 What's the tension don't I'll rush for your calculators at once or anything? Yes, I'm looking at you Sorry Oh, you already did it. Oh, I'm sorry. I thought you were sitting there like My bet the 13.1. Anybody else? Yes Okay alexis here's why I would have chosen mass one because I'm a lazy nerd ready Look at mass one Who's winning? Because it's accelerating to the right Who's losing? nothing You would get tension. Well, let's try that again, mr. Do it you would get tension Equals m1 a and you've already got tension by itself in a straight plug and chug But either one is fine and again I think I said to you to be honest on a test when I was checking my answers Because it takes all of when you get good at this it'll take you about 15 seconds. Maybe maybe 30 So it's pretty good. Oh, uh mass one was uh, what was mass one two? So if you go two times 6.53 Oh, I can do this in my head two times six is 12 two times point five three is One point Good gosh, mr. Do it oh six and you do get 13.1 Newton Nice Oh, it says to find tension isolate one mass. Sorry. I forgot that it does that on the next page So we did it right here because I couldn't see so See above now For problems that involve friction We can still use the same approach, but the tension calculations are a little bit tricky We'll be doing this in physics 12 quite a lot, but I'll just show you as an example You know Imagining things If we had friction right here All we would have is one more force in the loser direction We would have one more force right there loser friction. Oh and friction is what times what? Your time's a normal four. I don't know the normal force. Oh, but look look look look I know another force the same size as the number four yet one more force You got to do three more buttons on your calculator Easy to deal with just why I like this approach here The atwood machine Why is it called an atwood machine? I don't know it's just called that I'm guessing a scientist whose last name was atwood wrote a paper on it The atwood machine is a pulley system as seen below. So you have a pulley Mass two makes it easier for a person or a motor to lift m1. How many of you've been on an elevator before? All of you every single elevator has a system where this is the elevator And they have a counterweight that's the same mass as the elevator so that when the elevator is empty Essentially it weighs almost nothing to move up and down because it's balanced So that way all the motor has to do actually is lift the weight of the people The elevator itself is perfectly willing to move down or move up with a minimal amount of effort Atwood machine elevators are great examples except Brandon the atwood machine is mythical This pulley here has no friction Is that true in real life? No these ropes here have no mass Is that true in real life? No, uh, this pulley here has no mass Is that true in real life? No, but will it make our math easier here in high school? Yes Otherwise you need calculus because as this rope goes down its mass is changing all the time because it gets heavier and heavier You're gonna keep massive okay So here's the first question I suppose I have equal masses on both sides of the atwood machine And then I lift one mass up Now I go When I release it, how does the system move? A it stays this way B the higher mass goes down C the higher mass goes higher I want you to think about your answer and then in about 10 seconds I'm going to ask you to vote and once again how high you hold your hand up is how sure you are of the answer I'm only picking because I know mr. Dewick will make fun of me if I don't pick I kind of feel like I got a gut instinct. Oh, I've thought this through and free-bodied and I'm sure Are you ready think about what you think your answer is Who says a few of you? Who says B? Who says C? Convince me Yeah, this sounds like a job for a free-bodied diagram Let's prove it ready Matt What are the masses acting on mass one get the obvious ones? I'll call that mg Is it falling? No, what's holding it up? Tension What are the masses acting on mass two? Mr. Dewick you didn't call them mass one and mass two that's because the question says they're identical masses So why don't I just call them both math both mass and mass? Oh, what's preventing this guy from falling? Tension let's do a free-body diagram now Many of you felt that this was going to move higher So you must have felt that this was the winner because this would move lower So let's assume it's the winner winner winner or loser loser Now at first glance you might think because this tension is pointing up It's also a loser but follow it around the pulley and if you follow it around the pulley which way will it end up pointing? Down it's actually a winner And at first glance you might think that this gravity is actually a winner because it's down But Dylan follow it all the way around the pulley and which way when it gets to this side will actually end up pointing up Loser no the force not you And we're looking at both masses Look at the left hand side What does it work out to? Well, what happens to the tensions? What happens to the mass g's the mg's? Matt, you know left hand side works out to fact we get this zero equals m both times a How would I get the a by itself? You know what zero divided by anything is In fact, you know what I only need to go that I can say look for this to be zero One of these had to be zero because you're multiplying are the masses zero. No What's zero? So if you're not accelerating You ready? Who says a because you're not accelerating it must remain at rest There is no unbalanced force to cause this thing to move And we've done a lovely little says using principles of physics explain your answer I think we did a nice algebraic Oh, no numbers nerdy kind of a proof But we said, you know when we set up our free body weight or minus loser we get nothing zero equals ma Well, the m's aren't zero. There's gotta be the a's. These are identical masses Now there is some friction here is true, but trust me It's quite happy Says in the diagram below Assume mass two is bigger than mass one. In fact, what we're going to do is we're going to throw some numbers on here So Jacob make up a number for mass two in the teens somewhere And how about just to make this a bit fun add a decimal So we're going to go to three sig figs Something point something but in the teens somewhere give me a mass 15.4. So we're going to put a little m2 equals 15.4 kilograms Okay, and connor give me a mass for mass one Somewhere between five and 10, but give it to me to Three two decimal places three sig figs Something point something something make one up 7.23. That's an odd number. Why did you pick that out never mind? Let's label the forces acting on these masses What are the forces acting on this guy get the obvious ones? I'll call that m1g because they labeled it m1 Is it in free fall no tension What are the forces acting on this guy get the obvious ones? m2g Is it in free fall no Attention who's winning? Which way is this thing going to accelerate if it accelerates at all? I think m2 is going to move down because it's the heavier mass So i'm going to let down on the right side be winner Up on the right side is loser And we're going to walk down the entire rope You ready Winner winner or loser loser Now again at first glance you might say kody that this tension is a loser because it's pointing up But follow it all the way around when it gets to the right side which way will it end up pointing winner And which way will m1g end up pointing when it gets to the right side kody Up loser the force not you Did we look at more than one mass? Did we list the forces from more than one mass? Then it's not going to be m a it's going to be m1 plus m2 or the mass of both Times a By the way part a says write the force equations. There's the force equation What does part b say? Solve for Accelerate okay, you know what? Hey, everybody tension is just canceled you've all canceled tension Y'all feel more relaxed yes Are you gonna do the joke every time maybe Um, let's see if we can without rewriting this. Uh, how would I get the a by itself? Shannon Let's go straight to the final equation and I think a Is gonna be a m2g minus m1g All divided by m1 plus m2 And you think about how complicated a system this is but actually we get a pretty clean looking equation It's not really plug-and-chug. You gotta have to use brackets. I think but it's kind of close This is why I like this tug-of-war approach winner minus loser Because we have stuff moving up and down in this question to let up always be positive and down always be negative Would give us yucky math m2 15.4 times 9.8 minus m1 7.23 Times 9.8 all over 15.4 plus 7.23 Just remember when you type this in you'll have to put brackets there and there There and there, but you can do it all in one fell swoop If you've got a good calculator Let's see. Can I come up with an answer in my head? That's about half of that 0.9 0.3 ish what do you get? 3.5. I'm at that bad off really 3.54 3.54 units map meters per second square oh follow-up curveball questions Now that you know how fast this guy is accelerating or how fast this guy is accelerating If I said after two seconds, could you figure out how far it had fallen? Well, let's see vi would be zero the final I don't know t is two d equals vi t plus that Oh, I could use d equals vi to have a t squared and figure out how far this little system has moved in two seconds Or I could figure out how fast it was moving after two seconds Or so there's all sorts of all the previous stuff that we've done is now also fair game Although usually we do that in physics 12 What does b want us to find? Oh b is to find the acceleration. Sorry, mr. Duke. What does c want us to find? There was a c wasn't there? Yes. No, yes. Okay. What does c want us to find? Oh, it just says interpret the equation for the acceleration. That's kind of boring. Um, You know what? We're going to add a part d find The tension but I'm relaxed mr. Duke. I just got rid of tension. No, we're going to find it now So d By the way interpret the equation. Here's what it means. Did we get a positive answer here? Then we guessed right on the winner If we got a negative answer here Shannon, you know what that means you guessed the wrong winner Actually, your loser was the winner That so it built in there's an error check there too One negative answer would be nervous Okay, let's find the tension Which mass do you want to use? Justin pick mass one or mass two. I don't care Mass one Okay, why did you pick mass one? Did you have a reason here's why Mr. Duke lazy nerd would have picked mass one Is tension the winner in mass one? Then that means it's already going to be positive, which means it's going to be easier to get by itself That's me being really lazy. It really I mean, that's me being so lazy. I'm saying I don't even want to have to write an extra minus like that that lazy Okay, so looking only at mass one then who's winning Justin tension Who's losing m one g equals m? We're only looking at mass one. So it's going to be m one a How would I get tensioned by itself? How would I get tensioned by itself, Justin? Yeah, that's the other reason I would have picked mass one because I would have also said Oh, and then the other one is negative to plus it over. I don't have to worry about minusing I'm better at I make less dumb mistakes when I plus than when I minus so sure Tension is going to be m one a plus m one g. It's going to be what was mass one 7.23 We're still wondering why connor picked that number 3.54 Plus 7.23 kind of a weird number times 9.8 And just and if you're lucky you still have this number as a decimal right on your calculator to all its extra digits So you don't even need to you can just go times 7.23 plus 7.23 times 9.8 if you're lucky What's the tension? Where'd you get shot? 96 0.4 because one of the three sig figs 96.4 newton is anybody else I got one notter two notters rest you know 7.23 times 6.54 plus 7.23. I think you're right times 9.8 96.4. I think is what I got or do you get or do you get 96.5 if you carry the original value? 96.4 Nice extension two masses three masses Same approach except it's now Winner minus loser equals the mass of all of them times a carefully label your forces and then walk down the entire rope Example three says find the acceleration of these systems Example four says find the acceleration and the chord tension if m one equals that and m two equals that I think what i'm going to do because I've talked for a long time I got a short class tomorrow what i'm going to do is i'm going to pause here I'm going to give you some homework to work on tomorrow I'll do this example and this example and then the rest of the class will be finished the homework So what's your homework number one? I think right now you can try number one two three four Five is good By the way look up at number four for a second How many ropes? Two well three counting this one, but two you know how many tensions you're going to have Two, I would call this tension one tension one tension two tension two and you'll end up having to solve for both Don't see if you can handle that one. Otherwise, I'll be willing to go over that either next class or on tuesday Number six now you don't have to do this all tonight. This is what you're capable of doing now And then i'm going to assign the rest of this tomorrow But i'll give it to you now anyways if you want to try and get ahead of the game because you're busy friday and saturday or something like that I'm going to be assigning number seven I'm going to be assigning number eight Why don't you just say all of them because I wasn't sure I couldn't remember Okay So we'll pause there If you can't get some of these i'm going to say leave a big gap Maybe tomorrow when I do the last couple I might help you or I'll go over them a little later on tomorrow in class