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# Treating systems (the easy way)

## Video transcript

so in the previous video we solved this problem the hard way maybe you watched it maybe you didn't maybe you just skipped right to here and you're like I don't you want another hard way just show me the easy way please well as we're going to talk about now turns out there's a trick and the trick is after you solve this problem the hard way with a five kilogram mass and a three kilogram mass when you find the acceleration what you get is this that the acceleration of the five kilogram mass is just 29 point four divided by eight kilograms but when you do enough of these you might start realizing wait a minute twenty-nine point four Newtons that was just the force of gravity pulling on this three kilogram mass in other words the only force that was really propelling this whole entire system forward release the only external force propelling it forward and eight kilograms down here you might be like wait eight kilograms that's just five kilograms plus three kilograms is that just a coincidence or is this telling us something deep and fundamental and if not a coincidence turns out you'll always get this that what you'll end up with after solving this the hard way you'll get that in the very end you'll get all the external forces added up here where forces that make it go like this force of gravity end up being positive and forces that try to like resist the motion so if there was friction that would be an external force that tries to resist motion would be up top and then you get the total mass on the bottom and this makes sense the acceleration of this entire system if we think about it as a single object so if you imagine this was just one single object and you asked yourself what's the total acceleration of this entire system well it's only going to depend on the external forces and in this case the only external force making it go was this force of gravity right here you might object you might be like wait what about this tension right here isn't the tension pulling on this five kilogram mass making the system go it is but since it's an internal force now if we're treating this entire system as our one object since this tension is pulling trying to make it go you've got another tension over here resisting the motion on this mass trying to make stop that's what internal forces do there's always equal and opposite on one part of the object on the other so you can't propel yourself forward with an internal force so these end up cancelling out essentially the only force you have in this case was the force of gravity on top only external forces and the total mass on the bottom and that's the trick that's the trick to quickly find the acceleration of some system that might be complicated if you had to do it in multiple equations and multiple unknowns but much much easier once you realize this so the trick sometimes it's called just treating systems as a single object let me just show you really quick if that made no sense let me just show you what this means if we just get rid of this so what I'm claiming is this if you ever have a system where multiple objects are required to move with the exact same magnitude of acceleration right because maybe they're tied together by a rope or maybe they're pushing on each other maybe there's many boxes in a row and these boxes all have to be pushed at the same acceleration because they can't get pushed through each other right if there's some condition where multiple objects must have the same magnitude of acceleration then you can simply find the acceleration of that system as if it were a single object and writing sys for system by just using Newton's second law but instead of looking at an individual object for a given direction we're just going to do all of the external forces all the external forces on our system treat it as if it were a single object divided by the total mass of our system and so when you plug in these external forces these are forces that are external so external means not internal to the system so if I if I think of this five kilogram box and this three kilogram box as a single mass tension would be an internal force because it's applied internally between between these two objects between objects inside of our system but the force of gravity on the three kilogram mask that's an external force because that's the earth pulling down on the three kilogram mass and the earth is not part of our system similarly the normal force is an external force but it's exactly canceled by the gravitational force so even though those are external they're not going to make it in here I mean you can put them in there but they're just going to cancel anyway we only look at forces in the direction of motion and if it's a force that causes motion we're going to make that a positive force if it's a force in the direction of motion like this force of gravity is we make those positive forces so forces will be plugged in positive into here if they make the system go and that might seem weird you might be like wait how do I decide if it makes the system go we'll just ask yourself is that force directed in the same direction as the motion of the system so we're just saying the system is going to accelerate if there's forces that make it go and we're going to make negative we're going to plug in negative forces the forces that make the system stop or resist the motion of the system so maybe I should say resists motion of the system in this case for this one down here I don't have any of those so resist motion of the system I don't have any of those I could have if I had a force of friction then there'd be a external force that resists the motion I would plug in that external force as a negative because it resists the motion too even though this might sound weird it it makes sense if you think about it the acceleration of our system treated as a single object is only going to depend on the forces that try to make the system go and the forces that try to make the system stop or resist the motion so if we add those accordingly with positives and negatives we divide by the total mass which gives a total measure of the inertia of our system we'll get the acceleration of our system it makes sense and it works turns out this always works and it saves a ridiculous amount of time for instance if we wanted to do this problem if you just gave me this problem straight away and you were told do this however you want I would use this trick I would say that the acceleration of this system which is composed of this five kilogram mass and our three kilogram mass which is going to be equal to I'd ask myself what force makes this system go what force drives the system and it's this force of gravity on the three kilogram mass that's driving this system right if you took this force away if you eliminated that force nothing is going to happen here this is the force making the system go so I'd put in my three kilograms times 9.8 and at this point you might be like whoa okay that gravity made it go should I include this gravity too but now that gravity is perpendicular to the motion for one so this grip this gravity isn't making the system go that's just causing this mass to sit on the table and for two it's canceled by that normal force so those cancel anyway even though they're external forces this is it this is the only one that drives the system so I put that in here and I divide by my total mass because that tells me how much my system resists through inertia changes in velocity and this is what I get I get the same thing I got before I get back my three point six eight meters per second squared and I get in one line I mean this trick is amazing and it works and it works in every example where two masses or more masses are forced to move with the same acceleration so this is great this will save you a ton of time this is supposed to be a three here and to show you how how useful it is let's say there was friction let's say there was a coefficient of friction of 0.3 well now I'd have a frictional force so there'd be an external frictional force here it'd be applied to this five kilogram mass I'd have to subtract it up here so if I get rid of this so it's not going to be three point six eight anymore I'm going to have a force of friction that I have to subtract so minus mu K so the force of friction I'll just put force of friction and so to solve for the force of friction the force of friction is going to be equal to well I know three times 9.8 is let me just write this in here twenty nine point four Newtons minus the force of friction is given by so there's a formula for force of friction the force of friction is always mu K F n so the force friction on this five kilogram mass is going to be mu K which is 0.3 so it's going to be 0.3 times the normal force not the normal force on our entire system I don't include this three kilogram mask it's only the normal on this five kilogram mass that's contributing to this force of friction here so even though we're treating the system as a whole we still have to find individual forces on these individual boxes correctly so it won't be the entire mass that goes here the normal force on the five kilogram mass is just going to be 5mm kilograms times 9.8 meters per second squared I divide by my total mass down here because the entire mass is resisting motion through inertia now if I solve this for my acceleration of the system I get one point eight four meters per second squared so this is less less than our three point six eight and that makes sense now there's a resistive force a resistive external force trying to prevent the system from moving but you have to be careful what I'm really finding here I'm really finding the magnitude of the acceleration this is just giving me the magnitude if I'm playing this game where positive forces are ones that make it go and negative forces are ones that resist motion external forces that is I'm just getting the magnitude of the acceleration individual boxes will have that magnitude of the acceleration but they may have positive or negative accelerations in other words this five kilogram mask accelerating to the right going to have a positive acceleration in other words the acceleration of the five kilogram mass will be positive one point eight four and the acceleration of the three kilogram mass since it's accelerating downward will be negative one point eight four meters per second squared