 Hey guys, welcome to tutor terrific today. I'm going to be doing a problem-solving video in physics This is a video about force problems on inclines this is a commonly difficult problem for those of you who studying Newtonian dynamics and This one's going to involve friction to to make it extra saucy and complicated so The problem at hand requires you and this video really requires you to have a good facility in kinematics which involves these equations over here, which we'll be using today and in some basic Newtonian dynamics such as the sum of the forces equals mass times acceleration and Kinetic friction is an equation Coefficient of kinetic friction and the normal force multiplied together So you need to have a basic facility in these things to really understand or make use of this But if you do, this is the problem for you. So today we have a bar of soap that is accelerating down an incline that is 8.0 degrees and the length of the incline is 8 meters and It also gives you the problem the coefficient of kinetic friction very small 0.05 Zero between the soap and the ramp So there's some friction there. We can't ignore all friction But we are asked how long will it take the soap bar to traverse this entire ramp eight meters long? given that information so this is quite a lengthy problem, but The first step is always to draw a picture which we already have thankfully everything is already labeled for us that we know and After that if you already have the picture or you've drawn your picture based on the verbiage of the problem Your next step is to draw a free-body diagram a free-body diagram For the soap bar now that is a map which shows all the forces that are acting on The object not forces that the object imparts on other things But the force is imparted on the soap bar itself So I'm gonna draw the soap bar right here Let's talk about the forces acting on the soap Well, the soap has mass we weren't given mass which might scare some of you But we don't need it, but it does have mass and so it has a force of gravity Because it has weight That'll point straight down Okay, there's the force of gravity on the soap The soap is resting on a surface And if that's true then there's a normal force which is perpendicular to the surface itself So that would be at this eight-degree angle Like this. That's the normal force Okay In addition to these two forces We were told that there's a coefficient of kinetic friction Which means there's friction in the kinetic sense and that always opposes motion So if the direction of velocity as this soap bar accelerates is down the hill parallel to the ramp The force of friction will be pointing parallel to the ramp, but anti-parallel to that velocity so backwards So like this at a right angle to the normal force like so Alright, so this Designates a somewhat of a problem for us when we're analyzing this situation Because we have two forces at an angle. It is much easier to reduce the number of forces at an angle if you can And that is done by rotating the axes. Currently the axes are like this vertical straight up and horizontal It's straight left and right. Well, that means we have two forces at an angle. If we were to rotate the axes such that the y-axis pointed directly parallel to the normal force and the x-axis pointed Horizontal to that 90 degrees to that we have just one Vector at an angle that would be gravity. So what we're going to do is we're going to redraw this picture Where the axes themselves are rotated So it says the normal force points directly upwards Okay, so I'm drawing it like this and you'll see why in a second the friction force will point this way Okay, directly horizontal now and gravity will point at an 8th degree east of south direction Like this there's the force of gravity, okay After you've rotated your axes to the desired location You need to now resolve any vector at an angle still into components only gravity is at an angle And so I'm gonna go ahead and Resolve gravity into components and its components will be these dashed lines Okay, so show their components of an existing vector the horizontal component We will call f gx and the vertical component. We will call f g y the angle Which is right here using geometry is actually congruent to this eight degree angle here. So this is 8.0 Degrees, I'm not going to show that right now. If you do enough of these problems You'll realize that that is going to be the angle Between gravity vector and vertical is the same after you rotate the axes as the vertical of the incline originally now Something to keep in mind that I didn't already mention is the value of the acceleration due to gravity lowercase g That is 9.8 meters per second squared we are going to be using that in Our I'll put an extra zero on there. We're going to be using that in this problem Okay, now Why did we do this? Well, remember we need to find time. Okay, we're gonna have to since this is a force problem on inclines Get to time by using Newton's second law The force sum equals mass times acceleration We can actually split this up into two equations one for the y direction and one for the x direction and Everything is in components except for the mass So I could set up an equation for the sum of forces in the y direction I'm going to leave some space here, and I could set up a sum of forces in the x direction So what you literally do for each of these equations is you add up the vectors in those particular directions whether Up or down for y or right or left for x. So let's do it for a sum of forces y I usually set this one up first because it's a little simpler than the other one. There's less going on in this case That would be true So let's look at f y we need to determine a positive Direction before we can continue with this analysis, and I always choose when we are going down an incline I choose downward as Positive okay, but you could also set upwards as positive if you like it doesn't matter So if downward is positive that means f g y the vertical component of gravity which points straight down is a positive vector That I'm going to add to f n now normal force if positive is downward will be negative So I'm going to actually subtract its value This will equal ma, but here's the thing is there acceleration in the y direction Well, that would only be true remember. We're rotating our axes. So this is the y direction now It would only be true if the soap Is levitating off the ramp which it's not or if it's falling through the ramp which it's not So that means ma is m times zero or zero Okay, let's continue our analysis. I'm going to need an expression for the normal force in this problem So I'm going to separate the normal force right now F g y excuse me Not x f g y will equal f n when I add it to the other side Okay, now let's look at f g y itself If you look closely you notice that f g y is trigonometrically related to f g If you make this a right angle here, you realize that that's the same as f g x We've got a little right triangle Okay, f g y is the adjacent side to the angle and F g would be the hypotenuse. So you might notice a cosine relationship between them That's because f g y is equal to f g times the cosine of eight degrees. That is very true So we can write f g cosine 8.0 degrees Equals f n Now I want you to remember something from physics in addition to this the force of gravity Is always usually replaced with mass times g a 9.8 meters per second squared you can always replace f g with mg as we say so back here I'm going to replace f g with the expression m G so now I have mg cosine 8.0 degrees equals the normal force. I will be using this very soon Now let's move on to the sum of forces in the x direction okay This expression has two forces as well f g x and f k, but we'll do a little more Substitution than we did in the f y sum equation So we need to choose a positive direction. I'm going to choose the direction the soap is traveling which is to the right as positive Okay, and remember it's a little bit downward, but we've rotated our axis from here to here So now this is to the right in our rotated axis diagram So f g x is positive and f k then would be negative because it points in the wrong direction quote-unquote So the sum of the forces would be f g x plus negative f k Now is there acceleration in the x direction you betcha This f g x is a little bit bigger than the friction because this soaps Accelerates down the hill in order to get to the other end at a certain time The ray we also know that it accelerates is because it is set from rest at the top of the hill and allowed to slide under its own quote-unquote power So that means there's an acceleration So the sum of these forces equals mass times acceleration because acceleration is not zero Realized though that this is a horizontal acceleration. So we put a little x here just to remind ourselves This is horizontal acceleration only Okay, now it's time to do some substitutions The reason I'm doing all of these is because I don't know what f k is I don't know what f g x is and I don't know what the acceleration is and I don't even know the mass. I'm not even given that But what's the goal? We have to remember the goal. The goal is to get time. Okay? Now as you may have guessed and you'd be right none of these force equations have time in them none of them So we have to have a go-between So we can get back to the kinematic equations up here known as the big three which have time in them And it looks like according to those equations. We need a The acceleration and we can get that Through these force equations the acceleration is like the liaison or the go-between Between the force equations and the kinematics equations. So we have to find this a x. There's no Vertical acceleration. So all we need is a x So that's why we're going to do some substitution now F g x. Let's look at that f g x is here also here in our Triangle analysis this geometric analysis notice how it's opposite the eight degree angle And so we know that sine is opposite over hypotenuse. So f g x will equal f g sine 8.0 degrees That's the force of gravity in the x direction Next we have the force of friction luckily, I have this nice expression for the force of friction in terms of some things that I Could know I'm definitely no mu k 0.05 And I have an expression for the normal force Hopefully I can use that. Well, I'm going to substitute those in So the force of friction is excuse me. We're going to subtract force of friction, which is mu k Times f n. I'm going to use this expression right here for f n M g cosine 8.0 degrees That equals m a x Okay, almost done with the substitutions. Now remember what I said about f g Even if there's this sign next to it, you can still replace the f g with mass times gravity m g So I'm going to write that m g sign 8.0 degrees minus mu k M g cosine 8.0 degrees equals m a x. Oh my goodness Looks like I made this worse rather than better, but that's not true. Remember how I told you we didn't have mass Well, we don't need it. That's because if you look at all of these terms There's a mass in each and every one. So that means it cancels So you can divide out the m's from every term now all that's left is things that we know So what I'm going to do is I'm going to move things up a little bit I'm going to continue the problem So now the next step is to write that same expression without the m's in it. So we have g sign of 8.0 degrees Minus mu k g cosine 8.0 degrees And that equals The acceleration we're after Okay, now remember that g equals 9.8 Minus per second squared and that we learned earlier in physics and we also know mu k Mu k the coefficient of kinetic friction. That's 0.05 So we're going to plug this all in and get our expression for a of x Your calculator you want to make sure that you are first in the proper mode, which should be degree mode Because we're working with degrees. That's how you do it on the ti 83 or 84 calculators check out one of my calculator tutorial videos for more information then We're gonna start plugging this in so this is the first term in 9.8 times the sign of 8.0 degrees close those parentheses. That's one term that we're going to subtract from that A larger product of things first the coefficient of kinetic friction 0.05 times 9.8 times the cosine of 8.0 degrees close those parentheses and That should equal our acceleration. Let's see Aha 0.87 much less acceleration than due to gravity because that's a very Very flat angle very low angle and so that is our acceleration now Let's keep a lot of the digits because we don't want to round too many times and we're not had we're not at our final answer yet So I'm going to write the following down for a x Write down about four or five actually six significant figures point eight seven eight six six five meters per second squared So now I'm going to make room and we're gonna move on and figure out which kinematic equations We will need to use to figure out how long it takes to get down the ramp So I've made some room now and I've written down our Solved for acceleration partially rounded point eight seven eight six six five meters per second squared now I need to figure out what set or single Kinematics equation. I'm gonna have to use to get to time from here. That's my goal remember is to get to time Now realize also that this is the only acceleration. It's in one dimension. These are one dimensional kinematic equations So I don't really need the x near the a any more In order to figure out which one of these and I only have to use one of these kinematic equations I need I need to write down what else I know and I know a few things. I actually know the initial position The initial velocity Okay, that's really helpful the initial position since I know the entire length I'm traveling can be set to zero zero meters. That's really helpful In addition, I know that I'm starting the soap from the top at rest and that means the initial velocity is zero as well This is going to help me a ton Determine which equation is best to use to figure out the total time Now if you look at the first one that has time in it It's really nice, but the problem is I don't know the final velocity that the soap has at the end of the ramp If I did I'd be able to use that very nicely The third equation here does not have time in it However, I could use it to solve for V and then plug V into here and then get time However, that's two equations when I only really need one the middle one. That's right I can use the middle equation which looks scary, especially when it comes to T. However, it's not so bad Let me show you I'm gonna write that here again Let's talk about some things. I told you that the x naught is zero. So that means this first term is Zero we don't have to worry about it. I also told you that The initial velocity is zero So that means this term is also zero So now we don't have to worry about this being a nasty quadratic equation. It's very simple We just have x equals one half a T squared awesome Well now I just have to solve for T Now if with you using your algebra skills, we could see that this one half I'm gonna multiply both sides by two Then I got a divide by a and then I just have this expression equals T squared Which would require us to square root both sides. So that means T equals the square root of 2x over a Okay, so now I have to do is plug in the final x value, which is eight meters and the a which we derived Earlier, so we have two times 8.0 meters And that's all divided by 0.8 7 8 6 6 5 meters per second squared I always want to check and make sure that my units work out and Meters cancel and the second squared are on the bottom of the bottom. So they'd come to the top square root them. I get seconds perfect now when I Calculate this time in my calculator. I get 4.26 and lots more It's quite a long decimal actually But let's check out the significant figures that I'm allowed to have if you come back up to the top You can see all of my initial measurements have two significant figures since there's plenty of adding and multiplying Dividing we're gonna have to go by the accounting rule Which is just you can only have as many sig figs no matter where they are as your least accurate measurement And so two sig figs is what I'm limited to because of all my measurements being two sig figs So I have to round this guy and he's gonna round upwards. He's gonna round up to 4.3 seconds Perfect, this is our time Now that's an 8 meter ramp. So 4.3 seconds seems kind of reasonable for a rather moderate acceleration Awesome guys. Thanks for watching appreciate your views and your comments and your likes. So keep it up This is Falconator signing out