Hello.

In this video, we are going to study the effect of friction, and to consider

how a body can stay in equilibrium while it is on an inclined plane.

We are going to look at what is the effect

of the operating surfaces, what is the influence of the inclination.

We are going to look at, in which case there will be a sliding,

and in which case there will be equilibrium.

Obviously, it is the case which interests us

more, since our structures must be in equilibrium.

Let's start by considering our test bench.

This test bench will be characterized at each measure by

the maximal inclination, which will be tolerable for a surface condition.

That is to say for which angle

the block that we can see, will not slide on the surface.

We can note that there are two tracks on our

test bench, one Teflon track, on the left, white.

It is a synthetic material whose the

property is to offer very little sliding.

And, on the right, we simply have a

wood track that we will use to compare to.

And, you can see that

the block has several surfaces which

will enable us to study several surface conditions.

We start with the block, whose the lower face

is in Teflon, which slides on the Teflon track.

This is the condition where there will be the least friction.

Indeed, we only get 9 degrees of possible inclination.

When we make sliding the slab, the lower face

of the wood block, on the Teflon

track, we get an inclination of 11 degrees.

When we make sliding the block with the lower

face in rubber on Teflon, we reach 13 degrees.

Finally, if we put

the block whose the lower face is in sandpaper

on the Teflon track, we

reach a much larger angle of 34 degrees.

Let's look now,

quickly, at what happens when we make sliding on the wood track.

The Teflon block, with the lower part in Teflon on the

wood track, is able to stay in equilibrium with an angle of 15 degrees.

Wood on wood, we get 18 degrees.

Sandpaper on wood, we get 41 degrees.

Then, the best result is rubber on wood with 45 degrees.

Well, maybe you now understand why, when we make

hike in the mountains, we use

shoes with rubber soles.

It does not work very well on Teflon, but it

works very well when the surface is a little bit rougher.

We are going to summarize all the results in this table.

So, for the Teflon track,

we got an angle of 9 degrees with Teflon, of 11 degrees with

wood, of 13 degrees with rubber, and of 34 degrees with sandpaper.

For the wood track, we got 15 degrees with Teflon.

We can see that it is not the same value than with wood on Teflon.

Why ?

Because it is not exactly the same wood.

If you remember,

to have seen the wood block which was in a quite hard wood, while the track

was made by a wood a little bit softer. Wood on wood, we get 18 degrees,

sandpaper 41 degrees, and rubber 45 degrees.

We will come back, later, to this table to complete it.

We want, now, to look at the equilibrium of the block,

on an inclined plane.

So, it is subjected to gravity, to its weight G.

Then, there is a contact force which acts under the

structure and a friction force which prevents it or not to slide.

To solve this question of equilibrium, we simply

pass, a free-body under our block.

Thus, we include, all the weight, and

we are going, thanks to the free-body, to reveal

the contact force and the friction

force which act just under the block.

So, here, I draw again the free-body, on the right.

The gravity force which I had already drawn, it obviously acts in this way

we have G,

the weight of the block. The contact force just acts at

the interface, here, between the block and the inclined plane.

It is perpendicular to this inclined plane, and its magnitude

is equal to the projection of G on this perpendicular,

to the surface. Well, I am going to draw, here, G.

I represent it by a noncontinuous line, because G does not really act here.

When I am going to make its projection, on the perpendicular to the surface.

Well, we project it in this way, perpendicularly.

And I can read that this contact force C,

it has this magnitude.

So, we have here, again, a perpendicular.

And the third force which acts on this body, it acts, here.

It is the friction force.

Well, we can notice one thing, it is

that these two forces, the friction force

and the contact force, they intersect, here.

And there is no intersection with the weight.

Thus, we do not have what we had to get.

What we must get, to have the equilibrium,

is an intersection of the three forces on only one point.

This is a little mistake.

Note that the complete solving would not be extremely

complicated, it is simply not useful, at this stage.

What we are going to do, for the rest of this course, is that we are going to consider

a slab much thinner.

Thus, we are going to make a much smaller mistake.

In theory, we would need a slab with an infinitesimal

thickness, then we would not make any mistake anymore.

So, we have G which acts on the slab. We have the contact force.

Then, I am going to draw it on the center of gravity, as if it acted

on the center of gravity.

It is not totally right, but

the mistake is quite small, the friction force.

Now, let's look in the polygon of forces what it means.

We copy the force corresponding to G.

Then,

we add the force

corresponding to C, to the contact

force. So, the contact force, it

has a length which goes until the projection

of G on the vertical. You can see, that is approximately right.

If now, we trace the resultant

of these two forces. what we can notice is that this

resultant is here. It is parallel to the slope.

It is Rcg.

And it is a force which pushes our body downwards.

Well, that is this force which is going to make our body sliding.

If there was not any friction, our

body would slide, for example, like a skier on

a slope.

There is a very little friction, when we ski.

Thus we have, as result, a driving force.

What happens ?

This is the case which interests us, when the body does not slide.

Well, when the body does not slide, it means that there is

a friction force which acts in the upwards direction.

I draw it a little bit staggered,

simply for us to be able to see it.

Here, a friction force, which

opposes to the sliding, and which closes the polygon of forces.

Thus, our body is in equilibrium.

Let's now look at what are the reasons for

a body to slide, or for a body to be in equilibrium.

So, we will have

a body in equilibrium, if

the magnitude of the

resultant of C and G is less than or

equal to the magnitude of the maximal friction

force. Why the maximal friction force ?

Well, imagine precisely that we have the case of

the rubber block which can hold until 45 degrees.

If we put it on a slope of only

20 degrees, it does not mobilize all the force.

And it is not the friction force of 45 degrees which acts.

Because if this friction force would act,

then, our body would go up the slope thanks to the friction.

That is not right.

What is right, is that the friction force adapts to correspond

exactly to the force which is necessary not to have sliding.

This, until the maximum value that it can reach.

We have seen that for Teflon on Teflon, the force would not be large.

However, when we have rubber on wood,

well, the force is quite large.

We often write this value of the

maximal friction force, as (Greek) nu times

the magnitude of the contact force. We have, here,

a perpendicular. And, we can recognize,

here, the inclination angle alpha, that we find again,

here, in this construction. And we can notice that the

maximum friction

force will be

equal to the contact force

times the tangent of alpha max, which

is the maximum angle that we have been able to measure.

So, in other words, nu equal to the tangent

of alpha max. Let's come back to this table, to complete it.

We are going to add here, each time, alpha max.

Then, here, we are going to say nu equal to the tangent of alpha max.

And we have the values, here, zero point 158,

zero point 194, zero point 674,

zero point 231, zero point 268,

zero point 325, zero point 869,

and one point zero in this case, of 45 degrees.

In this video, we have seen the reasons for

which a body slides or not on an inclined plane.

That is that the friction. There is

no sliding if the maximum friction which can be mobilized

is larger than the driving force resulting

from the weight and from the contact force.

We have seen that the friction coefficient nu can

be deducted from a simple experiment taking the tangent of

the maximal possible angle of inclination.

We will not really come back on friction thereafter,

but it is important to understand well that

these friction forces do not have a nature

fundamentally different from the other

forces, with which we will be confronted.