In this video we are going to go through some of the equations that help characterize that
of rigid body motion.
This falls under fluid statics typically which is kind of confusing to use the word fluid
static and motion in the same sentence but when we take a look at what rigid body motion
implies you will understand why it falls under fluid statics.
We have this container that has a fluid in it.
If we go back to our general equation of motion.
This was developed for both fluids at rest and fluids in motion with the only condition
being that there are no shearing stresses occurring in the fluid.
We can take this and break it into the individual components for rectangular coordinates and
they look like the following.
If we do this for the x component we get the partial derivative with respect to x equal
to the density times the acceleration in the x direction.
We can do the same for the y direction and lastly for the z direction we have this term
With the specific weight, our gravity term, and I am going to bring it to the other side
so that we can lump these two together.
I want you to think of a time where you have a container filled with liquid and we quickly
accelerate it in a certain direction what is going to happen?
We know from experience that that slope of the fluid all of a sudden changes.
Maybe we would get something that looks like this and we would have the fluid move.
Of course this is under a constant acceleration.
As soon as you start changing the acceleration maybe you have some sloshing around of our
fluid.
Let's take this open container of a liquid and move it at a constant acceleration.
When we do that we get the following slope of our fluid.
We want to characterize what is going on.
Let me draw a coordinate system for our fluid.
We will say that this is the z direction.
Call this the y direction and call that into the page our x direction.
We are moving this container in the y direction and it is affecting the slope of our fluid.
So it has an affect on both the y and the z direction but not in the x direction.
We will say that ax=0.
If ax=0 then this means that our pressure gradient in the x direction has no change.
I am going to rewrite our pressure gradient in terms of a positive value.
This is equal to negative rho times the acceleration in the y direction.
When I rewrite this to have the similar form of dP/dz we will set it equal to negative
rho times the combination of acceleration due to gravity and any acceleration that we
might apply to it in the z direction.
Let's take a zoom in at our free surface of the fluid.
When we think about this slope of our fluid we have a differential y with respect to differential
z.
Say we want to take a look at the change in pressure between two points.
Maybe here and here.
At this first point we have (y,z) and at this second point we have (y+dy,z+dz).
Our change in pressure or differential pressure can be written as the following.
Our differential change in pressure with respect to y times our change in y plus or differential
of pressure with respect to z times our change in z.
If you recall a few seconds ago we had relationships for these differentials.
Here we have dP/dy and here we have dP/dz.
We are going to go ahead and plug those in and write this differential pressure.
Here is the nice this about this.
Along isobaric lines or lines of constant pressure dP is equal to 0.
Where are our lines of constant pressure?
At the free surface here the pressure every where at the free surface is just atmospheric.
Along a line of constant pressure we know the dP=0.
Now we have this relationship that we can rewrite into the following form.
I am just going to pull over the terms so that is looks like the following The first
term we can cancel out the densities and we get the following relationship.
As long as the acceleration in the y direction is not 0 the free surface of the fluid will
be inclined.
It will have a slop that is based on the ratio of these accelerations.
What does this mean?
It means that if we look at this free surface as this container is accelerated with some
constant acceleration in the y direction the slope of that line is just going to be the
acceleration in the y direction over gravity plus az.
If there is no acceleration in the vertical direction, the z direction, then it is just
the acceleration in the y direction over g.
Another special case is if the acceleration in the y direction is 0 then what we get is
that dP/dz is not the hydrostatic pressure.
It is going to be a function of the acceleration in the z direction.
If there is no acceleration then we can just get density times gravity which is what we
would expect when calculate hydrostatic pressure.
This is saying that for a container that is following or moving upwards that the pressure
gradient is going to to be a function of both gravity and that acceleration in the z direction.
Think about what happens when we have a freely falling film and the acceleration in the z
direction is negative g.
That is basically saying that the pressure gradient is 0 in all three direction.
This means that if the pressure surrounding the mass or the fluid here then the pressure
throughout the fluid would be 0 if it is in free fall.
Hence why if you look at a fluid in space the only thing holding it together is the
surface tension of the fluid.
One thing I want to show you is a simulation on the wolfram demonstration site.
On this site there are a number of simulations but the one that is useful in this regard
is on called pressure with in an accelerating container.
You can play around with which direction you want the container to accelerate.
You can see how it affects the free surface of this fluid.
You can also change the density of the fluid.
What that is going to affect as you can see in the screen is the gauge pressure.
The gauge pressure is being reported for this red dot.
What happens when we just move the acceleration?
The gauge pressure is going to change.
It is going to be more drastic towards the outside edges of the container because the
fluid level at this point with respect to that dot changes more drastically then it
does in the center.
It gives you an idea of how the pressure can change any where inside of the fluid when
we have our container acceleration in the horizontal direction.
Hopefully that give you a good idea of what we mean by a rigid body motion.