Treating systems (the hard way)

alright this problem is a classic you're going to see this in basically every single physics textbook and the problem is this if you've got two masses tied together by a rope and that rope passes over a pulley what's the acceleration of the masses in other words what's the acceleration of the three kilogram mass and then what's the acceleration of the five kilogram mass and if you're wondering what the heck is a pulley so the the pulley is this part right here this right here is the pulley so what a pulley does a pulley is a little piece of plastic or metal that can rotate and it's usually got a groove in it so that a string or a rope can pass over it what it does is it rotates freely so that you can turn what's a horizontal tension on one side into a vertical tension on the other or vice versa it turns vertical forces and horizontal forces it allows you to transfer a force from one direction to another direction so that's what these pulleys are useful for and if they can spin freely and if this pulley has basically no mass if there's no resistance to motion at all then this tension on this side is going to be equal to the tension on this side this vertical tension gets transferred fully undiluted into a horizontal tension and these tension values will just be the same if this pulley can spin freely and if its mass is really small so that there's no inertial reason why it doesn't want to spin so that's the problem let's say you want to you wanted to figure this out what is the acceleration of the 3 kilogram mass what's the acceleration of the 5 kilogram mass now I gotta warn you there's an easy way to do this and a hard way to do this now I'm going to show you the hard way first sorry no one ever ever wants to hear that but the reason is that it won't the easy way won't make any sense unless I show you the hard way first it won't make any sense of why the easy way works let's I show you the hard way and for two the hard ways aren't really all that hard so I'm calling it the hard way but it's not really that bad and for three sometimes teachers and professors just want to see you do it the hard way so you should know how to do this so what do we do we want to find acceleration well you know how to find acceleration we're going to use Newton's second law so we'll say that the Excel in a given direction is going to equal the net force in that direction divided by the mass now what do we do what mass are we going to choose we've got a couple masses here one thing we could do let's just pick the five kilogram mass just pick one of them so I'm going to say that the acceleration of the five kilogram mass is the net force on the five kilogram mass divided by the mass of the five kilogram mass and remember we should always pick a direction as well so do we want to pick the vertical direction or the horizontal direction well since this box is going to be accelerating horizontally and that's what we're interested in I'm going to put one more subscript up here X to let us know we're picking the horizontal direction so I can fill this out now I can plug stuff in the acceleration of the five kilogram mass in the X Direction is going to be equal to all right well forces do we have to figure out what goes up here you always draw a force diagram so what forces do I have on the five kilogram mass I'm going to have a force of gravity I'll draw that straight down FG and there's going to be an equal force normal force upward so this normal force up should be equal to the force of gravity in magnitude because this box is probably not going to be accelerating vertically there's no real reason why it should be if this table is rigid and there's one more force on this box there's a force to the right that's going to be the force of tension and if there's no friction on this table then I have no leftward forces here I'm ignoring air resistance since we usually ignore air resistance that's it the only horizontal force I've got is T tension and I divide by the mass of the five kilogram box which is five kilograms but we got a problem look at we don't know the acceleration of the five kilogram mass and we don't know the tension I can't solve this normally what you do in this case is you go to the vertical direction the other direction in other words that's not going to help me either that's just going to tell me that the normal force is going to be equal to the force of gravity and we kind of already knew that so that doesn't help so what do we do well you might note this is only the equation for the five kilogram mass so now you have to do this for the three kilogram mass so let's come over to here let's say that the acceleration of the three kilogram mass it's going to be equal to the net force on the three kilogram mass by the mass of the three kilogram mass and again which direction should we pick will this acceleration over here is going to be vertical so let's solve this for the vertical direction I'm going to add one more subscript Y to reline myself and you should do this too so you remember which direction you're picking so what forces do I plug in here you figure that out with a force diagram I'm going to have a force of gravity on this three kilogram mass and I'm gonna have the same size of friction or sorry the same size is tensioned that I had over here so the tension on this side of the rope it's going to be the same as the tension on this side assuming this pulley offers no resistance either by its mass or friction so assuming that it's mass is negligible and there's basically no friction that I'm going to have a tension and that tension is going to be the same size so I'll draw that coming upward but it's not going to be as big as the force of gravity is on this three kilogram mass I've got the force of gravity here this tension is going to be smaller and the reason is this three kilogram mass is accelerating downwards so these forces can't be balanced the upward force of tensions got to be smaller than the force of gravity on this three kilogram mass but this tension here should be the same as this tension here so I'll plug those in so let's plug this in a of the three kilogram mass in the Y direction is going to be equal to I've got two vertical forces I've got tension up so I'll make that positive because we usually treat up as positive I've got gravity down and so I'm going to have negative because it's downward force of three kilograms times the acceleration on earth is 9.8 meters per second squared now what do we do we divide by three kilograms because that's the mass but I've still got a problem I don't know this acceleration or this tension so what do I do you might notice if you're clever you'll say wait I've got my unknown on this side is acceleration and tension my unknown on this side is acceleration in tension it seems like I've got two equations two unknowns maybe we should combine them and that's exactly how you do these so I've got tension in both of these equations let me solve for tension over here where it's kind of simple and I'll just get the tension equals five kilograms time's the acceleration of the five kilogram mass in the x-direction so now I know attention is tension is equal to this and that tension over on this side the same as the tension on this side so I can take this and I can plug it in for this tension right here and let's see what we get we get that the acceleration of the three kilogram mass vertically is going to equal all right I'm going to have a big mess on top what am I going to get I'm going to get so T is the same as five ax so I'll plug in five kilograms times the acceleration of the five kilogram mass in the X direction and then I get all of this stuff over here so I'll get the rest of this right here I just bring that down right there all right now what do I have I've got three kilograms on the bottom still so have to put that here are we any better off yeah we're better because now my only unknowns are acceleration but these are not the same acceleration look at this acceleration here is the acceleration of the three kilogram mass vertically this acceleration here is the acceleration of the five kilogram mass horizontally now here's here's where I'm gonna have to make an argument and some people don't like this but it's crucial to figuring out this problem and the key the key idea is this if this three kilogram mass moves down let's say one meter the same moves downward one meter well then this five kilogram mass had better move forward one meter because if it doesn't then it didn't provide the one meter of rope that this three kilogram mass needed to go downward which means either the rope broke or the rope stretched and we're going to assume that our rope does not break or stretch that's kind of a lie all ropes are going to stretch a little bit under tension we're going to assume that stretch is negligible so the argument is that if this three kilogram mass moves downward a certain amount this five kilogram mass has to move forward by that same amount in order to feed that amount of rope for this three kilogram mask to go downward by that amount otherwise think about it if this five kilogram master sat here and the three kilogram moved or the three kilogram moved farther than the five kilogram mass then this rope is stretching or break so if you believe that if you don't believe it pause it and think about it because you got to convince yourself of that if you believe that then you can also commit yourself that well if the three kilogram master is moving downward at a certain speed let's say two meters per second then the five kilogram mass had better also be moving forward two meters per second because otherwise it wouldn't be feeding rope at a rate that this three kilogram meat needs to move downward at that rate and finally if you believe all that is not too much harder to commit yourself that this three kilogram mass no matter what its acceleration downward must be this five kilogram mass had better have the same magnitude of acceleration forward so that it's again feeding the rope so this rope doesn't break or snap or stretch because we're going to assume the rope doesn't do that so what I'm saying is that the acceleration of the three kilogram mass in the Y direction had better equal the magnitude so these magnitudes have to be the same the size the sign doesn't have to be the same so this three kilogram mass has a negative acceleration just because it points down and we're assuming up is positive down is negative this five kilogram mass has a positive acceleration because it's pointing to the right and we're assuming rightward is the positive horizontal direction so they can have different signs but the magnitudes that better be the same so that you're feeding this rope at a rate that the other one needs in order to move and so we can say that the magnitudes are the same in this case since one is negative of the other I can say that the acceleration of the three kilogram mass vertically downward is going to be equal to let's say negative of the acceleration of the five kilogram mass in the X direction I could have written it the other way I could wrote that a of the five kilogram mass in the X Direction is a negative a of the three kilogram mass in the Y direction they're just different by a negative sign is all that's important here okay so this is the link we need this is it so this allows us to put this final equation here in terms of only one variable because I know it I've got a three Y on this left-hand side I know a three Y should always be negative a 5 X so I take this and just plug it in for a 3 Y right here I'm going to get negative a 5 X equals oil all of this stuff so I'll just copy this save some time copy paste just equals all of that all I did was plug in what I know a 3y has to be equal to you because now look at I've got one equation with one unknown I just need to solve for what a 5x is it's on both sides but I'll need to combine these and then isolate it on one side so there's going to be a little bit of algebra here let's just take this let's give ourselves some room move this up just a little bit okay so what do we do we're going to solve for a 5x let me just get rid of this denominator let me multiply both sides by by 3 kilograms so I'm going to get negative 3 kilograms times a 5 in the x-direction if I multiply both sides by 3 kilograms then I get 5 kilograms times a 5 in the x-direction and I've still got - all right 3 times 9.8 is 29.4 Newtons so we'll just turn this into what's supposed to be 29.4 newtons so let's combine our a terms now let's move this negative 3 a to the right hand side by adding it to both sides and let's add this 29.4 to both sides so I'll get the 29.4 newtons over here with a positive if I add it to both sides and it'll disappear on the right hand side and then I'll add this term to both sides added all add a positive 3 kilogram times a to both sides it will disappear on the left and I'll get 5 kilograms times a 5 in the x-direction plus 3 kilograms times a 5 in the x-direction now we're close look at on the right hand side I can combine these terms because 5a plus 3a is the same thing as 8a so 29.4 Newtons point 4 equals 8 kilograms times I'll put the parentheses here times 585 in the x-direction now I can divide both sides by 8 and I'll usually put the thing we're solving for on the left so I'm just going to put that over here I'll get the 29.4 newtons 29.4 newtons over 8 kilogram is equal to the acceleration of mass five the five kilogram mass in the x-direction and if we calculate that just put that into my calculator twenty nine point four divided by eight I get three point six seven five so we'll just round will just say that's three point six eight three point six whoops three point six eight and it's positive that's good we should get a positive because the five kilogram mass has a positive acceleration so we get positive three point six eight meters per second squared but that's just the five kilogram mass how do we get the acceleration of the three kilogram mass well that's easy it's got to be have the same this gotta have the same magnitude of the five kilogram mass all I have to do is take this number now I know what a 5x is so I just plug that in right here well then I know that a three y is just going to be equal to negative three point six eight meters per second squared and I'm done I did it we figured out the acceleration of the three kilogram mass it's negative no surprise because it's accelerating downward we figured out the acceleration of the five kilogram mass it's positive not a surprise it was accelerating to the right the way we did it recapping really quick we did Newton's second law for the five kilogram mass that didn't let us solve we did Newton's second law for the three kilogram mass that didn't let us solve in fact it got really bleak because it seemed like we had three unknowns and only two equations but the link that allowed us to make it so that we only had one equation with one unknown is that we plugged one equation to the other first we had to then write the accelerations in terms of each other that's because these accelerations are not independent the accelerations have to have the same magnitude and in this case one had the opposite sign so when we plug that in we have one equation with one unknown we solve we get the amount of acceleration so that's the that's the hard way to do these problems so in the next video I'll show you the easy way to do these problems