In equilibrium what we do is ask ourselves
what happens if I'm applying many forces to some object,
as you can see this gray box here with the certain center of mass.
Is there a way I could put all this forces together
so that my object doesn't move.
Is there are some equilibrium that we can reach
even when forces are being applied.
There are few different types of equilibrium
so it's important to be familiar with this terminology.
First we talk about static equilibrium.
In a static equilibrium you can't have many forces on your object all acting together
but the object first of all does not have any net force acting on it,
meaning that if you added all the forces together,
you would end up with a zero so that your object has no acceleration.
But in static equilibrium we also impose that the object has no velocity.
So not only is it not accelerating but it's also sitting still.
It has no velocity, it's just motionless.
And this is important in many static's problem
if you have maybe a structure that you don't want to be moving.
You would analyze it using statics.
You would say it in static equilibrium hopefully not moving anywhere
and also not having a net force applied to it.
On the other hand we can talk about a dynamic equilibrium.
So if we add all these forces as we did with the case of a static equilibrium.
We again require that these forces add up to zero
so that there's no actual acceleration of our object.
It's not accelerating, but we do allow the object to move.
So this object here this box with the center of mass could be moving
but the total force acting on the box is actually zero.
If you put all these forces together. And so the box might be moving.
It might have a velocity but it has no acceleration.
We could also talk about a translational equilibrium or a translational motion,
meaning that the center of mass here is actually not moving
while the box itself might me moving.
For example rotating, so if I put the forces as I've shown here.
This box might start to rotate because what I've done is applied many torques to the box
so it could spin.
But notice that the center of mass didn't move anywhere.
So we would say that this box is in a translational equilibrium
because while there are forces on the box and those forces can cause the box to spin
there is no net translational force of the box.
On the other hand as you might expect we could define one more type of equilibrium
and that's a rotational equilibrium which is just the opposite of a translational equilibrium.
If I apply the forces on the box instead as you see here
the box might move all the way up to the side on its own.
But the box won't rotate and so you see without while the center of mass moved
there was no actual rotation of the box.
Here's an example of how we could find the torque being caused by
a particular one of the forces that we just saw with the box that could be rotating.
In this box, let's actually just focus on one of the forces.
Let's take this one on the top right and ask how can I find the torque
as being caused by that force where again we're going to use our torque equation
where the torque is force times R times the sine of theta.
So looking at this box, suppose maybe in a common problem
you might be given this distance the length of a side of the box,
it might be called A is a very common letter for the length of the side of a square like this.
And so you want to find an expression for the torque being caused by just this force.
Where again, we have that the torque represented by the Greek letter tau here,
is equal to R the distance between the pivot point and sine of theta times your force.
Now again sometimes geometrically instead of considering R times the sine of theta
where our center, a pivot point is here.
R is the distance between your origin and where the force is acting
which is here so our distance is this one.
And we can see an angle, if we wrote a line for our force
between the distance and the force that's being applied.
And so we could write this all out as torque equals R as I just written it here,
in which case you would need to find R for your problem times the sine of theta,
so you would need to also figure out what the theta angle is here times the force.
So all of those things are a little bit too time consuming,
so we want to find a simpler way to find the torque by using the idea of the lever arm.
Let's see where the lever arm would be in this problem.
So if you take a look, I'm going to extend the line of force as I've talked about before.
And I draw a line directly from the origin to the line of force.
So now this would be our lever arm L which I'm just writing as a cursive L
so that we don't get confuse with the number of 1 or the letter I.
So we have a lever arm, L which is right here.
And this is nice because L is simply half the length of the side of a box,
of the box that you are given and so for being a very idealistic here,
this forces is actually being applied right at the edge so don't get confuse by this gap.
L is going all the way to the end there.
So in this case we can rewrite the torque as torque equals the lever arm times the force.
Because again the lever arm is equal to R times the sine of theta.
And this is much easier to solve because if we're given the force of the object
and we can see geometrically that the lever arm is just the size of the box divided by two.
Then this is a much easier equation to use for the torque for this particular example.
Because in an example like this again you'll probably be given
something about the geometry of your object
as well as the magnitude of anyone of the forces
and you wouldn't want to try to find the diagonal of the box with R
or think about the sine of theta with the angle there.
So now that we have introduce the idea of a torque
in what quantities that torque depends on, that it depends on a force and a distance
we still have to more further develop the idea of our coordinate system.
How do you we describe objects that are in rotation?
So if we have an object like this one, where it's pivoting around a central point
at the top and the whole thing could rotate to one or the other side.
How do we describe this motion in a coordinate system?
We have a convention for this in terms of defining a positive and the negative
and that is that the positive direction is always considered to be counterclockwise.
While the negative direction is considered to be clockwise.
This can seem a little maybe arbitrary at first, a little counter intuitive
but it is a convention so it's something to remember
and it might help you to remember by looking at something like this.
If we have a simple X and Y or a horizontal and vertical axis for any reason.
We've often in trigonometric classes or similar setting
describe a line which is lifted up from the X access
and has an angle with the X access this angle theta.
So if I increase this angle theta notice what direction my line would move,
it would move counterclockwise.
It would move after the left like this and so just looking at
or thinking about the picture like this one,
you can imagine that an increasing angle, angle becoming more and more positive
will cause a line to rotate in a counterclockwise direction.
And so this direction, counterclockwise direction, the direction of the increasing angle
as what we will always call the positive direction.
And this is the convention we'll stick to, as we go through torque problems.