For more information, please see full course syllabus of High School Physics

For more information, please see full course syllabus of High School Physics

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### Friction

- Friction changes depending on the two materials involved. Wood on rubber is different than wood on wood is different than wood on ice. (This idea is captured by our coefficient of friction: μ.)
- Friction changes depending on how hard the two materials are pushed together. (This idea is captured by the normal force between the two materials: F
_{N}.) - Friction changes depending whether or not the two materials are already in motion relative to each other-
*static*vs.*kinetic*. (This idea is captured by having two different coefficients of friction: μ_{s}and μ_{k}.) - Friction always
__opposes__motion. Whatever direction the object has (the direction of →v), friction points the opposite way. - The formula for friction is
F _{fric}= μ·F_{N}. - Kinetic friction is just a continual force of F
_{fric}= μ_{k}·F_{N}, pointing opposite whatever the current direction of movement is. - Static friction is a little different. It opposes the force on the object until it is overcome, at which point it switches to kinetic friction. It can cancel out other forces, but it never exceeds them.
*Maximum*Static Friction = μ_{s}·F_{N}. - As usual, be careful when figuring out where all the forces go. A good free-body diagram goes a long, long way. And be extra careful when figuring out the normal force!

### Friction

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro
- Introduction
- The Normal Force and Friction
- Friction as an Equation
- The Direction of Friction
- A Quick Example
- Static vs. Kinetic
- How to Use Static Friction
- Some Examples of Î¼s and Î¼k
- A Remark on Wheels
- Example 1: Calculating Î¼s and Î¼k
- Example 2: At What Angle Does the Block Begin to Slide?
- Example 3: A Block is Against a Wall, Sliding Down
- Example 4: Two Blocks Sitting Atop Each Other

- Intro 0:00
- Introduction 0:04
- Our Intuition - Materials
- Our Intuition - Weight
- Our Intuition - Normal Force
- The Normal Force and Friction 4:11
- Two Scenarios: Same Object, Same Surface, Different Orientations
- Friction is Not About Weight
- Friction as an Equation 7:23
- Summing Up Friction
- Friction as an Equation
- The Direction of Friction 10:33
- The Direction of Friction
- A Quick Example 11:16
- Which Block Will Accelerate Faster?
- Static vs. Kinetic 14:52
- Static vs. Kinetic
- Static and Kinetic Coefficient of Friction
- How to Use Static Friction 17:40
- How to Use Static Friction
- Some Examples of μs and μk 19:51
- Some Examples of μs and μk
- A Remark on Wheels 22:19
- A Remark on Wheels
- Example 1: Calculating μs and μk 28:02
- Example 2: At What Angle Does the Block Begin to Slide? 31:35
- Example 3: A Block is Against a Wall, Sliding Down 36:30
- Example 4: Two Blocks Sitting Atop Each Other 40:16

### High School Physics Online Course

### Transcription: Friction

*Hi, welcome back to educator.com, today we are going to be talking about friction.*0000

*At this point, you have got a really strong grasp on the basics of Mechanics.*0005

*Force = mass × acceleration, we have talked about it in two dimensions, you have got a really good idea of how Newton's laws work.*0009

*But so far, we had to pretend that friction does not exist, as if something that we could not really deal with.*0015

*But no more, now we are finally going to tackle friction.*0022

*You have got enough understanding about mechanics, you will be able to understand how to use friction in our work.*0024

*First, let us get a sense of how friction works in two dimensions.*0032

*Imagine you have got a plank of wood that you are pushing along at a constant speed.*0037

*Here is some floor, here is some plank of wood on that floor, and we are pushing it along at a constant speed.*0042

*First thing to notice, is that in real life, we are used to the idea that if we want something to move, (since everything experiences friction), you have to push on it if you want to keep a constant speed.*0047

*It is not going to have that constant speed unless you push on it, because friction is going to sap the energy out of it.*0057

*So, for the first time, we are saying that we need a constant force to keep that constant speed.*0064

*Up until now, if we had any force at all, we would have had an acceleration automatically, because we have been talking about being on a frictionless surface.*0071

*It would be a small acceleration, but we would have had some acceleration because we would have had some force, unless all the forces are cancelling out.*0079

*Now, we are going to have all the forces cancel out, because we have friction cancelling out the forces we are putting in, so we can have a constant velocity.*0086

*With that out of the way, we have got this plank moving along at a constant speed, because we are putting in some force into it.*0092

*Now, which would be easier to push, which would take less push, which would take less force for pushing on the plank?*0098

*The plank was on a floor that is made of wood, or the plank on a floor made of rubber, which one of these will stick together more, which one will have more friction?*0104

*Just like you would expect, the wood.*0113

*The wood is going to stick less, and the rubber is going to stick more.*0116

*If we want to make it easy for ourselves, we are going to want that wood floor.*0120

*What if we were to put the plank on a piece of ice?*0123

*It is going to make it even easier.*0127

*Different pairs of materials have different connections.*0128

*They behave differently with one another because of material science and chemistry and stuff that we are not going to really talk about, but friction is a pretty complicated idea that we will experience in lots of further courses and there is lots of cool interesting things to learn about it.*0132

*But in our case, we just know that, if we have different materials, we are going to have different frictions.*0146

*Different PAIRS of materials, that is an interesting thing to keep in mind, it is not just one material, it is the pair of it.*0151

*If we had a rubber plank on top of that rubber floor, we would have experienced even more friction.*0157

*An ice plank on top of an ice floor, it would have been the least of all.*0161

*It is the pair together that gives us the friction between them.*0165

*Let us talk about another thing for our intuition to deal with.*0170

*Imagine that, that same plank of wood is on a wood floor, but this time, we are going to put some sack of sand on top of it.*0172

*So, we have got some sack on top of it, and there is going to be some amount of sand, it is either going to have 10 kg of sand, or 20 kg of sand.*0180

*Which one is going to be easier to push along?*0191

*The 10 kg sack or the 20 kg sack, which one is going to have more friction reacting with that?*0193

*More friction force for us to overcome, the 10 kg sack push on the plank or the 20 kg sack push on the plank?*0201

*Which one would you expect?*0210

*It is just like you would expect, it is the 10 kg sack.*0212

*More pressure means more friction, harder we push on something, the more the friction that we have to overcome.*0214

*The lighter something is, the lesser friction that we have to overcome.*0222

*Assuming the same object, and the same material for the incline, which of the following three situations will be the easiest to push?*0227

*It is similar to the pressure idea, now we are going to start talking about the normal force.*0234

*In all of them, gravity ('g') is pointing straight down.*0240

*Which one of them would be easier, which one of them would you expect to?*0246

*Just like you would expect, the steepest incline.*0248

*Why is that?*0250

*We can explore that idea by looking at two extreme scenarios.*0252

*The exact same object and the exact same surface, but very different orientations.*0256

*One of them is a horizontal orientation, the other one is a vertical orientation.*0260

*Which one of these is going to have more friction going this way?*0264

*Here is our friction force, here is our friction force, which one is going to experience more?*0267

*The one that is sitting on it.*0274

*Why is that?*0276

*That is because, this one has 'mg' down here, so it has got the pressure (the normal force) pushing that amount.*0277

*How much does this one on the right, how much is the force normal?*0289

*We have got 'mg' down here, but there is nothing this way, so our normal force, F _{N} = 0, because there is no pressure, no interaction, nothing holding it against the wall to cause friction to happen.*0293

*If you push really hard on something, it is going to have more friction.*0312

*If you do not have any push between the things at all, there is no way for the materials to interact, there is no friction between them.*0315

*If we have no normal force, we have no friction.*0321

*If we have a lot of normal force, if we push really hard on it, we are going to have more friction.*0324

*If we were to instead, come along and push crazy hard on this, then we are going to have a resultant normal force that is equal and opposite, we are going to have this normal force because it is not going to blow through that wall, assuming the wall is able to withstand that much force, we might actually to able to arrest the power of gravity, arrest the acceleration due to gravity, the force due to gravity will be canceled out because we will be able to make a really large friction by pushing really hard.*0329

*You can test this out in a real quick demonstration.*0356

*If you take just a normal book, and you go up to a flat wall, and you just put the book up the wall, and you take your hand away, of course the book falls to the ground.*0359

*It is like you would expect.*0366

*If you were to put the book up against the wall, and push really hard with the flat of your hand, not under it, because then you would be holding it up, it would not be friction, it would be just direct force applied through your finger tips.*0370

*But instead, if you were to push really hard against it, you will be able to keep it in place, because you put so much pressure on it, the friction of the book against the wall is going to be able to overcome the pull of gravity.*0380

*It is going to beat out gravity, and it is just going to stay still.*0392

*Just like you would expect, from all this talking, friction is not just weight, it is about how hard the object is pushed, it is about the pressure between the object and the surface, the two materials, the interaction, it is the normal force.*0397

*For those of you having trouble with calculating the normal forces on inclines, I would recommend you to refer to the 'Newton's second law in multiple dimensions lecture', it will give a good explanation.*0410

*You need to calculate just how much of the gravity is perpendicular and parallel to the surface.*0429

*To sum up, friction is based on the interaction between the materials involved in it, and the normal force of the object on the surface.*0445

*What kind of materials do we have, how hard the pressure is, the two things, the normal force.*0451

*If you want to turn that into an equation, that's going to become the friction = μ × F _{N}...*0458

*μ is a Greek letter, and it is the coefficient of friction between the two materials, and it is spelled 'm-u', it will change depending on what the materials are, and it is going to vary a lot depending on specifics, and we have to determine it experimentally.*0471

*There is no easy formula for determining what it is going to be. You just have to go into a lab, get it, or look it up in a table. *0493

*Even in looking up a table, it is going to vary, because depending on the specific condition of the object, whether it is dirty, clean, if it is wet, if it has grease on it, if there is a layer of air, if it is operating in vacuum -- what things are happening between it, it is going to vary a lot. *0499

*So, it is basically up to you to figure it out in a lab, or to be able to look it up in a table where it has some very, very similar situations to the way you are doing it. *0515

*Or, it is given to you precisely in the problem statement.*0522

*So, figuring out μ can be a little difficult, but normally that's what the problems will be about, or it will be given to us in the problem.*0525

*Once again, going back to the equation, friction = μ, the coefficient that represents the interaction between the two materials, times the force, the normal force, fn. So μ × fn.*0532

*One thing to keep in mind, is that we do not have to worry about the area touching it.*0548

*If we had a block of mass 'm', and we had a table of mass, 'M', but the same material on the bottom.*0556

*Same material here, same material here, same surfaces, it is not about the cross-section, the area touching the ground, it is just about the pressure.*0567

*Why is that?*0575

*That has to do with the way friction works, it is what is happening on a really microscopic thing.*0577

*If we have a lot of area, the pressure per square area, the force per square area, is going to wind up being much smaller in the case when we have got that large surface.*0583

*So, same pressure, but it is going to be extended over a large area, whereas in the table example, where we have got just the little weak contacting, it is going to be the same pressure, but it is going to be over a small area, so the total effect is going to be the same, either a small force per area, but over large area, or a high force oer area, but over a small area, the total effect of the pressure is going to be the same.*0600

*So you do not have to worry about the cross-section, you just have to worry about the interaction between the materials.*0628

*One last thing: Friction is a force.*0635

*We know forces come in vectors, so what direction does friction come in?*0637

*It is not going to go in the direction of the normal fore, that is why our equation in our previous page was not in vectors.*0641

*Because it is upto us to figure out what direction friction is going to go in.*0647

*Friction always opposes the movement.*0651

*Whatever direction it is moving in, keep in mind that it is the velocity , not the acceleration, whatever direction it is currently moving in, it is the opposite direction that the friction is going to point.*0653

*Friction always is fighting current motion, so the velocity, whatever the direction of velocity is, the opposite of that direction, is the direction that our friction is going to move in.*0665

*So with this point, we have got a pretty good understanding of how force works.*0676

*We have got this interaction between μ and the normal force.*0680

*Let us consider these two diagrams here:*0682

*We have got, the block is the same in both diagrams, and the surface it is resting on is the same on both diagrams.*0685

*Let us assume that F _{1} and F_{2} are both big enough to move the block.*0692

*But also that, F _{1} and F_{2} are equal in magnitude, they are the same number of newtons.*0696

*If F _{1} and F_{2} have different orientations, but same magnitude, which block will accelerate faster?*0707

*If we break down our forces into components (we can do that since force is a vector), we look at the vertical amount in F _{1} and the horizontal amount in F_{1}, and over here, the vertical amount of F_{2}, and the horizontal amount of F_{2}.*0714

*We see that the thing that is actually do the motion here, is this right here, it is going to be the actual horizontal motion is going to stem from the horizontal component of our force.*0736

*If we were instead looking at what the normal force is now, we need to figure out what the normal force is going to be.*0752

*Both these cases, we still have gravity to contend with.*0759

*We have not dealt with gravity.*0761

*So there is the force of gravity, and over here, it is going to be the exact same force of gravity, so force of gravity on both of them.*0762

*How much does the normal force has to be to cancel these things out.*0769

*Before when we were talking about the force of gravity and the normal force, they were going to be equal to one another (in the horizontal case), because the only thing creating the normal force is gravity.*0774

*But in this case, if you push through an object, and the object does not blow through the table, then that means that the table has to resist both the object's force of gravity, and in addition, the force that you put into the object.*0789

*So, the table, the surface has to resist both the forces, that has been put into it by ourselves, by the problem, and the force that is put into it by gravity.*0803

*In the first case on the left, it is going to have to fight both gravity, and the amount of the force, the normal force is going to be F _{N} over here.*0813

*What about over here?*0824

*In this case, we have already got this component over here, is going to cancel out this component over here, so the normal force over here, is just going to be this little smidgen, down here.*0825

*In F _{2}'s case, we lift off some of the effective weight, what the normal force has to be is much smaller.*0836

*So which one of these is going to have a higher friction, this one is going to have a much smaller friction because it has got a much smaller normal force.*0845

*But over here, we have got this huge normal force in comparison, so we have got this giant friction.*0862

*We have got the same equal force horizontally, so we know that the giant friction is going to wind up sapping more of the acceleration and so, F _{2} is the more efficient, easier way, it is going to cause more acceleration.*0866

*F _{2} will accelerate the block faster, because it will have the smaller F_{N}.*0881

*So it is really important to pay attention to the interaction between the force of gravity, then also the forces that we are putting into our object.*0886

*One more thing to talk about, is the idea of, an object being still, at rest on a surface, and an object moving along on a surface.*0894

*Which one of these will take more effort, more force from us?*0902

*Just start a refrigerator moving, sliding on a floor, just start that refrigerator up, or keeping an already sliding refrigerator go away.*0907

*If we want to just, just start it moving up in addition to creating motion requiring some amount of force from us to get that started, there is actually going to be this little thing, if you have to sort of like, unstick it, we have to pop it off of where it was already located.*0915

*It might seem like a trick question, but it really is not, it really cannot take more force to start something moving than to just to fight kinetic friction.*0932

*Kinetic friction is going to be different from static friction.*0941

*The friction of when it is moving, is going to be different from the friction when it is still.*0944

*Why does this happen?*0948

*That is a really complicated thing, it is something for future classes in chemistry, more physics, friction is something there is still doing lots of research into, so it is really complicated for right now, but it is definitely something interesting, but we do not have time to talk about it right now.*0949

*The exact reason is lots of complicated, but it suffices to say that on a microscopic level, the two surfaces interact differently between one another.*0969

*They are going to wind up interacting in a different way when they are going to be still, and when they are already moving against one another, slight differences happening microscopically , and sometimes major differences as we will see in some of the numbers that we are going to see soon.*0976

*Static versus kinetic, if we are going to be able to talk about two different kinds of friction, kinetic- the moving kind, and static- the still kind, we are going to have to use a different coefficient for each one.*0993

*So, μ is now going to split into two different categories: static is going to be μ _{s}, kinetic is going to be μ_{k}.*1006

*So, we have got μ _{static} and μ_{kinetic}.*1017

*One thing to keep in mind: In almost all cases, μ _{s} is greater than μ_{k},*1020

*There are a very few special cases where this is not going to be true, but as far as we are going to deal with in our course, it is almost always true, sometimes they will be equal and there is really freaky materials where μ _{k} is larger, but it is beyond this course, it is not something we are going to have to worry about.*1029

*If you get really interested in material science, it might be the kind of thing you have to deal with in graduate school, but not something that you have to worry about in high school physics.*1052

*Applying kinetic friction is pretty easy.*1061

*If we just want to have friction on an object, it is just going to be, μ _{k} × F_{N}, until the object stops moving it is going to be in the direction opposing the current motion.*1064

*What about static friction?*1076

*That is a little bit different.*1078

*If we have an object sitting still, and we push on that object, we have got an object like this, and it is giant, and a guy comes up, and he pushes on it, lightly.*1080

*It is going to be able to defeat him, but it is not going to go back with all of the friction, you know, if you have to push this lightly, if it is going to be able to cancel out this lightly, and this lightly, and say it is able to cancel out all the way up till this big, it is not going to react with the static friction force in the opposite direction of this big every time.*1091

*It is going to cancel out whatever is put into it.*1110

*Static friction is going to be able to cancel out up to the amount of force, up to it is maximum amount.*1113

*So the maximum static friction, static friction resists an object starting to move it, until it gets surpassed.*1119

*Until we get to that really extreme case, we are always going to have the case that static friction is going to oppose however much force is put into it.*1125

*It is not going to put in more than that, it is just going to oppose the amount put into it, until we suddenly get to the point where we are able to equal and then surpass static, just that equal point is the razor's edge of flipping over into kinetic friction, at which point the object lurches forward, unsticks, starts to move, and then kinetic friction comes into play, and in almost always μ _{k} will be less than μ_{s}.*1136

*So we have some slight acceleration, if we kept up a constant force.*1157

*The static friction cancels out the force that would cause acceleration, but it never exceeds them.*1160

*That gives us, the maximum static friction = μ _{s} × F_{N}, but keep in mind that it is the maximum static friction, not more than that, but just the maximum.*1166

*It is the top amount that it can be, we are not going to see that every time we put any small force into it, it is going to be the top amount, that is μ _{s} × F_{N}.*1179

*What is some basic examples of μ _{s} and μ_{k}?*1193

*These are some approximate values, this table here, keep in mind that these can vary depending on the specific situation, the condition of the materials involved, wet, greasy, air between them, perfect vacuum, there is certain material properties that can happen.*1196

*For the most part, these are going to remain the system, but it about the whole system interacting together, so it is really something that has to be experimentally determined, or given to us in the problem statement, or something we are solving for from the problem statement.*1214

*Take a look at these, these give us some idea how these things work.*1226

*Notice, μ _{s} and μ_{k} can change very greatly, the difference between cast iron when it is moving and when it is static, is vast, it is almost a tenth of what it start off as.*1234

*But rubber on concrete, it is not much of a change, it is still a change, but it is not giant.*1248

*Ice on ice, once again, pretty large change there.*1255

*Teflon on Teflon, Teflon starts off with a very low friction coefficient, but it stays the same whether it is moving or whether it is still, Teflon is the stuff that goes on to non-stick frying pans. (Teflon is actually a brand name, no one ever recognizes the chemical name, unless they learned it before in chemistry.)*1258

*This gives us some idea of what it is, we start to see that μ _{s} is almost always larger than μ_{k}, sometimes they are equal, and like I said before, there are few freaky cases where μ_{k} is larger.*1277

*It really can vary what it is, we see massive changes from 1.1 to 0.04, we can have even higher than 1.1, grip of a rock climbing shoes on rock is going to be even larger than 1.1, μ _{k} can get very small, μ_{s} can get very small, really depends on the situation.*1301

*You have to get it in the, either the problem statement, for most part we see numbers between 0.2 and 1 as the very highest, but for very slippery objects, we will see even lower, it has to do with what we are getting in the problem, and the specific materials we are working with in our case.*1324

*One special thing to talk about, is wheels.*1341

*How do wheels work!*1343

*So, you might think at first at wheel are going to have kinetic friction between the road and themselves, because they are moving.*1345

*Not actually true.*1353

*One special thing to note is that, when a wheel rolls along a surface, it is going to use its static friction, not the kinetic.*1356

*Now, why is that?*1362

*When the wheel is rolling, at the moment of contact, consider this sort of like flash forward thing, you have got some point here, and then that point is here, and then that point is here.*1366

*At the moment of contact, when it is right here, when it is on the ground, it is actually still because it gets laid down, and then it gets picked back up, it does not move relative to the ground until it is off of the ground.*1382

*If we have got this perfect circular wheel rolling, the wheel is not actually going to wind up having any friction on the ground.*1396

*In reality, the contact patch just moves slightly, but we are talking extremely small rolling resistance.*1403

*For instance, 0.001, that sort of scale, very small.*1408

*So for our purposes we can pretend that there is no friction from a rolling wheel, if it is able to stick to the ground.*1413

*Static friction is what you use for a wheel.*1420

*Notice, this does not mean that a car is being slowed by friction to the wheel.*1425

*Static friction can be very large, numbers like 1.0 for a wheel on concrete in dry conditions, but that does not mean that the car is taking all that out.*1429

*In fact, because it is being put down , and then it is moving off, it never moves, it is never trying to be moved around, when it is on ground, it is like it is practically still.*1440

*It is perfectly still from the point of view of the tyre at that moment.*1452

*That piece, that dot, does not start to move away until it is off of the ground.*1455

*Once it is off the ground, it can move around, because it is not going to have any friction.*1459

*So the only thing that creates friction is that tiny contact patch, and because that tiny contact patch is picked up before it moves relative to the earth, relative to the road, it is not going to give us any frictional force on our car.*1463

*So on the contrary, the fact that it is the static friction is what is going to allow the car to move smoothly, and experience practically no friction.*1476

*I have included bearings, and good oil, it being able to have a good wheel system, you are going to be able to have a almost frictionless motion, and you will be able to have all the motion to the car translated easily as it is running frictionless.*1484

*At least, that is what we would hope.*1497

*In reality, there is going to be some slight friction, because nothing is perfect.*1500

*But, it is going to be pretty darn good.*1504

*It is going to be way better than if we just had a metal body on the ground, that we are shoving along.*1506

*So, we will be able to experience effectively no friction, while it rolls along the ground in a straight line.*1511

*When the car turns, and tries to change its velocity, either by accelerating, so it is going to have those contact patches spinning up, because they are going to be moving faster than they were before, and this is a little complicated to think about.*1516

*But the acceleration, the force, it is the frictional force that allow the car to get that traction, which is why you sports cars, racing cars have really big flat large wheels, because they want a big contact patch, so they can get lots of force into the earth, where as cars that are trying for efficiency tend to have much thinner wheels.*1530

*They are going to have less contact patch.*1551

*If you want to be able to get a car that gets better fuel efficiency, you pump up the wheels a little bit heavier, because that will make them firmer, tighter, and will be able to have a less contact patch on the ground, which means they will have a little bit less friction.*1553

*Remember these are very small numbers, if you are driving at 100 miles, it can have an effect.*1565

*Or if you were to turn, that is when friction is going to come into play, normally you would have the wheel running like this, but then if we want to turn, the wheel is going to turn like this, but the motion of the car is going to be like this.*1570

*So, normally your wheels are going like this, and we have effectively no friction.*1591

*If instead we turn, the car has two choices, if it were to keep going in this path, then all of a sudden, friction will be breaking its contact patches with the earth, because that is not the direction wheel wants to roll in.*1594

*Instead, it is going to go this way.*1610

*So, if the car were to keep going this way, it would break friction, friction would fight it.*1612

*So instead, it goes this way, which means that friction is going to wind up actually pulling this way.*1618

*This is a little bit complicated to think about, but the force of the wheels, friction is the only thing that connects the car to the earth.*1622

*The car and the road are connected through the friction of the tyres.*1630

*So when you go into a turn, the thing that pulls you into the turn, is going to be the friction of the wheels on the ground, and it is going to be μ _{s}.*1634

*This is a lot of explanation for something that does not seem to make sense, but if you want to be able to understand how a car rounds a corner, like we will in the section when we talk about uniform circular motion and force, we are going to be actually understand this.*1641

*So this stuff actually matters, it is a little complicated to think about at first, but it will make sense.*1654

*If something is going to be rolling, it effectively has, static friction, it effectively has no friction, because it is going to be putting in its contact patch, and lifting it up.*1659

*But if it wants to have an acceleration, that contact patch only has to move relative to the ground, otherwise, the rotation movement of the wheel is going to change the speed that the wheel is moving along.*1667

*So its going to require friction to be the interaction, the interplay between those two things.*1678

*Let us finally start talking about examples for the normal basic friction.*1683

*We have got a block of mass 10 kg, resting on a flat surface, horizontal force acting on it.*1687

*It just barely begins to move, unsticks at the force F = 60 N .*1694

*What is μ _{s}?*1699

*First let us do a free body diagram.*1700

*What forces are acting on it?*1702

*There is Mg, pulling down, F _{N} = Mg, (flat surface, nothing else pulling down, so they are going to cancel one another out.)*1704

*Now we are going to work to figure out what the friction is, and we know that friction is going to pull this way.*1716

*If μ _{s}, (we are dealing with static friction, the moment of unsticking, that razor's edge between staying still and just beginning to move, is going to be the maximum static friction).*1721

*The force is going to be equal to the maximum friction, for it to unstick at 60 N.*1746

*What is the maximum static friction?*1751

*That is μ _{s} × F_{N} = μ_{s} × Mg.*1753

*What is F?*1764

*F = 60 N.*1766

*For it to just unstick, we know that the maximum static friction had to just be, barely on that razor's edge, where they are just equal.*1770

*As soon as you surpass it, you flip it to motion, you switch over to kinetic.*1778

*So, that precise moment when they are equal, is the moment of unsticking.*1782

*Now, plug in our numbers, so, μ _{s} = 60 / Mg , we know 'M' and 'g', so, = 60/(10 × 9.8) = 0.61.*1787

*So for this case, between this block and this block and this surface, we have got μ _{s} = 0.61.*1810

*Once it starts to move, it still has that force of 60 N on it, and now it has an acceleration of 1 m/s/s.*1816

*We know that the sum of the forces, = mass × acceleration.*1823

*We know what 'a' is, we know the forces operating on it.*1830

*The force, up here, = 60 N - μ _{k} × Mg = Ma = 10 × 1 .*1835

*We get, 60 - 10 = μ _{k} × Mg, 50/Mg = μ_{k} , μ_{k} = 0.51.*1865

*There is our answer for what μ _{k} is.*1893

*Next example: For this example, we have got a block resting on a surface that can be tilted.*1896

*We have got some tilt, θ on our surface, μ _{s} = 0.35.*1902

*What angle θ will the block barely begin to slide, what is that instantaneous, that razor's edge, that break over point between staying still relative to the incline and suddenly starting to move along the incline?*1912

*Notice, for this problem, we do not have the mass of the block, but it turns it we are not actually going to need it.*1920

*So, we have got a block, it is going to weigh some 'm', so mg is the pull of gravity on it.*1931

*How much of this is going to be perpendicular?*1938

*The perpendicular force, is going to depend on what θ is.*1942

*How much is the parallel force?*1950

*That is also going to depend on θ.*1953

*What is θ , we can figure it out by referring to that old lecture that I had on Newton's second law in multiple dimensions.*1954

*We can also see that in the extreme case (90 degrees), then we have that this is going to be sin(90), so we would have all parallel.*1963

*In the case of 0 degrees (other extreme), we would have cos(0) =1, so all perpendicular.*1974

*So we see that θ has to go here, but we can also figure that out by other geometrical means. (This can be a real confusion down the road, so it is good to refer to.)*1981

*And there is lots of problems that involve incline, so it is really important to have a good understanding of how this works.*1995

*This is mg, so we have, the perpendicular force of gravity = mg × cos θ .*2000

*How much is the normal force?*2013

*That is going to keep it from bursting through the incline, so F _{N} = F_{g}(perpendicular) = mg cos θ .*2015

*What is the parallel force?*2026

*The parallel force = mg sin θ .*2028

*(we are able to do this because we have a right triangle, so these two things are perpendicular, so we use basic trig.)*2036

*At this point, what are the forces acting on it?*2044

*At this point, we have the parallel force going this way, mg sin θ = gravity parallel.*2047

*What other thing is operating on it?, friction!*2057

*Friction is going to be pulling backwards.*2061

*We are going to be using static fricton, because the block starts off at rest.*2063

*What is the moment of flip, is going to be when that maximum static friction, is just equal to F _{g} (parallel), that is the razor's edge, the moment of flipping.*2068

*What is the maximum static friction?*2086

*That is μ _{s} × F_{N} = F_{g} (parallel) = sin θ × mg.*2088

*What is F _{N}, μ_{s} × mg × cos θ = sin θ × mg, (we get mg on both sides, that is why we do not need to know the mass, cancel on both sides.)*2101

*μ _{s} × cos θ = sin θ , collapse it into one trig function by dividing by θ*2122

*Since sin θ / cos θ = tan θ , we get, μ _{s} = tan θ.*2131

*Now, we have done this in general, if you want to know what the angle is, we plug in numbers.*2137

*We get, 0.35 = tan θ , taking arctan on 0.35, θ = 19.3 degrees.*2143

*That tells us what this specific angle is.*2166

*But it also tells us in general, if we want to do this for anything, that is a really easy to find out what μ _{s} is for any pair of objects.*2168

*For any material on some surface, you measure the angle and just keep very slowly tilting it until it just begins to move.*2176

*Really easy way to experimentally derive what μ _{s} is going to be.*2185

*Example 3: We have a block against a wall, and it is sliding down.*2192

*Between the block and the wall, μ _{k} = 0.2.*2196

*How hard do we have to push against the block, to cancel out gravity, to give it a constant velocity?*2202

*If we push in with some force here, what is the normal force?, it is just going to push back with the exact same amount, so F _{N} = whatever force we put in, in terms of magnitude.*2209

*What is going to be operating on this in addition?*2223

*We got gravity pulling down by mg, friction pulling up by some amount that is going to be connected to μ, and the normal force.*2225

*Are we moving, or are we not moving?*2242

*In this case, we started off knowing that the block is sliding down, that means we will be using μ _{k}.*2244

*What is the force of friction?*2252

*Friction = μ _{k} × F_{N} .*2253

*That means, 0.2 × F _{N} (F_{N} is the amount that we push in, that is the amount the wall has to resist.)*2261

*We get, μ _{k} × F .*2272

*If we want those two things to cancel out, we want an acceleration = 0.*2280

*That means that sum of the forces, is going to have to be equal to 0, because, ma = 0.*2287

*That is the way we are doing it right now, is we know that we are in equilibrium, because there is no acceleration.*2304

*It is going to have a velocity, it is sliding down, but we know that there is going to be no acceleration.*2309

*The net of the forces, we have gravity, friction; we also have in the horizontal direction, the force that we are pushing, and the normal force, but they cancel each other out.*2316

*We do not have to worry about that, because it is just staying parallel to that wall, so all we have to worry about is the things that can have an effect, in this case gravity and friction.*2327

*Let us say that up is positive, so, frictional force - mg = 0.*2339

*So, what is the frictional force?*2350

*It is μ _{k}F - mg = 0, μ_{k}F = mg, so, F = mg / 0.2.*2352

*In this case, we get, F = 5 × mg, so the amount of force that we need to push it, to keep a constant velocity or keep it still (then we use μ _{s}) is going to be dependent on the coefficient of friction.*2375

*In this case, 5 × (force of gravity).*2413

*Last example: This one will definitely require some thinking.*2417

*We will start off thinking about the problem and then actually approaching it.*2424

*It is a great way to approach a problem in general, think about it, then approach it, then actually do the math.*2427

*We have got 2 blocks of masses, M _{1} = 2 kg, and M_{2} = 1 kg.*2432

*They are sitting atop each other, they have μ _{s} = 0.7 between them.*2438

*The bottom block is resting on a horizontal frictionless surface.*2444

*What is the minimum force to keep the top block from slipping?*2447

*First of all, if they are moving at the same rate, what does that mean?*2452

*That means we have got some a _{1} acceleration, we have got some a_{2} acceleration, and they are both going to be moving in the direction of force.*2459

*These are our accelerations, going this way.*2468

*But, what about the fact that if they had different accelerations?*2471

*If they had different accelerations, then one of them is either going to be sliding off the other, or sliding behind the other, there is going to be a difference in their relative velocities, which means that they cannot be staying together anymore, they have to be slipping, by the definition of slipping.*2477

*That means, just to begin with, we know that the acceleration of the first block , has to equal the acceleration of the second block, so we can call them in general, 'a'.*2494

*What else do we know about this?*2506

*What keeps block 2 on top of block 1?*2508

*Ther is nothing, no forces we are putting in externally, the only force that is keeping it there, is the force of friction.*2514

*M _{1} is moving this way, that means that for them to stay attached, static friction wants them to stay in place, M_{2}' friction is going to pull this way.*2522

*So, we got friction moving this way.*2532

*So, friction is going to be pulling block 2 over, what about block 1?*2541

*Resultant force, so M _{1} is going to be reduced by that same friction, these will be equal in terms of magnitude, not direction.*2545

*So, M _{2} is going to be accelerated by friction, M_{1} is going to be decelerated, or at least lose some fo its force to friction.*2558

*That gives us an idea of what we are actually doing here.*2565

*We have got these 2 blocks, they are pulled along, only the bottom one is being pulled along, and the way it is able to communicate with the top one, the way that it is able to cause it to move, is by using friction.*2568

*The bottom one and the top one, they only communicate by friction, so friction has to be the way here.*2579

*If we are able to put so much force, this makes sense, I am sure you have seen it, if you have got 2 books on top of one another, we yank the bottom book really hard, the top book will just fall down, whereas if you yank the bottom book really slowly, they will both slide along easily together.*2584

*So it is going to be connected to the coefficient of friction, and the masses of the books.*2598

*What is going to be the minimum force to cause the top block to slip?*2602

*What is the maximum force to keep it in place?*2611

*It is going to be the same thing, that razor's edge once again between slipping and not slipping.*2613

*But, now we have got the understanding to actually approach this problem.*2617

*Final thing, now we can actually do the math.*2622

*Let us start looking at the two free body diagrams.*2626

*In this case, we have got some force.*2629

*What else is operating on it?*2632

*We have got friction, from the top, what about its own friction?*2634

*Does it have its own friction from the ground?*2640

*No!, remember, we said that it is on a frictionless surface, so in this case, there is only one friction, there is just the friction between the blocks.*2643

*What is mu; _{s}?*2656

*We know μ _{s}; what is normal force?*2658

*How hard does M _{2} push in!*2660

*M _{2} is going to push in with M_{2}g, so, F_{N} = M_{2}g, because it does not burst through the box, it does not move through it, it stays on top of it, so the normal force has to equal M_{2}g.*2663

*With this in mind, we can start coming up with our formulae.*2682

*Net force is, we know that, F - friction = M _{1} × a . (We can use vectors, but we do not have to be, because we understand the directions, because they have been dealt with.)*2685

*So, F - friction = M _{1} × a .*2717

*What is the forces on M _{2}?*2720

*Just friction, = M _{2} × a .*2725

*With that in mind, we can start to figure out what is F going to have to be equal to.*2730

*F - M _{2}a = M_{1}a, so, F = (M_{1} + M_{2}) × a .*2734

*That is how much force is necessary to give an acceleration of this, because it has to move both the objects, the whole system.*2755

*We can sub that back in, we can now figure out what is friction.*2763

*F/(M _{1} + M_{2}) = a , plug that into this formula right here,*2768

*We get, the friction = M _{2} × F/(M_{1} + M_{2}), so, friction = M_{2} × F / (M_{1} + M_{2}).*2780

*Now we could sub these things in, we could figure out what are the actual numbers, what also is friction.*2807

*In this case, friction = F / (1+2) = F/3, because that is the amount that it has to get, it is the share that the top block has to get, because it has one third the total mass of the system, so it has to get equal share for its mass, to be able to move it with the same acceleration.*2813

*So friction has to be equal to F/3, for the acceleration to be the same between both the objects.*2841

*What is the maximum amount of friction?*2847

*Remember, that it is going to have maximum friction, (static friction, because we are static here), is going to be the maximum velocity without slipping.*2849

*Once again, it is that razor's edge, so the minimum velocity for slipping is going to be that flip over point, the maximum velocity without slipping is going to be the same thing as the minimum velocity of slipping, if we just go an infinitesimal amount over, we are going to start to slip.*2863

*So, the maximum static friction = the maximum velocity that we can move at.*2886

*The maximum force, maximum velocity, maximum acceleration,*2891

*ACTUALLY I SHOULD NOT HAVE SAID THE VELOCITY, THE MAXIMUM ACCELERATION, my apologies, you can of course have any velocity, it could be whizzing along in space, a million miles per hour, it does not matter.*2894

*From its point of view, it is not experiencing any force, so it is about the maximum acceleration.*2906

*So, maximum force.*2912

*With all that in mind, it will slip at the moment, when, F/3 = max. static friction.*2915

*What is max. static friction?*2928

*μ _{s} × M_{2}g, so, F = 3 × μ_{s} × M_{2}g.*2930

*Plug in numbers, 3 × 0.7 × 1 × 9.8 = 20.58 N.*2951

*That is how much it is, to finally get the thing to just start moving, if we just barely see 20.58, that is the razor's edge, slightest bit f difference off 20.58, and it will just start to slip, because it will just exceed this maximum static force.*2976

*Hope friction made sense, if you got difficulty in understanding how an incline works, definitely refer to Newton's second law in multiple dimensions, it will give you an understanding of how to deal with parallel and perpendicular forces, it is important to understand that when you are dealing with friction.*2992

*Hope you enjoyed it, see you later.*3009

1 answer

Last reply by: Professor Selhorst-Jones

Mon May 15, 2017 11:32 PM

Post by sania sarwar on April 26 at 09:26:04 PM

which geometry lesson would you recommend in order to understand example 2's math?

1 answer

Last reply by: Professor Selhorst-Jones

Fri Mar 25, 2016 6:25 PM

Post by Peter Ke on March 7, 2016

At 49:29, why F_fric=m_2*a and not F_fric=u_s*m_2*g?

1 answer

Last reply by: Professor Selhorst-Jones

Mon Oct 6, 2014 11:59 AM

Post by Tori Carroll on October 5, 2014

I have a question on a problem similar to your third example. Say a box of mass m is held at rest against a vertical wall by a horizontal force FA. The wall has coefficient of friction ÃŽÂ¼. How would you solve for the minimum coefficient of friction ÃŽÂ¼ in terms of FA, m, and g?

1 answer

Last reply by: Professor Selhorst-Jones

Sun Jul 28, 2013 9:01 PM

Post by enya zh on July 27, 2013

Which type of objects have greater static friction than kinetic friction? Just curious.:)

Thanks!!!:):)

1 answer

Last reply by: Professor Selhorst-Jones

Sun Oct 28, 2012 9:49 PM

Post by varsha sharma on October 28, 2012

in example 3 shouldn't it be

mg-fric.= 0 ( because the object is moving down )

(though by doing your way ,the answer will be the same)

3 answers

Last reply by: Professor Selhorst-Jones

Wed Oct 17, 2012 1:58 PM

Post by Nik Googooli on August 30, 2012

50/m.g=

50/98=0.51 not 0.71

1 answer

Last reply by: Professor Selhorst-Jones

Thu Sep 6, 2012 4:48 PM

Post by Patrick Gomez on August 7, 2012

I love Physics! It's amazing how a person's whole way of viewing the world around them changes as they continue to learn more.