Brahe, Kepler, and then Galileo have made great strides in taking these celestial
objects and studying them as physical concepts that we can measure, and
discuss, and hopefully understand. But the real understanding waited a few
more decades for, as I promised, Issac Newton.
We're going to spend the next few clips following in the steps the, rather large
steps of Sir Isaac, and seeing what it was that he taught us.
And we're going to start at the beginning with mechanics, the science of motion,
which is going to underlie everything else.
We're going to do some physics, bear with me.
It's going to be intense. To study motion,
the first thing we need to do is make mathematically precise what it is that
Galileo said when he said, an object retains its state of motion.
So mathematically, an object's state of motion is represented by an object's
velocity. Velocity is a, physics concept.
It means the, speed with which something is moving, along with its direction.
So, we denote it, by this V with an arrow on top of it.
V stands for velocity and the arrow reminds us that this is a number, the
speed at which something is moving, along with the direction, so that, this ball
can be moving at one meter per second, to your, left or my left and at one meter
per second to my right or up or down or in or out.
And those, even if the speeds are the same, are different values of the
velocity. Velocity can be thought of, as we know,
speed is measured in meters per second because it is the change in position,
where you are, the distance divided by time.
And velocity can change. The ball can start slowly and speed up.
And its change is called the acceleration.
So acceleration is the rate at which velocity changes, and acceleration is
measured therefore, in meters per second. The difference, in velocity between the
beginning and the end of an interval, divided by the length of the interval.
The same way that velocity is distance divided by time, acceleration is
velocity, per time. And so the rate of change of velocity is
an acceleration, and is measured in meters per second squared, and like
velocity, acceleration has a direction. So that, if an object is moving to the
left and speeding up, then its acceleration points to the left.
If an object is moving to the left and slowing down.
That means its velocity points to the left but its acceleration points to the
right, because the velocity is less to the left later in the interval than it
was before. So, acceleration, like velocity, has a
direction. And, in addition, to expressing the fact
that an object can be speeding up or slowing down, there can be a change of
velocity that is associated with maintaining the same speed, but merely
change in direction. If an object is moving to the left and
starts moving towards your camera, then this is change in velocity even if speed
never changed and particularly useful example for us in studying astronomy is
going to be in fact, of motion without changing the speed.
Let's look at it and may be it will help clarify these concepts a little bit.
Example you're going to want to study is the example of an object, say maybe a
planet, moving in a circle about some center, say the sun.
At a constant speed, so in this first pane, you see the, black dot which is our
object. And it is set so that it is moving around
the circle, at a constant speed. Its speed is not changing.
Its velocity however is clearly changing, because if I freeze the animation here
the velocity of the object, manifestly, is.
Pointed in this direction, it's moving that way.
Whereas, if I let the animation go a little farther, and freeze it again here,
the object is moving at the same speed but in a different direction.
And therefore, there is definitely a change in speed, and, to, quantify and
understand that change in speed, we will move down, to this pane, where, I have
put in, the object's velocity. It is as an object with a direction so
we, we draw it with an arrow, and the arrow at the beginning points up because
that's the direction in which the object is moving.
And as the object moves, the velocity changes because, while the arrow is of
the same length its direction is continually changing.
So this is the way your velocity changes when you move around a circle at constant
speed. Coming down here I've got two circles.
The circle along which the object is moving and then this imaginary circle.
And if you look what the imaginary circle does is it I've copied the blue arrow
over to here and drawn it so that its tail is always at the origin.
So the edge of this arrow is giving me the direction and, of course, the
constant magnitude, constant size, of the velocity vector.
when the arrow points to the right the object is here and it's moving to the
right and so on. So this is my velocity vector V.
Now I can understand how to measure acceleration.
Acceleration is going to be the rate of change of this vector so that if I freeze
the animation at any given point I can see what's going on.
The object is over here, it is moving in this direction, that's why this velocity
vector points this way, but I know where the velocity vector is going.
It will next move over here to the right, so the vector that measures the change in
velocity will point along the circle just as the vector that measures the change in
position points along this circle. To make that clear, here's the same two
circles and on this here on the velocity circle I've added a green arrow to
represent the change and velocity. And as they play you can see that the
green arrow is constant in magnitude because the velocity is rotating at a
constant rate just because this thing is rotating at a constant rate.
And the green arrow points along the velocity circle and what this green arrow
is in fact is our acceleration. It's the rate of change of the velocity.
And so to make that explicit, I've here, done three circles.
This is the position circle, this is the velocity circle, and over here we have
the acceleration circle or we have taken the green arrow the rate of change of
velocity, and move its tail to the center, so that at any time, this gives
me, the rate at which, the velocity is changing.
So you can play with all three arrows and I'll post this file so you can try to run
it. The main thing I want you to notice here
is that if you look there's a relation between the direction of the acceleration
arrow and of the position arrow. Shen the position arrow points here into
the upper left the acceleration is to the bottom right.
And that's not a coincidence velocity is 90 degrees to position.
Acceleration is 90 degrees to velocity adding up to 180 degrees away from
position. So when your out here your acceleration
points that way and if I move it and freeze it again you'll see that this is
always maintained. When your at this position your
acceleration points in that position. So the acceleration at any given point is
in fact at any given time is in fact pointing directly to the center of the
circle. If you ask in which direction this object
is moving it's moving along the circle. In what direction is it accelerating?
That way towards the center. So we found that moving in a circular, in
circular motion at a uniform speed, you're experiencing an acceleration at
constant magnitude always directed towards the centre of the circle.
Now if, it's important for us to understand this circular example better,
so let's go through it. If the radius is R, and the speed V, I
want to know. We know the direction of your
acceleration. What is its magnitude in meters per
second squared? So I'm looking for an acceleration in
meters per second squared and it needs to be determined by.
What is there to determine? The fact that you're moving around a
circle of a given radius at a given speed.
And we all have an intuition, that if you're going faster around a curve you'll
need to accelerate more, to take the curve.
And if a curve is a tight curve, you need more of an acceleration.
So let's see if this is borne out. And we'll use a trick physicists use,
it comes from units. We're going to do what we call
Dimensional Analysis. We need to find some combination of R and
of v. That could in principle be an
acceleration. And it turns out there's only one because
if you think about it R is measured in meters, it's a length.
v being a velocity or a speed, is measured in meters per second.
We need to fashion out of these something that could potentially equal a.
And it's clear that the only way we're going to get seconds squared in here is
to take the square of the velocity. So write the square of the velocity, this
is a good idea but that is, has units of meters squared per seconds squared
because it's the product of two things with units meters per second, so that's
not quite right. This does not have the right dimensions
but we need to clean out the meters, and to clean out the meters, we can simply
divide by R, which is measured in meters. Now dimensional analysis doesn't always
give the exactly right answer. Eh, if the acceleration could easily have
turned out to be 3 Pi * v^2 / R. And the 3 Pi would not show up in
dimensional analysis. In the case at hand it turns out that
there is no constant, you can do the calculation, takes little
bit of effort, but, we're not going to do it.
The answer, is that we got the right answer for, precise, correct, expression,
using just this dimensional analysis. So the acceleration towards the center of
a circle, is given by v^2 / R. It has the fancy name, centripetal
acceleration. Because it is towards the center.
And this will show up often in what comes, in what comes next.
We now have a nice way to state Galileo's principle of inertia.
An object upon which no external influence acts, moves with constant
velocity or, equivalently, has zero acceleration.
Now we step it up. What happens when an external influence
acts? And Newton introduces a brilliant concept
here which is the external influence is quantified by something called a force.
What is a force? It's a tricky concept.
A force is that which causes acceleration.
The idea that Newton introduces that is that you can imagine applying the same
force to two different objects. And, we all have that intuit of idea.
I can give an identical push to a tennis ball and to a refrigerator.
And, we both, all have the intuit of idea that we'll get more of a motion out of
the tennis ball that, that of the refrigerator, because the refrigerator is
heavier or technically the refrigerator has a greater, has a larger mass.
And so, Newton quantifies this intuition in the idea that indeed there's such a
thing as a force The, something that measures the amount of external influence
being applied to an object, and the object's acceleration is related to this
force by a property of the object itself, known as its mass.
And the mass is roughly the amount of stuff in the object, all counted,
weighted appropriately. It's measured in kilograms.
And it roughly, an object that is massive is an object that is heavier.
We'll make that precise in a second. And force like acceleration has a
direction. You can push something to the left or
push it to the right. And so we have this new physical concept
of force. We need to understand what units we
measure force in. And that follows directly from this
equation, a force of one unit is a force that causes a mass of one kilo to
accelerate at one metre per second squared.
So the unit of force is this unit, kilogram times metre per second square.
And it is designated of course with the name newton, and denoted by an N.
So that would be a force of one N. Before we go on to clarify some details
here let's go onto Newton's third law. This was Newton's second law.
Newton's third law explains why there has to be an external agent applying the
force. And this is crucial.
Newton tells us that universally, when an object A is applying the force to
B, and the force is given by F. Which means B is being accelerated.
Something is applying the force. Go find the thing that is applying the
force. Then B, automatically, it reacts on A, we
call it. And that means B applies to A.
A force, that is equal in magnitude and opposite in direction.
We write it mathematically as negative F. Negative a vector means a vector of the
same size pointing in the opposite direction.
What this means is that when you push on the refrigerator the refrigerator pushes
back on you. This is not an action by the
refrigerator. This is not some act of will.
It's Newton's third law in action. Now, let's do an example of this.
it was known already to Galileo that, if we let loose some object, we all know
this, if I let something go, this ball right here, then an external force will
act on it. How do I know?
Because, the ball has an initial velocity of zero and clearly it does not maintain
the initial velocity of zero. It begins to move downwards, so it has a
downward pointing acceleration. And, in fact this acceleration was
measured by Galileo Is that people after him and it is, has the interesting
property that it is uniform for all objects.
So any object in the vicinity of earth accelerates down, the velocity of the
acceleration is called the acceleration of gravity and in our fancy units, it's
about 9.8 meters per seconds squared. This is my son's ten kilo shot put ball.
And when I hold this I'm preventing it from falling down with the standard
acceleration of gravity by applying a force directed upwards to the ball.
Newtons third law then tells us that if I'm applying a force to the ball point
directed upward then the ball is clearly applying a force to my hand directed down
and believe me I can feel it. Of course gravity is going to play an
important role in what goes on. It's also a force that acts all around us
so let's Quantify what we said, the weight of an
object is defined to be the force gravity applies to it.
We know that objects on earth fall with an acceleration, that I told you, 9.82
meters per second squared. So, the force of gravity on any object is
directed always down. It's, given by the mass of the object
times this constant, which is why we confuse mass with weight but separate the
two and there is a property of the object g is a property of earth.
We often say that I weigh 59 kilos. A more correct statement is that my mass
is 59 kilos. My weight here on earth is mg which is 59
kilos times 9.82 meter per second squared or doing the calculation, I believe I did
it right, 579 kilos times meter per second squared which is 579 Newtons.
My weight is 579 Newtons. We'll play more with,
The specifics of Newton's laws, but the first thing we need is to understand
some, very deep and important, consequences that are mathematically
derived, and Newton derived them from Newton's laws.
And, the most important of these take the form of thing called conservation laws.
There are quantities that you can define for a physical system that do not change.
So, let's look at this. And, the first conservation law Involves
this quantity called momentum. We define another vector called momentum.
It's essentially the velocity vector, but multiplied by mass so, roughly it
measures, the amount of umph, the amount of inertia some people say that the
object pass a, refrigerator moving at two meters per second, has, more momentum,
than a tennis ball, moving at two meters per second.
The reason this is important, is because, F being,
satisfying Newton's law, let me rewrite it again here.
Newton's second law, F is mass times the rate of change of the velocity.
Since mass does not change, it's a property of the object, F could be
thought of as the rate of change of this momentum thingy.
And so, if F is the rate of change of the momentum, we have something very deep.
Imagine that there are only two objects in the universe but there are forces
between them. So A acts on B and the velocity of B
changes and B acts on A and the velocity of A changes.
What we know, though, is that the force that A applies to B, whatever it is, is
equal in magnitude and opposite in direction to the force that B applies to
A. Which means that the rate of change of
A's momentum is equal in magnitude and opposite direction to the rate of change
of B's momentum with the consequence that the rate of change of the total momentum,
add up these two vector object, the total momentum cannot change.
Now add more objects and the same principle extends.
There is in the universe a total. Conserved object, the total momentum of
the universe and that cannot change. Moreover, if you could isolate some
subset of the universe so that it doesn't interact, there are no external force on
it, then the momentum of that little group of objects is completely conserved.
The other forces can do is allow objects to exchange some momentum, some momentum
can be transferred from one object to the other so that the momentum of the two
objects change but their total is unchanged and we say that momentum is
conserved. We call this a conservation law and such
conservation laws are critically important.
Let's take a look at what this looks like.
This is a toy appropriately enough called Newton's Cradle.
It has, a row of five steel balls hanging from,
framework and we will pull one ball off to our left so that when I release it, it
will impact the real balls with the horizontal momentum pointing to the right
and let's see what happens. What happens is that the incoming ball
comes to a complete halt, transferring all of its horizontal momentum to the
next ball down the line, which transfers it to the next, and so on, until the last
one leaves the pile at essentially the same momentum as the one with which the
initial ball came in. We can repeat the same process with two
balls and see a similar interesting process.
We can do three, and if we are ambitious, even four balls, and reproduce the same
results. I hope I convinced you that we can see
mathematically where Newton's laws lead to momentum conservation very directly.
A little more math, that we're not going to follow through, shows that there's
another way to find a conserved quantity that is associated deeply with circular
motion. So it involves picking a center for
motion and it's called angular momentum because circles are parameterized by
angles. And it's a quantity that is given, it's
related to momentum, here's our friend momentum, but it's the momentum times the
radius of a circle in which one moves. And what one finds is if you have a
collection of objects that all are moving together, you sum up, just as with
momentum, the total angular momentum. The mass of each of them times the
velocity of each of them times the radius at which each of them is moving around
the circle. This is assuming everybody is moving in a
circle. There are more complicated expressions in
another cases. Then this angular momentum is also
conserved in that it can be traded between different parts of a system but
the total is conserved. This is going to be extremely important
to us because things in space tend to spin.
So let's demonstrate that with our valiant demonstrator.
Standing on a platform and holding out some weights, I have my friend Derek
start me spinning slowly, and what we see is that as I pull the weights in they
have a significant factor of m and I'm making their r smaller.
My angular momentum is conserved by making all of me spin faster and I can
control my speed when I pull my arms out I slow down, when I pull them in, I speed
up again. This is a great demonstration of the
conservation of angular momentum, if not perhaps of exceptional physical grace.
There's another conservational law that you can show,
follows from Newton's equations. Imagine an object, upon which the only
force that acts is gravity like this, rubber ball when I throw it up and down.
So as long as the ball is in the air essentially gravity is the only force
acting. And what we know is that when I throw it,
as it moves up it'll slow down, and then as it comes down it'll accelerate, moving
down. This is formulized in the following
statement. It turns out, that if you form the
combination m times g times h where m times g is the force that gravity applies
and h is the height of the object above something.
My hand, the floor, it doesn't really matter what, we'll show why.
And if you take this quantity which we call gravitational potential energy and
add to it this combination of speed and mass m * v^2 / 2, this combination is
called the object's kinetic energy. Then the sum of both of these is
constant. As h increases the object slows down
because, that's how gravity works, and as it falls down, it speeds up, and this
mathematically expresses this. Notice that it's clear from here why I
could have measured the height from the floor or my hand, or the bottom floor of
the building it doesn't matter, that just adds a constant that never changes.
So, adding a constant to the energy, until very late in this class will be
completely irrelevant. Now this is very nice but it's only true
if the only force acting is gravity and in general there are other forces that
act. And so in general this conservation of
energy is violated. But not really.
What really goes on is that whenever there is another force other than gravity
that is acting, a typical example is friction.
If I roll my chair back, it slows down. I move it, it had kinetic energy and the
energy disappeared. Where did the energy go to?
Well, if you listened closely, you could have heard that some of it turned into
sound energy. It's also true that there is friction in
the bearings of the wheels, and that converts some of the energy to heat.
Heat is a form of energy and friction is the force that translates,
converts kinetic energy very happily in the heat.
So there are many different forms of energy: sound, light, heat, chemical
energy, electric energy, nuclear energy. We will talk about all of them in turn
they'll all show up. When you add them all up together it
turns out, in any process, the total energy.
Is in the universe is conserved. And again, if you isolate a chunk of the
universe from the rest, then the total energy in that chunk is conserved.
Energy is neither produced nor destroyed. this is, might come as surprise to all
the politicians who talk about the need to produce energy or conserve energy.
Both of those are political terms, but scientifically energy can neither be
produced nor destroyed. And it can be conserved, in fact that's
always is. It's a very important concept to us.
Let's remember the units in which we measure energy.
So They follow from this equation, energy is measured in units of kg * m^2 / s^2.
If you plug it in you'll see that both of these terms have the same units, which is
good because otherwise adding them up would make no sense.
And, this is dignified by the name joule, and indicated by a J.
So this is joule. And, it's hard to have exactly a sense
for how much a Joule is. So, let me help you by perhaps suggesting
the following idea. our body, to push this chair back and
forth to move my arms I am producing kinetic energy by converting chemical
energy from food I ate. And, we measure the energy content of off
food in calories. We all have some sense of what calorie
is. So one Joule is about 4.2 calories.
Fun demo of conservation of energy is this bullying bar pendulum.
I am holding it up against my face, its got potential energy because the angle
elevates it from the floor. I release it, it requires potential
energy swinging to the other side converting it back to potential, and when
it comes back, if energy is conserved, it will not go any higher than it did before
and therefore will not smash my face. Let's try it.
This is it. Those are Newton's laws.
I don't know if you appreciate what I mean by this is it but by the end of this
class I think you'll have a deep understanding of this.
In a very real sense, this is all of science.
Certainly all of physical science. the equation that governs science is F =
ma. The rest of, certainly a century of
physics, is figuring out the details in the sense of figuring out m, what kind of
objects are in the universe? What do forces act on?
And figuring out F, what are the forces between various objects?
One way I want you to think about this equation, fancy name is a differential
equation, but what this really means is it's a prescription for figuring out what
the universe will do next at any given moment.
And the way it works is this. If we know the forces that act on things,
and we happen to know at some instant where all of the relevant objects in our
system are. And because this only determines the
acceleration, we need to know how they're moving, because if a ball is here,
clearly there are several different motions it can do under the influence of
gravity. It could just fall.
Or if it is initially moving up, it will go up and then go down.
But once you know where it is to begin with and which way it's moving, then
gravity takes over and tells you the rest.
In other words you can take the positions and velocities in a given instant, use
those to figure out where the forces are, perhaps where it is might influence what
forces act, is it touching my hand, is it not touching my hand.
That allows you to figure out the accelerations.
Those tell you in turn how the velocities will change.
So you can figure out the velocities an instant later.
Using those velocities, you can now figure out the new positions, figure out
the new forces and repeat this process is called solving a differential equation
and in essence what the universe does is that it solves Newton's equation all the
time. What this allows you do, do is that if
you have some knowledge of what we call the initial data, all of the position and
velocities of the parts of the system at any given instant, you can predict what
everything is going to do into the infinite future and also you can roll the
clock backwards and figure out where everything was at any given time all the
way into the indefinite past. And as I said, most of a century of
physics goes into filling in the details of F and m and applying Newton's laws in
various situations and finding the consequences.
Let's start that process with the force that's most important to us, which is
gravity, in the next clip.