Three box system problem

so check out this problem you've got a 12 kilogram mass sitting on a table and on the left-hand side it's tied to a rope that passes over a pulley and that rope gets tied to a three kilogram mass and then on the right side of this 12 kilogram box you've got another rope and that rope passes over another pulley on the right it is tied to the five kilogram box over here the question is what's the acceleration of the 12 kilogram box let's make it even harder let's say there's a coefficient of kinetic friction between this 12 kilogram box and the table of 0.1 now you're looking at a really hard problem if you try to solve this the hard way and by the hard way I mean using Newton's second law for each box individually and then trying to solve what you'd end up with is at least three equations and three unknowns because you have three different accelerations for each of these you'll have two different tensions because this left rope is under a different tension from the right rope now it's only when you have a single rope can you say that it's the same tension this problem can be hard I mean there's gonna be tons of algebra mistakes potentially and so you get to avoid that we can solve this the easy way and if you remember the easy way is just by saying well let's treat all of these boxes as if they're a single object and we can do that because they're all going to have the same magnitude of acceleration that I'm just calling a system that's going to be the magnitude the acceleration of our system all these boxes will accelerate with the same magnitude some may have negative accelerations some may have positive accelerations like these but they're going to all have the same magnitude of acceleration because we're going to assume these ropes don't break and if they broke then they'll be different magnitudes or if they stretch but we're assuming that doesn't happen and the way we can find this is by just saying well if this is just a single object I don't have to worry about any internal forces now these tensions become internal forces and those don't make a system accelerate only external forces are going to make it a system accelerate so I'll have to do is find out what are all the external forces that try to make this system go try to accelerate it and ones that try to prevent acceleration I'll call this F external and then I divide by the total mass because this is just simply Newton's second law as if this were one big object so what are my external forces well the force that makes it go is going to be this five kilograms force of gravity so I'm have a force of gravity over here that tries to propel the system forward this is the one that's going to be driving the system if I let go of these boxes it's going to start shifting in this direction because this five kilogram mass has a larger force of gravity than this three kilogram mass so I'm going to include that as a positive but I'm just going to define direction of motion as positive because it's easy you can do it differently if you wanted to you could find the other ways positive so five times 9.8 meters per second squared is how you find this force of gravity are there any other forces that are propelling this forward no no external forces are so what are there any forces that are trying to reduce the acceleration yeah there's this force of gravity over here this force of gravity on the three kilogram mass is trying to prevent the acceleration because it's pointing it's pointing opposite the direction of motion the motion of the system is up right and down across this direction but this force is pointing opposite that direction this force of gravity right here so I'm have to subtract three kilograms times 9.8 meters per second squared and are my going to have any other forces that try to prevent the system from moving you might think the force of gravity on this 12 kilogram box but look at that that doesn't really in and of itself prevent the system from moving or not moving that's perpendicular to this direction I've called the direction of motion that's positive direction if it were a force in this way whoops if it were force this way or a force that way it'd try to cause acceleration of the system this force of gravity just gets in negated by the normal force so all you have to worry about that force so are there any forces associated with the 12th kilogram box to try to prevent motion it turns out there is there's going to be a force of friction between the table because there's this coefficient of kinetic friction so I've got to force this way this kinetic frictional force that's going to be have a size of mu K times FN that's how you find the normal force and so this is going to be minus the MU K is zero point one and the normal force will be the normal force for this 12 kilogram mass so I'll use 12 kilograms times 9.8 m/s^2 you might object you might say hey hold on 12 times 9.8 that's the force of gravity why are you why are you using this force I thought you said we didn't use it well we don't use this force by itself but it turns out this force of friction depends on this force so we're really using a horizontal force a force that tries to prevent motion which is why we've got this negative sign here but it's a horizontal force it just so happens that this horizontal force depends on a vertical force which is the normal force and so that's why we're multiplying by this point one that turns this vertical force which is not propelling the system or trying to stop it into a horizontal force which is trying to reduce the acceleration of the system that's why I subtract it and then I divide by the total mass and my total mass is going to be 3 plus 12 plus 5 is going to be 20 kilograms and now I can just solve if I solve this I'll get that the acceleration of the system is going to be 0.392 meters per second squared so this is a very fast way look at this is basically a one-liner if you can put this together right it's a one-liner there's much less chance for error than when you're trying to solve three equations with three unknowns this is beautiful when you apply this though be careful the acceleration of the five kilogram mass would be negative zero point three nine two because it's accelerating downward the acceleration of the 12 would call positive zero point three nine two because it's accelerating to the right and we typically call rightward accelerations positive and then the three kilogram mass also would have positive 0.392 because it's accelerating upward