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Lecture Comments (14)

1 answer

Last reply by: Professor Selhorst-Jones
Fri Mar 25, 2016 6:25 PM

Post by Peter Ke on March 7 at 07:22:23 PM

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 μ. How would you solve for the minimum coefficient of friction μ 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.:)

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


50/98=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.

Related Articles:


  • 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: FN.)
  • 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: μs and μk.)
  • Friction always opposes motion. Whatever direction the object has (the direction of v), friction points the opposite way.
  • The formula for friction is
    Ffric = μ·FN.
  • Kinetic friction is just a continual force of Ffric = μk ·FN, 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 = μs ·FN.
  • 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!


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 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

Transcription: Friction

Hi, welcome back to, 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, FN = 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 × FN...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 F1 and F2 are both big enough to move the block.0692

But also that, F1 and F2 are equal in magnitude, they are the same number of newtons.0696

If F1 and F2 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 F1 and the horizontal amount in F1, and over here, the vertical amount of F2, and the horizontal amount of F2.0714

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 FN 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 F2'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, F2 is the more efficient, easier way, it is going to cause more acceleration.0866

F2 will accelerate the block faster, because it will have the smaller FN.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 μs, kinetic is going to be μk.1006

So, we have got μstatic and μkinetic.1017

One thing to keep in mind: In almost all cases, μs is greater than μk,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 μk 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, μk × FN, 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 μk will be less than μs.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 = μs × FN, 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 μs × FN.1179

What is some basic examples of μs and μk?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, μs and μk 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 μs is almost always larger than μk, sometimes they are equal, and like I said before, there are few freaky cases where μk 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, μk can get very small, μs 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 μs.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 μs?1699

First let us do a free body diagram.1700

What forces are acting on it?1702

There is Mg, pulling down, FN = 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 μs, (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 μs × FN = μs × 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, μs = 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 μs = 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 - μk × Mg = Ma = 10 × 1 .1835

We get, 60 - 10 = μk × Mg, 50/Mg = μk , μk = 0.51.1865

There is our answer for what μk 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, μs = 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 FN = Fg(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 Fg (parallel), that is the razor's edge, the moment of flipping.2068

What is the maximum static friction?2086

That is μs × FN = Fg (parallel) = sin θ × mg.2088

What is FN, μs × 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

μs × cos θ = sin θ , collapse it into one trig function by dividing by θ2122

Since sin θ / cos θ = tan θ , we get, μs = 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 μs 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 μs 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, μk = 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 FN = 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 μk.2244

What is the force of friction?2252

Friction = μk × FN .2253

That means, 0.2 × FN (FN is the amount that we push in, that is the amount the wall has to resist.)2261

We get, μk × 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 μkF - mg = 0, μkF = 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 μs) 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, M1 = 2 kg, and M2 = 1 kg.2432

They are sitting atop each other, they have μs = 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 a1 acceleration, we have got some a2 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

M1 is moving this way, that means that for them to stay attached, static friction wants them to stay in place, M2' 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 M1 is going to be reduced by that same friction, these will be equal in terms of magnitude, not direction.2545

So, M2 is going to be accelerated by friction, M1 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 μs; what is normal force?2658

How hard does M2 push in!2660

M2 is going to push in with M2g, so, FN = M2g, 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 M2g.2663

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

Net force is, we know that, F - friction = M1 × 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 = M1 × a .2717

What is the forces on M2?2720

Just friction, = M2 × a .2725

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

F - M2a = M1a, so, F = (M1 + M2) × 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/(M1 + M2) = a , plug that into this formula right here,2768

We get, the friction = M2 × F/(M1 + M2), so, friction = M2 × F / (M1 + M2).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

μs × M2g, so, F = 3 × μs × M2g.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