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Current time:0:00Total duration:8:50

let's solve some more of these systems problems if you remember there's a hard way to do this and an easy way to do this the hard way is to solve Newton's second law for each box individually and then combine them and you get two equations with two unknowns you try your best to solve the algebra without losing any signs but let's be honest it usually goes wrong so the easy way to do this the way to get the magnitude of the acceleration of the objects in your system that is to say if I want to know the magnitude at which this five kilogram box accelerates or that this three kilogram box accelerates all I need to do is take the net external force that tries to make my system go and then I divide by my total mass of my system this is a quick way to get what the magnitude of the acceleration is of the objects in my system but it's good to note it'll only work if the objects in your system are required to move with the same magnitude of acceleration and in this case they are what I have here is a five kilogram mass tied to a rope and that rope passes over a pulley pulls over and connects to this three kilogram mass so that if this five kilogram mass has some acceleration downward this three kilogram mass has to be accelerating upward at the same rate otherwise this rope would break or snap or stretch and we're assuming that that doesn't happen so this rope is the condition that requires the fact that this rope doesn't break is what allows us to say that the system is just a single big total mass with external forces exerted on it so how would we solve this I just say that well what are the external forces keep in mind external forces or forces that are exerted on the objects in our system from objects outside of our system so one external force would just be the force of gravity on this five kilogram mass so I'm going to have a force of gravity this way and that force of gravity is just going to be equal to five kilograms times 9.8 meters per second squared because that's how we find the force of gravity should I make it positive or negative well this five kilogram is going to be the one that's pulling downwards so if the question is I hold these masses and I let go what's the acceleration this five kilogram mass is going to accelerate downward it's going to drive the system forward that's the force making this system go so I'm going to make that a positive force and then I figure out are there any other forces making this system go no they're not you might say well what about this tension over here isn't the tension on this three kilogram mass isn't that tension making this system go not really because that's an internal force exerted between the objects in our system and internal forces are always opposed by another internal force this tension will be pulling the three kilogram trying to make a move but it opposes the motion with the five kilogram mass and if we think of this three plus five kilogram mass is a single object these end up just cancelling on our single object that we're viewing is one big eight kilogram mass so those are internal forces we don't include them they're not a part of this trick we have to figure out what other forces will try to make this system go or prevent it from moving another force that tries to prevent it from moving is the force of gravity on the three kilogram mass or one force that tries to prevent the system from moving would be this force of gravity how big is that that's three kilograms times 9.8 meters per second squared and that's trying to prevent the system from moving this five kilogram mass is accelerating downward and this force is in the opposite direction of motion that trips people out sometimes they're like I don't understand they're both pointing down shouldn't they have the same sign they would when we're using Newton's second law the way we usually use it but when we're using this trick what we're concerned with our forces in the direction of motion this is an easy way to figure it out forces in the direction of motion we're going to call positive and any forces opposite the direction of motion we're going to call negative so forces that propel the system forward we'll just call that the positive direction forces that resist the motion we're just going to call that the negative direction and since this is on this side of the motion of the system this system is everything in this system is going this way the three kilogram mass goes up the string over here goes up the string up here goes to the right the string right here goes down the five kilogram mass goes down because all the motion of the system is this way we define that way as positive this force of gravity on the three kilogram mass is the opposite direction it's opposing the motion of the system it's preventing the system from accelerating as fast as it would have that's why we subtract it and now we just divide by the total mass and the total mass ajiz five plus three is going to be eight kilograms and I get the acceleration of my system so if I just add this up I get two point four five meters per second squared so this is a really fast way to get what the acceleration of our system is but you have to be careful if the question is what's the acceleration of the five kilogram box well technically that acceleration of the five kilogram box would be negative two point four five what we really found here since we were just finding the magnitude was the size of the acceleration since it's five kilogram box is accelerating down and we usually treat down as negative you won't want to forget that negative in putting in that answer the acceleration of the three kilogram box however would be positive two point four five meters per second squared so when you're applying this to an individual box you have to be very careful and make sure you apply that acceleration with the correct sign for that particular box and if you wanted to find the tension now now it's easy to find the tension I can find this tension right here if I wanted to if the next step was find the tension in the string connected to the boxes now I can just use Newton's second law but the way we always use it I'm done with the trick the trick is just a way to get the magnitude of the acceleration now that I have that I'm done treating it as a system or a single object I'll look at this single five kilogram mass all alone and I'll say that the acceleration of the five kilogram mass which is Newton's second law is going to equal the net force on the five kilogram mass divided by the mass of the five kilogram mass I know the acceleration of the five kilogram mass but if I'm going to treat up as positive now I got to plug this acceleration in with a negative sign so negative two point four five meters per second squared is going to equal the net force on the five kilogram mass I've got tension up you might be like wait we said that was an internal force it was an internal force and we didn't include it up here but we're doing the old rules now normal second law in the vertical direction so I use vertical forces and if their upward I'm going to treat them as positive and if their downward like this five times 9.8 I'm going to treat it as a negative because it points down five times 9.8 m/s^2 and I divide by the five kilogram mask because that's the box I'm analyzing I'm not analyzing the whole system I'm just analyzing the five kilogram box now and I can solve and I can get my tension the alternate way to do this would be to say all right let's just treat down as positive for this five kilogram mass I then plug my acceleration in is positive and I plug my force of gravity and positive then my tension would be negative I'd get the same value out here I'm just solving for the magnitude of the tension anyway so if I solve this if I plug this into the calculator and solve for tension I'm going to get thirty six point seven five Newton's which is less than the force of gravity which it has to be because if the tension was greater than the force of gravity this five kilogram mass would accelerate up we know that doesn't happen the tensions got to be less than the force of gravity so that this five kilogram mass can accelerate downward so that's a quick way to solve for the magnitude of the acceleration of the system by treating it as a single object we're saying that if it's a single object or thought of as a single object which we can do because these are required to have the same acceleration or same magnitude of the acceleration then if we're treating it like a single object only external forces matter and those external forces that make the system go are going to accelerate the system and those external forces that resist the motion are trying to trying to reduce the acceleration and we divide by the total mass of the system that we're treating as one object we get the acceleration if that still seems like mathematical witchcraft if you're not sure about this whole idea I encourage you to go back and watch the video we solved one of these types of problems the hard way and you see you really do end up with the force that tries to make the system go externally and the external force that tries to stop it divided by the total mass gives you the acceleration essentially what we're saying is that these internal forces cancel if you're thinking of this system as one single object because these are applied internally and they're opposed to each other one tries to make the system go one tries to make the system stop