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Hello, in this lecture, we will see how a man, standing on
Earth is in equilibrium. We will use
a new construction, called a free-body diagram,
which will enable us to identify all the forces which act on this person,
and we will determine how to define equilibrium
on the basis of precise criteria.
When a man is in equilibrium on the ground, it is obvious that, due to universal
gravity, his mass induces a weight which is activated by Earth.
Earth itself acts on this person, so that he is in equilibrium,
and, under the person's feet, acts a contact force.
We are going to use this construction
of a free-body diagram to evidence these effects.
Let's look at a man on the ground.
The first force which acts at his center of gravity, which is approximately at
this height, is his weight G, which acts vertically downwards, and this
weight has a value of approximately 800 Newtons in this case.
We can see lots of other things
around this person, which do complicate our thinking.
Are there some effects induced by the red
construction behind, or by the benches, or by the flower display cases ?
It is not clear.
So, what we want to introduce is the principle
of a free-body, which will enable us to isolate what interests us,
it will often be a structure, here it is a person, from
the environment, to only focus on what is
significant. I then draw a free-body which
surrounds the person, which includes everything which is significant, and then
which cuts, however, just under the person's feet,
since we want to reveal what happens under the feet.
What happens around him, however, is not very significant, and thus
disappears.
Thus if we look at this sky blue free-body edge,
inside this free-body acts a force of 800
Newtons downwards, and if it was the only
force which would act, well, the man would go down, would fall down.
There is necessarily another force, it is neither
on the left, neither on the right, the man is not borne
by air, it is not either on the top, the man is not
borne by his hair, but it is under his feet.
I draw here
a force which corresponds to the effects of both feet together, and which is the effect
of Earth on the man.
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Just where our free-body passes under the feet,
it passes at the boundary between the man and the ground, and this
kind of force is a contact force which
always acts perpendicularly between
two bodies which touch one another. We now want
to know how this body can be in
equilibrium, how this free-body can be in equilibrium.
Well, for it to be in equilibrium, it is necessary that
the ground has enough capacity to bear the person.
We can easily understand that this force must be equal to and
in the opposite direction that the person's weight, that is to say to be equal to 800 Newtons.
What happens if this person, instead of standing
on a firm ground, stands on quicksand ?
Well, the quicksand can exert a
force upwards of, let's say, only 300 Newtons.
So, the result of this is that there is a force
of 500 Newtons which pushes the person downwards, until,
once the friction acting along his legs increases,
the force which can be transmitted by the quicksand is significant enough to
equilibrate the 800 Newtons of the person,
it is thus necessary for the equilibrium that the
contact force of the ground on the person should be able to be equal to the person's weight.
We can also express it saying that it
is necessary that the sum of the forces acting
on the free-body should be zero. We can represent
this graphically, we have here the weight of
800 Newtons, weight G, and then
we have here the contact force
which is also equal to 800 Newtons.
Then, we can see that it is... so, I drew
these two vectors a little bit staggered from each other so that
we can see them, actually they are
really layered over each other. We can also say,
since they are two vectors, that the vectorial
sum of the forces must
be zero for a body to be in equilibrium.
Let's consider another case, the one of a person
who leans forward. I draw a free-body which surrounds
this person, we pass just under his feet,
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the forces which act on this free-body are on the one hand
inside the free-body, the weight G of the person and then, on the other hand,
under the free-body, there, the bearing force, the
contact force of the ground on the person.
What we notice is that these two forces
share the same line of action, that is to say that they are aligned.
If we look at the person's feet on the right, we
can draw here the path of the free-body, well, I am only interested
in one of both feet, but the reasoning I am doing
is also valid for the sum of both feet, well, we can
see that in this configuration, the sum of, the contact force, it acts
somewhere just at the front of the feet, but it is still possible.
If the contact force, however, had to
act even more on the left, it would cause
a problem, we could arrive at the end of the big
toe, and then after, there would be a problem
about the strength of the foot, and then, if we go further, this is the case
of someone who leans forward too much, this person here have a center of gravity which is
situated on the
left of the line of action of
the contact force,
In the first picture on the left, it is located slightly to the left, whereas in
the picture on the right, much more,
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in this case here, we deal with a loss of equilibrium
since the ground is still ready to provide 800 Newtons to resist to the weight of the
person, but however, it is impossible that the line of action of the
weight of the person coincide with the line of action of the contact force.
The person who loses his balance is a case that does not interest us anymore
in this course, since we want to
focus on the structures in equilibrium, but
it is obvious that it also means that, if it happens to a man,
it can also happens to a structure, and then we need to be very
careful not to get structures which are not able to be in equilibrium.
In this video, we have seen how to use a subsystem, to
evidence, not only the weight of the person, but also a contact
force between the ground and the person, we have seen that the equilibrium of
two forces is conditioned by two conditions ; the first one is
that the vectorial sum of all the forces which act on the
free-body must be zero, and the second one,
is that the forces have
an identical line of action. And
finally we have seen that a loss of equilibrium
results from a violation of one these two conditions.
Olivier L. BurdetDr
Aurelio Muttoni
