 We're now going to take a look at a category of fluid mechanics where there is no motion within the fluid and we refer to this area as being fluid statics. So what we'll do we'll begin with exactly what is the pressure distribution like within a fluid with no motion but before we get there let's take a look at what pressure is itself. So that there is a definition of pressure. We see that pressure is a normal stress that acts within a plane of a fluid and in normal that means that it's perpendicular to the plane and it's for a fluid element at rest and the other thing that we need to be aware of is that we will view it as being positive for compression. So what we're going to do we're going to begin by considering a chunk of fluid and we're going to take a look at what the pressure distribution is like within that in order to do this we're going to consider a static wedge of fluid. So I'm going to draw out our chunk of fluid it's a wedge so it's going to look something like this and I'm now going to draw the coordinates that we have and z or z will be in the vertical x will be in the horizontal and what we're going to do is assume that y goes into the plane of the of the screen and that will be width b into screen. Now this chunk of fluid the origin will be here let's assume that it has some mass it will say the weight is dw because we're going to look at a differential chunk of fluid. So that is the weight of the fluid element and I'm assuming that delta x is the size of the wedge in the horizontal and delta z is the size in the vertical. Now the pressure distributions we have px acting in the x direction on the z or z face and we have pz acting on the x face and the last pressure that we have is going to be our normal pressure distribution to the surface length delta s that forms the hypotenuse of this triangle or this fluid wedge and we're going to assume that the angle with respect to the vertical here is theta. So that is the scenario and the fluid wedge also has an angle of theta there through geometry. So what we're going to do we're going to examine the pressure distribution essentially what we're going to do is a force balance in the x and in the z direction but to begin with if we look at the fluid as being stationary which we said it was and if we sum forces in the x direction if the fluid is stationary the fluid is not moving so we say this is zero and that's going to be the pressure on the x surface multiplied by the area the area of the surface that px acts on is delta z times b and then we have to subtract off the component of pressure in this direction here from pn and so using trigonometry we know that pn sine theta that would give us the component in the x direction multiplied by the size of the surface which is delta s times b and then similarly forces in the z or z direction so I have that and I've used a cosine theta to denote the component of pressure which is in that direction now one last thing we have here let me clean up that b it's a little messy okay so that's supposed to be a b and the last thing we have we have to address the fact that the fluid element has weight and that would be acting in the z direction due to the gravity vector so we add that in okay so that is our force balance that we get by summing the forces in the x and in the z direction now what we're going to do we're going to replace sine theta delta s and cos theta delta s and we're going to do that using trigonometry and so what we can write is that delta z is equal to delta s sine theta and delta x is equal to delta s cosine theta and that's just using the angle that we have within the fluid wedge here and delta z and delta x now with that we can take this and we can plug it into our equations down here and remove the delta s sine theta and the delta s cos theta so let's do that in the next slide and see what we get so our x component force balance translates into something that looks like this now delta z delta b it's equates to 0 so this and this go and what we are left with is we're left with px equals pn and similarly for the z direction and that also will equate to 0 so what I'm going to do I'm going to isolate pz and we get this expression here now what I'm going to do I'm going to assume that our fluid wedge becomes infinitesimally small so I'm going to write as delta z goes to 0 we can then write that pz is equal to pn so that is similar to what we had for px is equal to pn and consequently what we can write is that px is equal to pn is equal to pz and what does that tell us it is that pressure at a point in a fluid is independent of direction in a stationary fluid okay so that is the first thing that we're going to look at in fluid statics and we'll continue on