 We're now going to take a look at a form of fluid statics where the object or the fluid itself is in motion but it's in rigid body motion and consequently all the fluid particles are moving at the same velocity in these scenarios or they may be rotating but we're going to take a look at pressure distribution in rigid body motion. So in rigid body motion all the particles are in combined translation and or rotation and consequently there's no relative motion between them so if we look back and take a look at the force balance equation when we did the force balance on a fluid element to derive the hydrostatic equation at the beginning we had this. So when we performed the hydrostatic balance we came up with an equation we had the gradient of pressure on the left and then on the right and this was in the most general sense and then we also had a term here which we said was with the viscous shear so we had the pressure forces we had gravity acceleration and the viscous shear forces this here is equal to zero for rigid body motion and consequently it goes away we don't need to worry about it and what we're left with then is the gradient of pressure is equal to the density times the difference between the gravity vector and the acceleration vector that our fluid may be going through so what I'm going to do let's expand that and when we expand the vector components we have the gradient of pressure on the left and I'm going to rearrange it so we start with the negative of the gradient of pressure so those become the equations that we're dealing with when we have rigid body motion and if you recall typically for the problems that we're looking at we're dealing with the gravity vector as being the Earth's gravity vector and it is negative and so that sometimes simplifies gravity appearing in these different equations here so what we're going to do we're going to take a look at a fairly simple case and and that is the shape of a free surface under uniform linear acceleration you can also look at it under rotation but what we're going to look at is the simplifying case of uniform linear acceleration and for this we have a is the time rate of change of the velocity and this is happening macroscopically so all of the particles undergoing some form of acceleration so what I'm going to do I'm going to draw out a container we'll try to visualize what's happening here originally when the fluid is at rest that is what the surface looks like and I'm going to draw a coordinate system X in that direction and Z in the vertical and normally when the fluid is at rest gravity vector is acting in this direction and we have pressure gradient going in the vertical direction there is no acceleration within our fluid container if we're looking at a case where we do have acceleration I'm going to arbitrarily just assume that we have acceleration something like this and what I'm going to do is break that down into the Z component of acceleration and an X component of acceleration and if we look at the hydrostatic equation let's go back to here if we look at this equation what we have is the gradient of pressure is equal to density times the difference between the gravity and the acceleration and so we have something like this gradient of pressure is proportional to the difference between gravity minus acceleration in the case of fluid at rest there is no acceleration and what we have is gradient of pressure is then proportional to the gravity vector and that's why as we go down in the fluid we have a gradient of pressure and that is going in the vertical direction when we add acceleration things change a little bit and what happens is when we add acceleration our fluid at rest is no longer a horizontal surface but it actually might be an angled surface which could look like this and so that's what happens when we can have acceleration and I will say that the angle of the surface becomes theta with respect to the horizontal and looking at our relationship we have gradient of pressure is proportional to the gravity vector minus the acceleration vector so let's evaluate what that looks like and if we have our gravity vector here and we have an acceleration vector we take the acceleration vector off we're basically going to add these two or subtract acceleration from gravity so what I'm going to do now I'm going to put a negative or a z and then an ax and what we then get is something that looks like this and that would be the resultant that we're seeing on the right-hand side of the equation and another way you could visualize this is that we're just adding the negative of the acceleration vector to the gravity vector and when you add those up you get the vector that we're showing here but the thing to note this is g minus a and what we see is that it is perpendicular to the new surface the new surface that is angled and now what we have that surface might be p equals p atmosphere but then as we go down our pressure gradient is then in this way and that would be a constant line of pressure that would be a constant line of pressure and that would be another constant line of pressure so in this case the gradient is acting in the direction of this g minus a vector and and so with this what we can do using trigonometry we can then figure out what the angle theta is and how that relates to the acceleration of the fluid so what we're going to do we write out tan theta is equal to and theta it turns out by geometry is also this angle here is equal to a x so it's the x component here and it's going to be divided by the vertical which is going to be this and this so when I add those two together we have g oops sorry let me change color we have g plus a z so we get an equation then that we can determine theta and that would be the angle of the surface we find that the angle of the surface is equal to the inverse tan of a x the x component of our acceleration divided by the gravitational constant plus the z component of the acceleration and this equation then specifies the free surface or the angle of the free surface for a fluid undergoing a linear acceleration and so what we're going to do is we're going to take a look at an example problem in the next segment I should apologize and my container was not exactly horizontal it's supposed to be horizontal here but I'm noticing it's at a bit of an angle so I apologize for that that's just the way that I drew the container container was supposed to be flat from the beginning and that was the fluid at rest thing for the free surface that we're looking at so what we're going to do next is we're going to solve an example problem using this and it will be for a container undergoing uniform linear acceleration