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check out this fine-looking sneaker right here we're going to use this shoe to illustrate some more challenging normal force problems and we're going to take this as an opportunity to discuss a lot of the misconceptions that people have about the normal force so one misconception is that people forget normal force is a contact force you only have a normal force when two surfaces are in contact so when the shoes in contact with the floor there'll be a normal force on the shoe and a normal force on the floor or if the shoe were in contact with the wall there'd be a normal force on the wall and a normal force on the shoe but if the shoe were just falling through the air here's what happens for a lot of people let's say the shoe just falling and you got a question and the question said draw the forces that are exerted on the shoe while it's falling through the air people get so used to having normal forces that they make a mistake they do this they say all right let me draw it over here they say there's a gravitational force and that's just fine there will be a gravitational force there's always gravity the Earth's always pulling down and it pulls down with an amount mg but they're so used to having normal forces I mean normal forces pop up in so many different questions it's almost just like a reflex people just automatically put what there's a normal force there's got to be a normal force right there's always a normal force but there isn't always a normal force if the shoe is not in contact with the surface you don't have a normal force it's not until this shoe makes it to the ground or touches another surface that you'll have that normal force so if we stick the shoe right here and we let it rest on the ground now you'll have a normal force and that normal force will point up and this is what people want to say and it's true when the surfaces are in contact but if they're not in contact you don't have a normal force and then here's another misconception people think the normal force is always equal to mg because again it's equal to mg in so many different scenarios that people just want to say well it's always equal to mg and again it's just like a reaction people see normal force they just automatically replace it with mg and that'll be true in this simple case but I'll show you coming up how that's not going to be true and what you do if it's not true so for instance if we wanted to find what's the normal force if this shoe has a mass M let's assume the shoe has a mass M what would the normal force be we can use Newton's second law we can always use new second law so we'll say that acceleration equals the net force divided by the mass and in this case since these are vertical forces I'm going to consider the acceleration in the vertical direction and the net force in the vertical direction and so what is the acceleration for the shoe vertically if it's just sitting here in a room sitting on the ground at rest and not changing its motion not changing its velocity the acceleration is just going to be zero so the vertical acceleration acceleration excuse me should be zero for the net force I've got an upward normal force so I'm gonna make that positive if FN represents the magnitude of the normal force this would be positive FN I'm just going to put a positive here even though I don't really need it but to show you that it's upward we're going to consider upward to be positive and then I've got this downward gravitational force and if mg represents the size of the gravitational force I'm going to put a negative here to represent that that gravitational force is down and then I divide by the mass of the shoe and if I do this I get that these two forces this net force divided by the mass has to be zero according to Newton's second law but I can multiply both sides by the mass and if I do that the left hand side is still zero now get that this is equal to the normal force minus mg so I'll have normal force minus mg and if I finally solve for the normal force I'll get that the normal force is going to equal mg and a lot of people are like yeah I already knew that duh normal force is always equal to mg but it's only equal to mg in this case because those were the only two forces look at the assumptions we made only two forces were the normal force and the gravitational force and we assumed that the acceleration was zero if you relax any of those requirements normal force is no longer going to be equal to mg and it was on a horizontal surface if you relax that requirement again there's no reason to think this has to be in the Y direction you could have normal forces in the X direction so let's slowly one point at a time try to relax some of these requirements and see what that does to the normal force in other words what if we just added another force what if we let the shoe sit here on the ground and I push in down on it so I'm pushing down on this shoe I'm going to say I'm pushing down with a force I'll just call it f1 so us a force of magnitude f1 and it's pointing downward how would that change this now so this is we're stepping it up this is going to be a little harder what do we do well the acceleration is still zero let's say it's still just sitting there so we don't have to do anything with the left hand side that's still zero multiplying by M still makes that zero but now up here in this force up here I'm going to have another force I'm going to have f1 that points down so in my force diagram I'd have another force that points down f1 that means I'd have to subtract it when I find the net vertical force I'd have f1 this would be a negative f1 right here and when I solve for FN I'd add mg to both sides to cancel it and then I'd add F one to both sides to cancel this f1 this negative f1 and I'd get mg plus f1 so I get the normal force is going to be bigger bigger by an amount f2 and that makes sense if you push down on oh not f2 Wow f1 sorry about that it's going to be bigger by an amount f1 so if I push down with an extra 10 Newton's of force there's more pressure right that makes sense the pressure between the ground and the shoe is going to be greater you're squashing these two surfaces together with greater force so the ground is got to push up to keep the shoe out of the surface that's what this normal force does it exerts a force to keep the object out of the surface to keep the object from penetrating that surface so if I push down on an object into a surface that normal force increases and it increases by the amount you're pushing down so that makes sense if you had an upward force let's say you had an upward force someone's pulling up on the shoe while you push down you're fighting over the shoe you're wrestling over it with somebody because they just they love the shoe they recognize the beauty this shoe well-crafted shoe so if there's an f2 pointing up we now have another force on our diagram that force would point up we'd call this f2 over here how would this change again still acceleration is zero but I'd have an upward force now so I'd have to add f2 vertically because that's another force I'd have a plus f2 right here and then over here when I solve for this I said I add mg to both sides f12 both sides and I have to subtract F 2 from both sides so now I'd have FN is mg plus F 1 minus F 2 this also makes sense if you pull up on a shoe you're relieving some of the pressure between the shoe and the other surface the shoe and the floor so if I pull up with 20 Newtons I'm going to reduce the normal force by 20 Newtons because I'm relieving some of that pressure between the shoe and the floor let's make it even harder let's make this thing scary sometimes you get really crazy problems you don't know what to do let's say we have another force let's say this force is going to be a diagonal force so we're going to pull this way oh that was not a well drawn force let me draw it like this so we got a force this way at an angle now how now we're talking this way this is f3 f3 at an angle of we'll call it Phi so the angle from this horizontal line here is Phi now what do we do so I've got this crooked angle in here now this f3 is going to be pointing this way so I'll add another force to my force diagram and I can figure out how to include this into my vertical force version of Newton's second law I can include the entire f3 force here's a mistake people make they want to just add f3 or subtract f3 but I can't do that this is the vertical form of Newton's second law this is only applying to the vertical direction the Y direction but f3 is pointing both vertically and horizontally so I have to only include the vertical part of f3 in this formula so what I have to do is say that alright f3 is going to have a vertical component that vertical component I'll call it F 3 y 4 F 3 in the vertical direction and it's also going to have a horizontal component I'll just call that F 3 X 4 F 3 in the horizontal direction so if we want to solve for F 3 Y I'll just use the definition of sine and I know to use sign because this side is the opposite to this angle I know sine relates opposite side so I'm going to write this as sine of Phi is going to equal the opposite side is f 3 Y so F 3 in the Y Direction divided by three total the total magnitude of f3 and if I solve this for f3 why I get f3 and the y direction is going to equal F 3 times sine of Phi now I can include this in my force my net force because this points upward so since it points up this vertical component it's going to add a plus f3 sine theta over here or sorry not theta it's going to be F 3 sine Phi and I'll have a plus F 3 sine Phi right here and when we subtract this F 3 sine Phi from the other side to get it over to here we're going to get minus F 3 sine Phi and that makes sense because if we pull up we know we're reducing some of the pressure will reduce the normal force so this component look at this component points up it's reducing some of the pressure on the bottom of the shoe the normal force goes down and so we subtract the F 3 sine Phi and one more way to step this problem up to the next level would be to say that this room isn't really just a room maybe it's an elevator and this elevator is accelerating upward with some acceleration a zero in that case nothing would change on this right hand side sometimes people think if there's acceleration there's going to be some new force but if these are the forces those are the forces the only thing that changes over here if this was an elevator that's accelerating up is it instead of zero you'd replace this with a zero or whatever the acceleration is that's it that's the only change you can still solve for FM the same way when you multiply by M it wouldn't be zero on the left anymore you'd have an M a zero and then a plus M a zero when you solve for the normal force alright I think we've pretty much exhausted this example of a shoe on the floor that's probably harder than any example you'll see but now you know how to handle any type of force you might meet or acceleration