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.