0:11

Okay gang, lets roll.

Â And welcome back to Analyzing the Universe.

Â Today I want to talk to you about a big problem that astronomy has.

Â And that is determining the mass of astronomical objects.

Â You see astronomy has a big disadvantage when it comes to analysis in general.

Â Why?

Â Because we can't change things.

Â Unlike a terrestrial physics or chemistry lab, where one

Â vary many conditions that might affect a given experiment.

Â Such as voltage, or PH, or a magnetic field.

Â In astronomy, we have to accept the experiments that nature gives us.

Â Thus, fundamental quantities that we take for granted on the Earth,

Â such as the mass of an object and its distance away from

Â us, become exciting challenges for us in the celestial realm.

Â 1:07

Why is the determination of mass so challenging?

Â Primarily it's because gravity is such a weak force.

Â And the only means by which astronomically sized objects can be measured.

Â Let's examine this in detail.

Â To do this we travel back in time, and visit Europe in the

Â 17th Century.

Â As we shall soon see, time itself was part of the problem.

Â Galileo, and then Newton, were hard at work.

Â The Holy Grail of the nature of forces in motion was being sought.

Â But what is a force? And on what does it depend?

Â Aristotle chimed in first, noting that if a force,

Â or impetus as it was called, was stopped, the

Â object to which the force was applied also stopped.

Â So, it was obvious, the natural state of an object was at rest.

Â Thus, force Is proportional to velocity, force

Â must be proportional to the velocity, since even to maintain a constant

Â speed, you seemed to need a force. And so

Â an error propagated down through the centuries, until the great

Â Galileo in a brilliant series of experiments involving incline

Â planes put forward the positively absurd notion

Â that an object could move forever at a constant velocity with

Â no force applied at all. Uniform, perpetual motion was

Â indeed possible. In the absence of a force, the velocity

Â of an object would not go to zero. But stay constant.

Â How did he show this? Galileo noticed the following.

Â If you have a ball on an inclined plane and

Â the ball rolls down the plane, it always goes back to the same

Â height when it rolls back up the plane. Always

Â to the same height as it started. Now, look and see what he did.

Â If the ball rolls down the plane.

Â 3:29

And now the plane has a slightly different angle.

Â It will roll further along the plane. Back to the same height as

Â it was initially and you can see where he's going with this now.

Â The ball rolls down the plane.

Â And now, the plane goes up like this, here's the height the ball

Â rolls down, the ball rolls over, all the way over to here, and

Â now for the piece de resistance, he rolls the ball

Â down the plane, and now instead of having any angle at all.

Â That has an upward swing, you can now imagine that if

Â you get that ball rolling all along a plane that's

Â exactly horizontal it will never

Â ever stop.

Â Even Galileo commented that this seems hard to

Â believe Yet this is what his experiments showed.

Â And is one of the first to usher in the scientific age, he believed that the

Â universe must be examined in the light of

Â data, and not understood according to pronunciations by authorities.

Â Be they philosopher like

Â Aristotle, or theologians like Pope Paul V.

Â 4:57

And so this simple idea, experimentally determined,

Â that motion could be maintained in the absence

Â of a force Finally led Newton to

Â the understanding that force was proportional to acceleration.

Â Force is

Â proportional to acceleration, and mass as the

Â resistance to motion completes the idea of inertia.

Â 5:29

But what about all those little arrows?

Â What do they mean, and why are they important?

Â To understand this, we need to take a

Â little side street down the road marked vectors.

Â 6:02

medium sized one, this being a fairly substantial

Â length. The direction is indicated by an

Â arrowhead. So, if this quantity is pointed in that

Â direction, we draw an arrowhead there. If this quantity

Â is drawn in this direction, we have an arrowhead there.

Â And if this quantity has a direction associated in

Â this fashion, we draw the arrowhead over in that

Â position. Thus, two vectors, A and B,

Â are identical as long as their lengths and their directions

Â are the same. So if this is vector A.

Â 6:54

This can be vector B, looks pretty good.

Â You might have to use a little bit of imagination there.

Â But the fact that they are written in

Â a different part of the blackboard is irrelevant.

Â However. To choose some other combinations.

Â 7:34

Even though their directions are the same, because the length of D

Â is not equal to the length of C. And also, a vector

Â E That might be written like this is not the same

Â as the vector F, even though their lengths are the same.

Â The magnitude of E and the magnitude of F might be equal, but the directions

Â are not the same. Now, we define vector addition.

Â By taking the tail of one vector and placing it on the head of another.

Â Drawing the resultant from the beginning of the first one

Â to the end of the second. If this is our vector A.

Â 8:48

in exactly the same direction as it has, preserving its direction.

Â Put it on the head of A, and then, we draw

Â the resultant from the beginning of A, and I'm going to, just, put the,

Â indication that A vector is this big

Â guy on the right here so that we don't get confused.

Â This becomes A plus B.

Â Okay, tail to head, tail to head, tail to head.

Â 9:32

This is our vector. Now, we can call

Â that vector c, so instead of calling

Â it A plus B let's denote it by another

Â letter C and now you see that in a way we

Â have been able to define subtraction.

Â Look at what happens here. C is equal to

Â A plus B. That means

Â that B is equal to C

Â minus A. So what happens

Â in subtraction is you take the two tails,

Â C, the tail of C, the tail of A, and you connect the heads

Â and that defines your vector B, which is a subtraction.

Â Of C minus A. Okay.

Â So now where is all this going? Why do we even bother with this stuff?

Â The reason we bother with it is because

Â experimentally, it has been determined that force, believe

Â it or not, depends exactly like a vector.

Â It has all of its characteristics associated with it with vector

Â addition and vector subtraction and other operations that we can also define.

Â 11:13

Let's look at a

Â simple example, a planet moving in a circular orbit around a star

Â or, one star going around another star. The situation is

Â as follows: here is our circular orbit,

Â here's the center where this big mass is supposedly,

Â and here is a place where our planet is located.

Â The position of the planet can be given by a

Â vector, r, and we'll call it r1 because it's a moment in time.

Â 12:17

time two.

Â And in fact, these are supposed to be the same,

Â it's supposed to be the same size circle, but, you know.

Â Now you can see that there has been a change in

Â the position of the planet from time one to time two.

Â And, we can see what that change is by just drawing these two vectors,

Â r1 looks reasonable. R2

Â looks reasonable.

Â And now, as we did before with subtraction, this

Â becomes our change in r. If

Â you go from r1 to r2 you have to add

Â this little amount. Which is delta r.

Â 13:22

Now, what's happening as this planet is moving around?

Â Not only is it changing its position, the

Â reason that it's changing its position is because

Â it's moving. And I think you can convince yourself that

Â the velocity of the planet must be denoted by

Â a vector that is perpendicular to the position.

Â Why does it have to be perpendicular, hm?

Â 13:54

Well, I'm not going to answer that question for you.

Â I want you to see on your own if you can convince yourself that it has

Â to be perpendicular, and it can't be something

Â going in this direction or in this direction.

Â It must be at a right angle to the position R1.

Â We can call that vector, the velocity vector.

Â 14:26

And just like we have a velocity at

Â time one, we also have a velocity at

Â time two. Also perpendicular

Â to r2, okay? And, if it's in

Â uniform circular motion, the lengths of these

Â two vectors must be the same. If they

Â weren't the same, then the motion, would not be in

Â a circle. Something would be happening, to the

Â position of this point, versus this point, so that you would not be able to get

Â from r1 to r2 if the speed of the object

Â were actually changing. But now let's look carefully at v1

Â and v2, if we draw v1,

Â over here, and we draw v2

Â 16:05

Convince yourself that this

Â angle has to be the same as this angle for r.

Â So what we are faced with are two similar triangles.

Â In which we can now look at various sides and

Â see if we can come up with a relationship for the acceleration.

Â 16:50

And this will be a key point when we

Â look at how this relates to the gravitational force.

Â So our

Â vector and as v1 gets closer and closer to v2, namely a

Â smaller and smaller interval of time of, is, observed to elapse.

Â Delta v gets closer and closer to being

Â pointed exactly towards the center of the circle.

Â Okay, so let's look at these two triangles, what we're after,

Â is an expression for the acceleration. And the acceleration is

Â 17:32

the change in velocity divided by the change in time.

Â Now, velocity already contains a time, right?

Â It's just the time rate of change of position.

Â So now we can use these triangles and see if

Â we can come up with a interesting

Â formula for what the so-called centripetal

Â acceleration is. If we look at these triangles, we notice

Â that delta v is to v, we're looking at the part

Â of the triangle here versus one of the legs, and since the triangles

Â are similar, that has to be equal to delta

Â r over r, where now I'm dropping the

Â subscripts here because we're just looking at the magnitude of the

Â acceleration. And now we can work with

Â these particular quantities. You can see that we're going to have to

Â have a delta t in here somewhere, so what we're going to do

Â is, just divide both sides of the equation by delta t.

Â 18:56

If we put a delta t on this side of the equation, then we can put

Â a delta t on the other side of the equation and still

Â maintain equality. But now, look at this.

Â [SOUND]

Â Delta v delta t is nothing more than our acceleration.

Â 19:18

And delta r, divided by delta t, is

Â nothing more than, the velocity, or the speed.

Â At least the magnitude of the velocity.

Â 19:35

So now, you can see that

Â delta r divided by delta t which is equal to V

Â implies that delta V divided

Â by delta t equals V. We're going to bring

Â this V over to this side of the equation. To Delta R

Â divided by delta T is another factor of V. So, we get V

Â squared and we're left with an R in the denominator.

Â And so now we have established that in circular motion,

Â the magnitude of the acceleration is equal to v squared over r.

Â 20:25

We have also established that by the direction of v

Â being in towards the center of the circle. It's

Â in the opposite direction of our vector r so that we can

Â write the acceleration in general

Â as minus V squared over r.

Â 20:52

In other words, the magnitude given by V

Â squared over r and the direction given by the

Â minus sign of r in towards the centre of the

Â circle. And that's all we really need to know to

Â be able to determine a lot about the dynamics

Â of planets going around stars, stars going around yet more massive

Â stars, or for instance, the Sun going around the center of the galaxy.

Â And in order to pursue that, we will

Â continue with a discussion of the gravitational force.

Â 21:45

I want to tell you why it took so long, for us to

Â understand what the nature of forces were. Why

Â did it take till the 17th century before we could figure it

Â out? And the key problem, was the quantity

Â associated with Delta t. Notice, that Delta

Â t appears in both an expression for velocity, and

Â also in the expression for acceleration. How do we

Â measure delta t? Well, we all know how we do it now.

Â We look at our watch and we say one, two, three, four.

Â Or, we have a stopwatch that we can push a button and get evermore accurate

Â divisions of any unit of time that we want.

Â But in the 17th century, it wasn't until around 1750 that Christiaan

Â Huygens, was able to invent an accurate pendulum clock.

Â And all of a sudden we didn't need to use the Earth's

Â rotation or sand dripping in an hourglass in order to figure out

Â what the elapsed time was.

Â So, starting in the 1750s when clocks were invented that gave us enough

Â precision into which we could measure and divide a second.

Â That was when the era began that we were able to actually figure out what the

Â difference was between a result that gave us a velocity or

Â a result that might yield an acceleration.

Â