Okay gang, lets roll. And welcome back to Analyzing the Universe. Today I want to talk to you about a big problem that astronomy has. And that is determining the mass of astronomical objects. You see astronomy has a big disadvantage when it comes to analysis in general. Why? Because we can't change things. Unlike a terrestrial physics or chemistry lab, where one vary many conditions that might affect a given experiment. Such as voltage, or PH, or a magnetic field. In astronomy, we have to accept the experiments that nature gives us. Thus, fundamental quantities that we take for granted on the Earth, such as the mass of an object and its distance away from us, become exciting challenges for us in the celestial realm. Why is the determination of mass so challenging? Primarily it's because gravity is such a weak force. And the only means by which astronomically sized objects can be measured. Let's examine this in detail. To do this we travel back in time, and visit Europe in the 17th Century. As we shall soon see, time itself was part of the problem. Galileo, and then Newton, were hard at work. The Holy Grail of the nature of forces in motion was being sought. But what is a force? And on what does it depend? Aristotle chimed in first, noting that if a force, or impetus as it was called, was stopped, the object to which the force was applied also stopped. So, it was obvious, the natural state of an object was at rest. Thus, force Is proportional to velocity, force must be proportional to the velocity, since even to maintain a constant speed, you seemed to need a force. And so an error propagated down through the centuries, until the great Galileo in a brilliant series of experiments involving incline planes put forward the positively absurd notion that an object could move forever at a constant velocity with no force applied at all. Uniform, perpetual motion was indeed possible. In the absence of a force, the velocity of an object would not go to zero. But stay constant. How did he show this? Galileo noticed the following. If you have a ball on an inclined plane and the ball rolls down the plane, it always goes back to the same height when it rolls back up the plane. Always to the same height as it started. Now, look and see what he did. If the ball rolls down the plane. And now the plane has a slightly different angle. It will roll further along the plane. Back to the same height as it was initially and you can see where he's going with this now. The ball rolls down the plane. And now, the plane goes up like this, here's the height the ball rolls down, the ball rolls over, all the way over to here, and now for the piece de resistance, he rolls the ball down the plane, and now instead of having any angle at all. That has an upward swing, you can now imagine that if you get that ball rolling all along a plane that's exactly horizontal it will never ever stop. Even Galileo commented that this seems hard to believe Yet this is what his experiments showed. And is one of the first to usher in the scientific age, he believed that the universe must be examined in the light of data, and not understood according to pronunciations by authorities. Be they philosopher like Aristotle, or theologians like Pope Paul V. And so this simple idea, experimentally determined, that motion could be maintained in the absence of a force Finally led Newton to the understanding that force was proportional to acceleration. Force is proportional to acceleration, and mass as the resistance to motion completes the idea of inertia. But what about all those little arrows? What do they mean, and why are they important? To understand this, we need to take a little side street down the road marked vectors. What is a vector? It is nothing more than a mathematical quantity, a number. That has a direction associated with it. The quantity is indicated by a length. This being a small quantity, this being a medium sized one, this being a fairly substantial length. The direction is indicated by an arrowhead. So, if this quantity is pointed in that direction, we draw an arrowhead there. If this quantity is drawn in this direction, we have an arrowhead there. And if this quantity has a direction associated in this fashion, we draw the arrowhead over in that position. Thus, two vectors, A and B, are identical as long as their lengths and their directions are the same. So if this is vector A. This can be vector B, looks pretty good. You might have to use a little bit of imagination there. But the fact that they are written in a different part of the blackboard is irrelevant. However. To choose some other combinations. The vector C, is not equal to the vector D. Even though their directions are the same, because the length of D is not equal to the length of C. And also, a vector E That might be written like this is not the same as the vector F, even though their lengths are the same. The magnitude of E and the magnitude of F might be equal, but the directions are not the same. Now, we define vector addition. By taking the tail of one vector and placing it on the head of another. Drawing the resultant from the beginning of the first one to the end of the second. If this is our vector A. And this is our vector B. What we do is, is we take B and put it in exactly the same direction as it has, preserving its direction. Put it on the head of A, and then, we draw the resultant from the beginning of A, and I'm going to, just, put the, indication that A vector is this big guy on the right here so that we don't get confused. This becomes A plus B. Okay, tail to head, tail to head, tail to head. This is our vector. Now, we can call that vector c, so instead of calling it A plus B let's denote it by another letter C and now you see that in a way we have been able to define subtraction. Look at what happens here. C is equal to A plus B. That means that B is equal to C minus A. So what happens in subtraction is you take the two tails, C, the tail of C, the tail of A, and you connect the heads and that defines your vector B, which is a subtraction. Of C minus A. Okay. So now where is all this going? Why do we even bother with this stuff? The reason we bother with it is because experimentally, it has been determined that force, believe it or not, depends exactly like a vector. It has all of its characteristics associated with it with vector addition and vector subtraction and other operations that we can also define. Let's look at a simple example, a planet moving in a circular orbit around a star or, one star going around another star. The situation is as follows: here is our circular orbit, here's the center where this big mass is supposedly, and here is a place where our planet is located. The position of the planet can be given by a vector, r, and we'll call it r1 because it's a moment in time. A little bit later, the planet has moved its position. To a place r2 at time two. And in fact, these are supposed to be the same, it's supposed to be the same size circle, but, you know. Now you can see that there has been a change in the position of the planet from time one to time two. And, we can see what that change is by just drawing these two vectors, r1 looks reasonable. R2 looks reasonable. And now, as we did before with subtraction, this becomes our change in r. If you go from r1 to r2 you have to add this little amount. Which is delta r. And, the angle between these two positions can be denoted by theta. Now, what's happening as this planet is moving around? Not only is it changing its position, the reason that it's changing its position is because it's moving. And I think you can convince yourself that the velocity of the planet must be denoted by a vector that is perpendicular to the position. Why does it have to be perpendicular, hm? Well, I'm not going to answer that question for you. I want you to see on your own if you can convince yourself that it has to be perpendicular, and it can't be something going in this direction or in this direction. It must be at a right angle to the position R1. We can call that vector, the velocity vector. And just like we have a velocity at time one, we also have a velocity at time two. Also perpendicular to r2, okay? And, if it's in uniform circular motion, the lengths of these two vectors must be the same. If they weren't the same, then the motion, would not be in a circle. Something would be happening, to the position of this point, versus this point, so that you would not be able to get from r1 to r2 if the speed of the object were actually changing. But now let's look carefully at v1 and v2, if we draw v1, over here, and we draw v2 over here. We're going to have in exactly the same fashion as we had a delta r in this situation. We're going to have a delta v right over here. And guess what that angle is, gang? That angle is theta. Why? Convince yourself that this angle has to be the same as this angle for r. So what we are faced with are two similar triangles. In which we can now look at various sides and see if we can come up with a relationship for the acceleration. I want to point out one very important thing. Look at the direction of delta v. Delta v is pointing back in towards the center of the circle. And this will be a key point when we look at how this relates to the gravitational force. So our vector and as v1 gets closer and closer to v2, namely a smaller and smaller interval of time of, is, observed to elapse. Delta v gets closer and closer to being pointed exactly towards the center of the circle. Okay, so let's look at these two triangles, what we're after, is an expression for the acceleration. And the acceleration is the change in velocity divided by the change in time. Now, velocity already contains a time, right? It's just the time rate of change of position. So now we can use these triangles and see if we can come up with a interesting formula for what the so-called centripetal acceleration is. If we look at these triangles, we notice that delta v is to v, we're looking at the part of the triangle here versus one of the legs, and since the triangles are similar, that has to be equal to delta r over r, where now I'm dropping the subscripts here because we're just looking at the magnitude of the acceleration. And now we can work with these particular quantities. You can see that we're going to have to have a delta t in here somewhere, so what we're going to do is, just divide both sides of the equation by delta t. If we put a delta t on this side of the equation, then we can put a delta t on the other side of the equation and still maintain equality. But now, look at this. [SOUND] Delta v delta t is nothing more than our acceleration. And delta r, divided by delta t, is nothing more than, the velocity, or the speed. At least the magnitude of the velocity. So now, you can see that delta r divided by delta t which is equal to V implies that delta V divided by delta t equals V. We're going to bring this V over to this side of the equation. To Delta R divided by delta T is another factor of V. So, we get V squared and we're left with an R in the denominator. And so now we have established that in circular motion, the magnitude of the acceleration is equal to v squared over r. We have also established that by the direction of v being in towards the center of the circle. It's in the opposite direction of our vector r so that we can write the acceleration in general as minus V squared over r. In other words, the magnitude given by V squared over r and the direction given by the minus sign of r in towards the centre of the circle. And that's all we really need to know to be able to determine a lot about the dynamics of planets going around stars, stars going around yet more massive stars, or for instance, the Sun going around the center of the galaxy. And in order to pursue that, we will continue with a discussion of the gravitational force. I want to tell you why it took so long, for us to understand what the nature of forces were. Why did it take till the 17th century before we could figure it out? And the key problem, was the quantity associated with Delta t. Notice, that Delta t appears in both an expression for velocity, and also in the expression for acceleration. How do we measure delta t? Well, we all know how we do it now. We look at our watch and we say one, two, three, four. Or, we have a stopwatch that we can push a button and get evermore accurate divisions of any unit of time that we want. But in the 17th century, it wasn't until around 1750 that Christiaan Huygens, was able to invent an accurate pendulum clock. And all of a sudden we didn't need to use the Earth's rotation or sand dripping in an hourglass in order to figure out what the elapsed time was. So, starting in the 1750s when clocks were invented that gave us enough precision into which we could measure and divide a second. That was when the era began that we were able to actually figure out what the difference was between a result that gave us a velocity or a result that might yield an acceleration.
