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if you're face-to-face with a sophisticated Newton's second law problem you're going to need a sophisticated understanding of Newton's second law and that's what I'm going to try to provide you with here so that no matter what scenario you're faced with you can apply this law in a correct way most people know Newton's second law is F equals MA which is fine it's a simple way to understand it and it's fine for simple problems if I had an asteroid for instance of mass M out in outer space so there's no air resistance or friction and there was only one force on it a force F and that force pointed to the right let's say that force was 50 Newtons well I could plug the 50 Newtons into the force I could plug the mass of the asteroid let's just say it's 10 kilograms into the mass and I'd find the acceleration of the asteroid in this case 50 over 10 would give me 5 meters per second squared but what if we had extra forces on this asteroid what if there was another force that pointed to the left that was 30 Newtons so let's call let's name these now let's call this F 1 is 50 Newtons let's say that's the magnitude of that force let's say F 2 was the magnitude of the 30 Newton force it points to the left yes that's the negative direction but let's just say these forces here just giving the magnitude of it and then the direction is specified by the direction of the arrow now what would I do well to handle this we need to understand that the left-hand side here isn't just force it's the net force or you can call it the sum of the forces so denote to denote the net force we often write this Greek letter Sigma and Sigma is a mathematical symbol that represents the sum of whatever comes after it so this is the sum of the forces because F comes afterward if I had G it'd be the sum of the g's and if I had eight should be the sum of the h's and this is a little confusing already people here some of phonetically and they think oh sometimes they're like oh so sum of like a few of no no no we mean all of all of the forces that's what this means you add up all of the forces that will equal the mass times acceleration so in this case we take this 50 Newtons I could take 50 Newtons because it goes to the right and I mean we can call leftward positive if we really 1 & 2 if there was a good reason but unless otherwise specified we're going to just choose rightward as positive and upward is positive so this 50 has to be positive and I can't now buy some of I add up the forces but I have to add them up like vectors this force here is a vector forces or vectors and so I have to add them up as vectors this is a vector equation I can't just take 50 plus 30 to get the answer because vectors the point to the left we're going to consider negative vectors the point to the right we'll consider positive and so I'll take 50 Newtons minus 30 Newtons that's what's going to be equal to the mass times the acceleration so I could plug in 10 kilograms if I wanted to multiply by a and in this case I'd get 20 over 10 is 2 meters per second squared so you have to add these up like vectors and if I had more forces it'd be just as easy to deal with I can just add them up as vectors so if I had another force here that was maybe that's maybe 25 Newtons we'll call that F 3 and let's say there's another force that points to the left this one's going to be 40 Newtons so we'll have 40 Newtons to the left we'll call this one F 4 well I can just keep including these I can just add them up as vectors the 40 Newtons points to the left so that's got to be a negative I'll put negative 40 Newtons and then this 25 points to the right I'll make that positive so positive 25 Newtons and I can find my total force my net force my sum of the forces and that would allow me to figure out the acceleration so there's one problem here though a lot of times physicists don't like writing this anymore at least physicists that are interested in education don't like this form of Newton's second law as much a lot of them don't and the reason is there's a misconception students have they think that as they're drawing forces here that M times a is also a force they want to draw an extra force on this asteroid maybe it points to the right that is mass times acceleration but mass times acceleration is not a force mass times acceleration is just what the net force happens to equal so if you add up all the net forces or sorry if you add up the net force on an object which is adding up as all the forces on the object that will just equal MA it happens to equal MA but it is MA is not a force in and of itself so you you cannot draw this as a force don't draw any of that that is not a force that's just what the sum of the forces happen to equal so upon realizing this physicists were like okay this is causing confusion so let's just write an equally good form a Newton's second law in terms of algebra but that also makes it so that people aren't so susceptible to falling into this misconception this alternative version of Newton's 2nd law looks like this the acceleration equals the net force divided by the mass you might be like off so what we just divided both sides by mass who really cares well here's why it's better because people are much less likely to think that acceleration itself is a force they're much less likely to say Oh acceleration is a force over here I mean people might do that but they're less likely so acceleration is not a force it's a vector but it's not a force vector and the other reason why this equation is nice it shows us the relational dependence of acceleration it shows us that the net force is what will give us acceleration and the more net force we have the bigger the accelerations so it shows us that the acceleration is proportional to the net force or the sum of the forces and it shows us that the acceleration is inversely proportional to the mass so the bigger the mass the less acceleration you have so it's another reason this equations nice shows us what acceleration actually depends on in terms of net force and mass so there's one other problem here so this is better this is already better now we know it's net force now we can write it this way and not fall into the misconception that MA is a force there's a problem now what if I introduced another force over here let's say I introduced the force that points downward let's say this force was 28 Newtons and this is fo well we did f4 this is f5 so we have f5 28 Newtons downward you might think oh I know how to deal with this now can't I just sneak this 28 in over here and write it as a negative 28 because it goes down can't I just put neg twenty-eight right there and it turns out you cannot do that that is not allowed and the reason is just like we couldn't take fifty plus thirty because we're adding these up as vectors and the leftward negative and rightward net positive we can't take a vertical force and just add that magnitude or subtract it from the magnitude of the horizontal force that's not allowed if a horizontal force and a vertical force added up will not equal the sum of or the difference of the magnitudes what I'm saying is this think about it this way if you had a certain amount of force to the right and a certain amount of force upward to add these up it would not equal this value plus this value you'd have to add them up with the Pythagorean theorem to add vectors this way you'd get this vector right here that would be your total vector so you'd have to do a squared plus B squared equals of the total four squared over there between those two vectors and you might be thinking oh great I don't want to have to do trigonometry here and it turns out you don't have to not yet at least if these are only forces if these are the only forces we have we don't have to do it this way I'm just trying to show you that you cannot simply naively add up 50 minus 28 and expect to get the total answer right but here's which good do you can take only horizontal forces deal with those in the horizontal direction first and only vertical forces and deal those deal with those in a vertical directions was the same trick we always play as physicist we say all right we're going to divide and conquer we're going to take all horizontal forces and put those into their own equation because of the horizontal forces should only affect the horizontal acceleration so if I just want a horizontal acceleration I can take only horizontal forces add those up and get the horizontal acceleration or I could take only vertical forces add those up and I get the vertical acceleration so if I take this I'll make an equation for each direction independently and I know I can find each component of the acceleration by just using the forces in that particular direction so this is a nice trick we're going to deal with each direction independently and then if we really wanted say say these were acceleration actors say we wanted the total acceleration we were talking about forces before but all vectors add up the same way you can use the Pythagorean theorem if you figure out a in the x-direction the total acceleration in the x-direction and you figure out the total acceleration in the y-direction you can figure out the total acceleration the magnitude of it by again using the Pythagorean theorem and so this is just a way to deal with this is a handy way of dealing with these forces that point in multiple directions let me stick one more force in here just because we should have one that points up and then we have all possible directions here all right so if this force F what are we on six is going to be maybe that's about 42 it's a good number Newton's upward how do we deal with this well we already dealt with the x-direction this was the x-direction this gives us all the forces in the x-direction that equals mass times acceleration in the X it's not written in this form it's just multiplied with the ten on the right hand side instead of divided here but this is the form you could use to relate forces in the x-direction to the acceleration in the x-direction we're essentially just taking all of these forces plugging them into the net force and the x dividing by the mass and we'd find the acceleration in the X direction for the y-direction we could say that acceleration in the y-direction would be the net force in the y-direction so how do we deal with this only vertical forces are going to be affecting the vertical acceleration so I can take this F 6 which is 42 Newtons it points up we're going to treat that as positive because we don't have a good reason to treat down as positive 42 up and that's the convention we usually pick 42 minus 28 28 points downward we typically choose that as the negative direction now we can divide by the mass well divide by 10 kilograms this gives us our acceleration in the vertical direction and now that we have both of these we could if we wanted to do a x squared plus a y squared equals the total a squared to find the magnitude of the total acceleration all right let's step it up one more notch and see what happens let's make it one step harder we're going to move this out of the way I'm going to make a little room so let's say this 40 Newton's is still applied but I'm just going to put it here so it's out of the way let's say there was one more vector involved one more vector that points this way and let's say the size of that vector we'll call this one F seven let's say F seven is 45 Newtons okay 45 Newton's applied at an angle let's just say from that point of 30 degrees how do you deal with this this is an even more sophisticated Newton second law problem here's where it starts to frighten people they don't know what to do a lot of times they try to just throw this 45 Newtons into one of the equations they think the 45 maybe should go in the X equation because it points the left that's horizontal that's X but it also points vertical sometimes they throw the whole 45 into the vertical equation maybe they do plus 45 over here but that's wrong you can't do that and you can't do that because only vertical forces and components of forces can go in this vertical equation and only horizontal forces and horizontal components of forces can go into this horizontal equation so we have to do at this point I think you know what we have to do we have to break this up you have to take this 45 Newtons that points up and to the left we have to break this up and do how much of this force points left how much of this force points upward so we're going to figure out what is this component of this force that way in the x-direction what is this component in the vertical direction I'll call this F 7 in the Y direction and this component here would be F 7 in the X direction it's getting a little cluttered there sorry about that we have to figure out what these components are once we figure out what those are I can associate the F 7 X into the X equation the f7y into the Y equation but I can't put the whole 45 into either equation because all 45 Newton's is not directed vertically or horizontally only part of it is so we have to figure out with more trigonometry we're going to use the same rule or the same idea over here but instead of Pi instead of the Pythagorean theorem we're going to take this 45 Newton's and break it up into components and the way we do that is with the definition of sine and cosine so if this is I'm just going to make it bigger over here 45 Newton's if this side is 45 Newtons then this side would be the adjacent to this thirty degrees this f7 in the x-direction and this side would be the opposite to that thirty degrees so that's f7 in the y-direction now we use sine and cosine let's use the definition of cosine the definition of cosine theta is going to be the adjacent over the hypotenuse so the adjacent to this 30 degrees is the side touching that angle which is f7x so f7 in the x-direction over the hypotenuse is the total magnitude of this force vector which is 45 Newtons so I can solve now for f7 in the x-direction f7 and the x-direction is going to be 45 Newton's times cosine of 30 degrees now I can take this f7 in the x-direction i take take this whatever it is this is just a number you can calculate it if you want but i'm going to take this and i'm going to plug this straight into the x-direction I want to put it right into here as let's see should it be positive or negative here's where it's tricky the f7y points up the f7x points left this is the X component so the leftward that we care about the fact that it points up doesn't matter in terms of X but this component points left so we have to include it as a negative 45 Newton's times cosine of 30 and now for the y-direction we can use the definition of sine of theta sine of theta is f7 in the Y direction which is the opposite because sine is opposite over hypotenuse so in this case the opposite is f7 in the Y direction over the hypotenuse the hypotenuse is the total vector which is 45 Newton's the magnitude of the total vector so we get that F 7 in the Y Direction is going to be 45 Newton's times sine of 30 degrees so now I can take this this is the Y component I can plug this over into how am I going to get there without crossing a line I'm going to take this and plug it over here right into this Y directed equation it should be positive or negative the Y component points up so it's going to be plus 45 Newton's times sine of 30 degrees well okay so Newton's second law now you're equipped you know how to use it these forces might not be asteroid forces it forces maybe their tension or gravity or normal forces or frictions maybe there's forces in all different directions but these rules still apply no matter what the force whether it's up-down left-right or diagonal now you know how you can figure out how to use an insect law no matter what direction the force is pointed