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So now that we know the S.I. system is the choice system to use, let's look at how the
primary units of time, length and mass are defined in the S.I. system. Time, for example.
You know, before the 1960's, the standard of time in the S.I. system, which is one second,
has to do with what's called 'mean solar day'. So basically we take what is called a mean
solar day and divide this into 24 pieces, that gives you one hour and each hour is divided
into 60 minutes and each minute divided into 60 seconds. But of course, that is not all
very precise because it depends on, you know, how the earth moves around the sun, that sort
of thing, which is fairly constant but not quite that constant. Then, in 1967, a new
definition came upon for seconds and that has to do with the famous atomic clock. You
know, what they did with, they chose a particular atom called cesium-133. This particular atom,
is used as what's called a reference clock. So what's a second? Well, there is a particular
radiation coming from the cesium atom and it vibrates at a very, very high frequency,
okay, many, many times a second. And so what you do is, you take this vibration, this particular
vibration comes from the cesium atom which will never change with time, it's very very
much a constant and you say,"Okay, this vibration is very very fast so the period of this vibration
is extremely short, its a very very small fraction of a second." What fraction? Nine
billion, one hundred ninety two thousand, 6 hundred ninety, thirty one... well, I lost
track actually. It's nine billion, one hundred ninety two million, six hundred thirty one
thousand, seven hundred seventy (9192631770) of a second! Actually, that's in your text
book, you don't have to remember these particular numbers, but just remember, the definition
of a second is now tied to the vibration of a certain spectral line from this particular
atom called cesium-133 and this highly accurate definition will never change with time. It's
very, very constant and the measurement of time using an atomic clock is the most precise
measurement human beings have learned to use, it's highly, highly accurate. Now, what about
length? The measurement, the standard of length in the S.I. system, historically, what the
French did, to establish the unit system was they took a certain metal bar, they took a
certain metal bar and they precisely cut this metal bar at both ends and they said,"Alright,
the distance between this end and that end of the bar is called a meter!" And this metal
bar still exists today, it is housed behind a double glass jar in the international bureau
of standards, outside Paris. And that definition, which is, you know, the one meter distance
between this end of the bar and that end of the bar, was kept until the 1960's. But you
think about it, this measurement, this particular definition, is not all that precise, right?
Because, first of all, you can say, the length of the bar can actually change with temperature
and pressure right? Okay, you say, let's fix the temperature and fix the pressure. Even
so, do you really think the two ends of the bar are precisely sharp so that the distance
between one end and the other end is well defined? Well, not under a microscope! Isn't
that so? If you take a powerful microscope, you find, this edge, that edge, it's not really
flat and smooth, rather it's not well defined at all. It's rugged, if you look at the situation
from a microscope. So you see, that definition is not very precise at all. Plus, there is
one more problem and that is, it is not easy to duplicate a bar. Now what if, for what
ever weird reason, god forbid, something happened to that bar so that the world wakes up the
next day and finds, gee, that bar is gone! We cannot find it anymore! Does that mean
we're going to lose that scientific definition of one meter permanently? Well, that would
be a disaster right? Then everybody can claim now, I can define a meter, you can define
a meter and then there would be chaos right? If you do not have a standard, universal definition
of a meter, for length, that of course, would be trouble. So, let us redefine one meter
such that, first of all, it is scientifically accurate and secondly, it is easily reproducible.
It does not depend on one particular bar or one particular piece of substance that's well
kept, no matter how well kept it is, it's still dangerous. So believe it or not, the
length unit one meter, the most recent scientific definition was as close as the year 1983.
There is an international committee who's sole job is to refine the definition of these
basic units. Okay, so in 1983, October 1983, the definition of one meter was rewritten
and this time, it is tied to a universal constant of nature, the so called, 'speed of light'
in vacuum. As you will learn from Einstein's theory of special relativity, the speed of
light in vacuum is a universal constant, it's the same here, it's the same, everywhere in
the universe, it never, ever changes. Now, the speed of light is a constant and light
travels roughly about 300,000,000 meters per second, roughly. It's a little bit less than
that. So roughly speaking, what's a meter? A meter is the distance traveled by light
in the very very small fraction of a second, right? 'Cause see, in every second if moves
about 300,000,000 meters and therefore, one meter is the distance traveled by light in
roughly 1/300,000,000 of a second. Of course you don't want to use 300,000,000 even because
that will conflict with the original definition of a meter, somewhat. So, you want to be more
precise, right? So, the more precise number, defined in 1983, is that a meter is the distance
traveled by light in vacuum during not exactly 1/300,000,000 of a second but rather 1/299,792,458
of a second. So it is tied to the speed of light in vacuum, which is constant. Why is
this a better definition? Well first of all, it is tied to a universal constant, which
is the speed of light in vacuum so you never have a problem of reproduction, you can always
recover that definition. It is not tied to a particular metal bar or anything like that,
or anything on this planet for that matter, it's a universal standard. Secondly, of course
in order to measure one meter, you must be able to measure a very very tiny fraction
of a second, in this case, like 1/300,000,000 of a second, but the measure of time as I
said, is the most precise measurement human beings can ever take with today's technology.
So this is not only universal but also very practical because we know how to measure time
very precisely. So this gives us the best definition of one meter. What about mass?
Well, originally, the French, during the establishment of the S.I. system, they made a specific platinum
uranium alloy cylinder and they said,"Alright, the mass of this cylinder is called, is defined
at one kilogram." And this cylinder today, is still kept at the International Bureau
of Weights and Measures, outside Paris. And believe it or not, this is still the definition
of a kilogram, currently, in the S.I. system. But of course you say, well, that would be
a problem right? But we're not going to worry too much about it, this cylinder has been
duplicated around the world and secondly, we have some sort of auxiliary measurement
for mass. We can, for example, take 12 grams, we take a large number of carbon 12 atoms,
very common carbon, carbon-12, the most common type of carbon, and we take roughly 6.022
x 10^23, we take that many carbon atoms and we put them all together, we call the mass
of that many carbon-12 atoms 12 grams, exact. So that gives us some idea how to find a gram.
Once we know how to find a gram, of course, we know how to find a kilogram. So even though,
the primary definition of one kilogram is still tied to a particular cylinder, there
are other ways to recover that definition if there's anything that can happen to it.
Okay, the definition of these basic units, like what is a second? What is a meter and
what is a kilogram? We don't really think about it right, usually? It doesn't mean these
are easy to obtain, okay? The definition has to be very precise and it also has to be easily
duplicatable and it's not always easy to achieve these things. That's why scientists spend
decades and actually centuries, trying to define these definitions. But we do have precise
definitions for these base units of course, based on that, we can also have derived units.
What is a derived unit? Let me give you and example. Primary units, that of... in mechanics,
that of length, time and mass. What about a derived unit? Well, for example, speed.
It is measured in meters per second, so you run at 3 meters per second so every second
you move forward by 3 meters. Your speed is 3 meters per second, that unit of speed is
the unit of length divided by unit of time so it's a combination of primary units, that's
called a derived unit. So with 3 primary units, meters, kilograms and seconds, we can get
a whole bunch of derived units. That of speeds, acceleration, force, momentum, all of these
things that we can find later, so we don't just have 3 physical quantities in mechanics
of course, we have many, many quantities but they can be all taken care of with derived
units. Again, there are only 3 primary units in mechanics; meters, kilograms and seconds,
in S.I. system that is. Alright, let us do a little review of what's called scientific
notation and orders of magnitude. It is clear that in physics we are often dealing with
a very, very wide range of quantities in terms of magnitude. Sometimes we deal with very
large quantities, sometimes very, very small quantities and you learned before that when
you have to deal with very large or very small quantities, it is a good idea to write that
quantity in what's called scientific notation. An example, the radius of the earth is about
4,000 miles. Now to be more precise, you want to write it in meters. It's about 6.37 x 10^6
meters. That is an example of scientific notation. What is the advantage of scientific notation?
Well if you don't do that, you have to write 6,370,000 meters. Why is this a better notation?
Well, a couple of things, first of all, you have to count how many zeros you have, 4 zeros
so this is, six million, three hundred seventy thousand meters. So here you don't have to
count, immediately you know it's 10 to the 6, it's a million, it's roughly 6 million,
6.3 million, immediately you know how big it is. It's the, order of magnitude, 10^6
meters, immediately you know this from the scientific notation, you know immediately
how big, how small it is. You don't have to count all the zeros. That's not a very large
number. If you look at, for example, how many meters light can travel in a whole year, that's
called a light year and you have to deal with a lot more zeros right? It's difficult to
count so that's why we use scientific notation. Another advantage of scientific notation is
you can immediately see how accurate the measurement is. For example, if you write it this way,
you never know whether this zero is accurate, that zero is accurate, what ever. Do you really
know the radius of the earth down to a meter or ten meters? You know it doesn't make any
sense right? It's not possible to measure the radius of the earth that accurately, but
with this notation it's clear that you know these three significant figures, 6.37. You
don't know what the next one is so your accuracy is down to .01 x 10^6 meters is about 10,000
meters or 10 kilometers, that makes sense. So you know how big it is and you know the
accuracy immediately by using scientific notation. Let's look at a couple examples how we might
estimate the orders of magnitude. One of the most interesting cases is to find the number
of seconds in a year. This is not hard to do, we can easily do that. First of all, how
many seconds are there in an hour? 60 minutes times 60 seconds per minute, so that's 3,600
seconds per hour then times 24 hours. That will give you the number of seconds in a day
and you multiply 3,600 and 24 and you get, trust me, 86,400 seconds per day. Now, how
many seconds do we have in a year? Well, no problem, you take that number, multiply by
the number of days in a year, 364.25 or 365 roughly and then you get 364.25 days in a
year. That will give you the number of seconds in a year and to the first 3 sig figures,
if you work this out to the first 3 sig figures, that will give you 3.15 x 10^7 seconds per
year. That is an interesting number. It's not hard to calculate but it's a number that
has a lot of interesting uses. You may not remember this number easily because it doesn't
really resemble anything, it's not something you have a particular attachment but if you
really look at it, 3.15, it is very close to a number that everybody knows by heart.
What do you think that is? Very close... how about pi! What is pi equal to? 3.14 right?
To the first 3 sig figures. Now that's not exactly pi, it's a little bit more than that,
but don't worry about it, it's close enough. So we say, one year is roughly pi times 10^7
seconds! Why do we replace it with pi? Because we know pi very well. So roughly speaking,
there's 31 and a half million seconds in a year. Okay, so that's an interesting observation,
with that, we can easily estimate, for example, how many heartbeats does a person go through
in a lifetime, roughly speaking? Let's see, you have, let's say, a person that lives to
be a hundred years old. That's not too bad right? Each year, how many heartbeats are
there? For simplicity, let's estimate the heart rate to be 60, one beat per second,
that's not a bad approximation for most people and then you say, there are this many seconds
in a year so that, roughly speaking, you have pi times 10 to the 7 heartbeats per year.
Multiply this by one hundred years, so roughly speaking you will have pi times 10 to the
9 heartbeats. So roughly speaking, if you, let's say, okay it's not quite one hundred
years, maybe ninety years, what ever, you're looking at a total heartbeat of about 3 times
10 to the 9. In other words, 3 billion heartbeats, in a lifetime. That's a lot by the way and
if you can make a dollar with every heartbeat, then you'll be 3 billion dollars rich after
nearly a hundred years! So that's an interesting number to think about. So you also can estimate,
roughly, how many seconds does an average person live. Of course, you'd say it's kind
of scary to measure your time in seconds right? But don't worry, it's a lot of seconds. So
roughly speaking, 3 billion seconds, that wouldn't be bad at all. Way over 80 years
old, so 3 billion seconds isn't bad at all. So that's an interesting number and there
is a little story associated with that conversion here. Once upon a time, there was an international
scientific meeting and during one session of the meeting, the chairman was in charge
of allocating hours or minutes to each scientist who wished to talk about their scientific
discoveries. So before each person starts to talk, the chairman will ask them,"How long
do you want to speak?" Normally they would say,"I want to use thirty minutes." or "One
hour." or whatever. Now, then he asked this particular scientist and the scientist was
interesting, he said,"Okay, I need one micro-century." Gee, what is a micro-century you wonder. That
sounds like a very long time because you say century instead of seconds, but really, think
about it. The chairman was very clever, he thought about it for a few seconds and he
said,"Okay, go ahead!" and then the scientist spoke for how long? For about 50 minutes.
So now tell me, what is a micro-century? Don't use your calculator, what is a micro-century?
It's not hard to do it at all because you know how long a year is, in terms of seconds.
A century is ten to the two years right? It's a hundred years, right? A micro, what's a
micro? A micro is ten to the negative six isn't it? So what's a micro-century in terms
of years? It's ten to the two times ten to the negative six. So how much is that? It's
ten to the negative for years. That's a micro-century right? Each year, you have pi times ten to
the seven seconds. So how many seconds are there in a micro-century? You take this number
and you multiply this by ten to the negative four, in other words you get about pi times
ten to the 3 seconds. Let's say pi is about equal to 3. So that's basically three thousand
seconds! Three thousand seconds, divided by 60 seconds per minute, how many minutes would
that be? That would be fifty minutes! You see, actually it would be a little more than
that because pi is more than 3, if you want to be picky, you get about 52 minutes for
a micro-century, but 50 is a good enough approximation. It's obvious the chairman knew this conversion,
otherwise he couldn't do it that fast. With that conversion, you easily know what a micro-century
is. So that's an interesting little story to tell you so that you will have a better
memory of this interesting conversion.