Just like friction is a special case of rigid-body equilibrium:
there are certain problems within friction that we want to talk about
-- friction applications. We're going to start with wedges.
If you want to lift a heavy block,
one of the best ways to do it is to stick a wedge underneath it.
That allows us to use a horizontal force to produce a vertical movement.
So if I have mu_s as 0.3, how hard do I have to push
in this particular instance to move that block?
Like any other equilibrium question, the first thing you do is read the problem.
The next thing you do is draw a free-body diagram.
So here's my free-body diagram of the whole system.
Remember that friction has to act along the surface to oppose motion.
So if you look at the wedge itself, the wedge would be moving to the left,
so the friction force has to point to the right.
Similarly if this thing is to move at all, then that blue block is going to go up,
which means the friction force has to go down.
Unfortunately when you write all that together, you've got five forces.
Equilibrium is only going to give you these two equations --
the sum of the forces of the x and the y.
Even if you assume it slips in both places, that only gives you two more [equations].
You can't handle that.
Which brings us to our very first example of a system.
So we did particles and rigid bodies.
A system is where you actually have to take the object apart
to consider the equilibrium of each of its individual pieces.
If I consider this as two separate things and I disassemble it,
I can look at the free-body diagram of the blue block and the red wedge.
They're both rigid bodies.
So if I draw the free-body diagram of this one or that one,
I should get equations of equilibrium for each of them.
That allows us to have four equations of equilibrium and in fact it would have to slip at three spots.
It would have to slip here, it would have to slip here,
and it would have to slip at the wall.
So now we have enough equations as long as I haven't added too many unknowns.
So let's look at what the free-body diagram for the wedge looks like.
If you start here, this --
as it moves to the left, friction has to oppose that motion.
So the friction at the floor (we talked about already)
is going to be in this -- just to the right, opposing P.
Well this now is a new friction force.
It's not the same magnitude.
You can't call them both F.
But this magnitude will also oppose P.
It's going to be along that surface now with this 30-degree angle.
And similarly there's a normal force here.
It's going to be normal to this surface.
Now that's considering that this is part of the world.
So I'm just considering the wedge as my object.
This has to be equal and opposite to this.
So if I consider now just the block, whatever is happening here it's the same spot.
So I can consider them all together, but if I take them apart,
whatever the block is doing to the wedge, the wedge has to do to the block and vice versa.
So if I picked this up and I put it back on top of that,
I have to get back to what I started with.
So these have to cancel if you consider them together.
That's equal and opposite.
So I have Nb and Fb here; I have to have Nb and Fb here.
They have to be in the same directions.
And remember that any force that isn't vertical or horizontal needs angles.
So this is what I have now.
Because I only added two new variables, I now have seven total forces:
1, 2, 3, 4 -- same -- 5, 6, 7.
Seven forces with these two equations of equilibrium and the slip at three places
will give me what I need to know.
So what does it look like when you look at those equations?
This is the sum of the forces in x and y for the block.
This is the sum of the forces in x and y for the wedge,
just bearing in mind what angle those forces act at.
And if I want to find P to move that box, it has to slip at all three of these locations.
It wouldn't make any sense otherwise.
You can't move just the wedge because the block is in the way.
So those are my equations.
Now let me say a word about that:
you will find books and people on the internet that simply go straight to these formulas
where there are no friction forces and all of the friction forces have already become 0.3*Nb.
I would prefer that you not do that in this class.
Yes, I agree: it has to slip in all of those three cases,
but it can be very confusing if you jump right into that.
So you'll see places where these are your only equations of equilibrium.
You will see free-body diagrams that are all labeled with mu times N.
But what you have here the way I set it up,
these equations will hold as long as this is in equilibrium
even if you've got it actually being in the opposite direction.
One of the questions you'll get with a wedge is,
"What P is necessary to keep the block from scooching down so far
that the wedge scooches out this way?"
In either case, these equations of equilibrium are sound,
but these assumptions you have to actually deal with carefully.
So what I would like to see is I'd like to see
that you write your equations of equilibrium
and then you talk about what's moving and what you can actually assume.
Once you've done that here, you've got these four.
That's all you've got.
The first two only include Nb and Nw. This one only includes Nf and Nb.
So once you have Nb, you can do Nf.
And this one only includes P. So now I have a number.
Now this number is almost twice what my block weighed.
So my weight -- my block was 600 Newtons, and this is almost twice that.
Clearly, that is not ideal.
I mean, the goal here is to lift heavy blocks by means of wedges
where I'm going to use less effort, not more.
In fact we used the 30-degree wedge because it's easy to draw pictures of.
If you make a much smaller wedge or a more slippery surface,
these numbers all play into here.
So that's just one more example where P ends up being only 391.
Having said all that, here are some points to remember.
Each surface has to have equal and opposite forces.
We talked about that a little bit.
You may not have an F without an N, because after all F_max is equal to mu*N.
If you don't have an N, you don't have an F.
Each separate surface has to have unique labels.
So I had Nb, Nw, and Nf. You can have 1, 2s, and 3s.
You can have Fred, George and Susie; whatever makes you happy.
But they have to be different labels, because they're going to have different magnitudes.
Any force that isn't vertical or horizontal has to have direction.
Bear in mind that F is always going to oppose motion.
So if you ask yourself,
"With what I have assumed here, which way is this going to move?"
Friction opposes that motion.
If you get the wrong direction --
for example, if you go through that analysis that we just did
and you've got an assumption that the block is actually sliding down --
-- so all you did is switch the signs on Fw --
you will, in fact, get the wrong answer.
So this is a situation where it matters.
N is always going to act perpendicular to the surface of the wedge.
So if you have a wedge that's sitting on a block like this,
the friction force has to act along the surface of the wedge,
not along whatever angle your block is sitting at.
And I want to talk for a minute about directions.
I've said it over and over again:
you can't have a free-body diagram without distances and directions.
Now I've got all the directions on my free-body diagram I wanted.
But I didn't have any distances.
So how is it that I can get away with that?
Bear in mind that the whole point of a free-body diagram
is so that I can write equations of equilibrium
from your free-body diagram without any extra information.
That's the gold standard.
Most of our wedge problems do not include whether the wedge is going to tip
because a wedge is crammed in between two things.
That's the purpose of a wedge.
So whether a wedge is going to be in equilibrium from a turning standpoint doesn't make a whole lot of sense.
In general, we also do not know where this normal force acts.
So all you would end up doing is saying,
"Well, I'm going to have this normal force, and it's going to act somewhere along this surface,
but I don't know where."
So I've just added a variable which I'm not going to be able to solve [for].
So if you are in a situation where nothing can tip,
you will find that we draw free-body diagrams without the distances.
Generally a no-no, but often happens in the wedges.
Bearing that in mind, if you have a block that might tip --
so if you have a wedge like this where what I'm doing is I'm putting this force in here
and this could tip over,
then you have to absolutely consider where the distances are
so that you can write the sum of the moments.
Last but not least, weights.
Almost always one of your objects will have a weight.
But usually the wedges don't.
Sometimes I'll give you a weight on a wedge, but rarely.
If there is no weight specified in the problem, you neglect it
or once again you'll end up with more variables then you can solve for.
So those are some things to keep in mind with wedges.
At the end of it all, read the problem, draw a free-body diagram,
put the forces on there as they need to be ... normals and frictions.
Friction is gonna oppose motion.
Write your equations of equilibrium.
After you've got that done, try to figure out where the slip is going to occur.
Write F = mu N for the places where slip occurs and then solve your problems.