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Video transcript

let's go let's say I have some type of a block here and let's say this block has a mass of M so the mass of this block is equal to M and it's sitting on this you could view this as an inclined plane or a ramp or some type of wedge and we want to think about what might happen to this block and we'll start thinking about the different forces that might keep it in place or not keep it in place in all of the rest so the one thing we do know is this if if this whole set up is near the surface of the earth and we will assume that it is for this for the sake of this video that there will be the force of gravity trying to bring trying to bring or attract this mass towards the center of the earth and vice versa at the center of the earth towards this mass so we're going to have some force of gravity let me start right at the center of this mass right over here and so you're going to have the force of gravity the force due to gravity is going to be equal to the gravitational field near the surface of the earth and so we'll call that G we'll call that G times the mass let me just write it the mass times the gravitational field near the surface of the earth and it's going to be downwards we know that or at least towards the surface of the earth now what else is going to be happening here well it gets a little bit confusing because you can't really say that you know it that you can do can't say the normal force is acting directly against this force right over here because remember the normal force acts perpendicular to a surface so over here the surface is not perpendicular to the force of gravity so we have to think about it a little bit differently than we do when we if this was sitting on level ground well the one thing that we can do and frankly that we should do is maybe we can break up this force the force due to gravity we can break it up into components that are either perpendicular to the surface or that are parallel to the surface and then we can use those to figure out what's likely to happen and what are the potentially netting forces or balancing forces over here so let's see if we can do that let's see if we can break this force vector the force due to gravity into a component that is perpendicular that is perpendicular to the surface of this ramp and also another component that is parallel to the surface of this ramp let me do that in a different color that is parallel to the surface of this ramp and I'll use a little so this is a little bit unconventional notation but I'll call this one over here I'll call this the force due to gravity that is perpendicular to the RAM that little upside-down T I'm saying that's perpendicular because it shows a line that's perpendicular to I guess this bottom line this horizontal line over there and this blue thing over here I'm going to call this the force the or the part of force due to gravity that is parallel that is parallel I'm just doing those two upward vertical bars to social something that is parallel is parallel to the surface so this is the component of force due to gravity this perpendicular component of force that is parallel so let's see if we can use a little bit of geometry and trigonometry given that this this this wedge is that a is that a theta degree third theta degree incline relative to the horizontal if you were to measure this angle right over here you would get theta so in future videos we'll make it more concrete like 30 degrees or 45 degrees or whatever but let's just keep it general if this is Theta let's figure out what these components of this of the gravitational force are going to be well we can break out our geometry over here this I'm assuming is a right angle and so if this is a right angle we know that the sum of a of the angles in a triangle add up to 180 so this angle if this angle this angle and this 90 degrees right angle says 90 degrees add up to 180 then that means that this one and this one need to add up to 90 degrees or if this is Theta this angle right over here this angle right over here is going to be 90 minus theta now the other thing that you may or may not remember from geometry classes if I have two parallel lines if I have two parallel lines here if I have two parallel and then I have a transfer so I'm going to assume this line is parallel to this line and then I have a transversal so let's say I have a line that goes like this we know from basic geometry that this angle is going to be equal to this angle it comes from alternate interior angles and we go and we prove it in the geometry module or in the geometry videos but hopefully this makes a little bit of intuitive sense and you could even think about how these angles would change is the transversal changes in all of the rest but the parallel lines makes this angle similar to that angle or actually makes it identical makes it congruent this angle is going to be the same measure as that angle so can we apply that anywhere over here well we have and all that's not obvious this line is perpendicular to the surface of the earth right over here that I'm kind of shading in blue and so is so is this force vector it is also perpendicular to the surface of the earth so this line over here and this line over here in magenta are going to be parallel I can even draw that that line and that line are both parallel when you look at it that way you'll see that this big line over here can be viewed as a transversal or you could have these this angle and this angle are going to be congruent they're going to be alternate interior angles so this angle and this angle by the exact same idea here it just looks a little bit more confusing here because I have all sorts of things but this line and this line are parallel you can view this right over here as a transversal so this and this are congruent angles so this is 90 minus theta degrees this two will be 90 minus theta degrees 90 minus theta degrees now given that can we figure out this angle well one thing we're assuming that this yellow force vector right here we're assuming that it is perpendicular to the surface of this plane or perpendicular to the surface of this ramp so that's perpendicular this right here is 90 minus theta so what is this angle up here going to be equal to this angle let me do it in green what is this angle up here going to be equal to so this angle plus 90 minus theta 90 must be equal to 180 or this angle plus 90 minus theta must be equal to let me write this down I don't want to do too much in your head so let's call it X so X plus 90 minus theta plus Plus this 90 degrees right over here Plus this 90 degrees needs to be equal to 180 degrees we can let's see we can subtract 180 degrees from both sides so you subtract 90 twice you get your subtract 180 degrees you get X minus theta is equal to 0 or X is equal to theta so whatever the inclination of the plane is or of this ramp that is also going to be this angle right over here and the value to that is is that now we can use our basic trigonometry to figure out this component and this component of the force of gravity and to see that little bit clearer let me shift this force vector down over here the parallel component let me shift it over here and you can see the perpendicular component plus the parallel component is equal to the total force due to gravity and you should also see that this is a right triangle that I have set up over here this is parallel to the plane this is perpendicular to the plane and so we can use basic trigonometry to figure out the magnitudes of the perpendicular force due to gravity and the parallel force due to gravity so the magnitude let's think about a little bit the magnitude of the I'll do it over here the magnitude of the perpendicular force due to gravity or the compare I say the component of gravity that is perpendicular to the ramp the magnitude of that vector a lot of fancy notation but it's really just the length of this of this vector right over here so the magnitude of this over the hypotenuse over the hypotenuse of this right triangle well what is the hypotenuse of this right triangle well it's going to be the magnitude of the total of the total of the total of the total a gravitational force I guess you could say that and so you could say that is that is M that is mg we could write it we could write it like this but that's really the well I could write it like that and so this is going to be equal to what we have the if we're looking at this angle right here we have the adjacent over the hypotenuse remember so let me just in a new color induce in a new color so a Toa cosine is adjacent over hypotenuse so this is equal to cosine of the angle so cosine of theta is equal to adjacent over the hypotenuse so if you multiply both sides by the magnitude of the hypotenuse you get the component of our vector that is perpendicular to the surface of the plane is equal to the magnitude of our four of our of our gravity or the force due to gravity the magnitude of the force due to gravity times the cosine of theta times the cosine of theta we'll apply this in the next video just so you can make the numbers a little more concrete sometimes just the notation makes it confusing you'll see it's really actually pretty straightforward and then this second thing we could use the same logic we have if we think about the parallel vector right over here the magnitude the magnitude of the force of the component of the force due to gravity that is parallel to the plane over the magnitude of the force due to gravity which is the magnitude of mg that is going to be equal to what well we have this is the opposite side to the angle and then so the blue stuff is the opposite side or at least its length is the opposite side of the angle and then right over here this magnitude of M G that is the hypotenuse so you have the opposite over the hypotenuse opposite over hypotenuse sine of an angle is opposite over hypotenuse so this is going to be equal to the sine of theta this is equal to the sine of theta or you multiply both sides times the magnitude of the of the force due to gravity and you get the component of the force due to gravity that is parallel that is parallel to to the ramp is going to be it is going to be the is going to be the force due to gravity total times cosine sorry times sine of theta times sine of theta so if you have these and hopefully you should see where this came from because you could all if you ever have to derive this again when you're you know thirty years after you took calculus or physics class you should be able to do it but if you know if you know this right here if you know this right here and this right here we can all of a sudden start breaking down the forces into things that are useful to us because we could say hey look this isn't moving down into this plane so maybe there's some normal force that's completely netting it out in this example and maybe if there's nothing to keep it up if there's no friction maybe this thing will start accelerating due to the parallel force and we'll think a lot more about that and if you ever forget these think about them intuitively you don't have to go through this whole parallel line and transversal and all of that if this is if this if this angle went down to zero if this angle went down to zero then we'll be talking about essentially a flat surface there is no inclination there and this angle goes down to zero then all of the four should be acting perpendicular to the surface of the plane so if this going to 0 if the perpendicular force should be the same thing as the graph of the total gravitational force and that's why it's cosine of theta because cosine of zero right now is 1 and so these would equal each other and if this equaled zero then the parallel component of gravity should go to zero because gravity will only be acting downwards and once again if sine of theta is zero so the force of gravity this parallel will go to zero so if you ever forget just do that little intuitive thought process and you'll remember which one is sine in which one is cosine
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