Fundamental 3 Quantities,
Forces, and Energy
Perception Cycle V4 Env Att Stim.png
Fundamental Concepts in this Lecture
1. Operational Definitions tell us how we measure something
2. Basic quantities: length, time, mass, charge
3. Basic quantities are combined to produce derived quantities
4. In measuring motion, we have displacement, velocity, and acceleration as derived
quantities
5. In measuring space, we have length, area, and volume as derived quantities
6. Density is a derived quantity formed by dividing the amount of mass by the amount of
volume of an object
Fundamental quantities and how to define them
Understanding light, sound, and their associated perceptions, requires that you recognize
they have powerful physical principles and concepts at their roots. We begin here with
some quantities or properties that we can measure, and laws and principles that relate them.
We rarely view terms like position, mass, and force as being full of romance and mystery.
Yet, they are the foundations for opening the door on the truly fascinating and mysterious
world around us. Even alone, they harbor deep mysteries. We describe light and sound in
terms of properties like position, pressure, or forces that repeat, or oscillate over time.
Concepts like mass, position, and wavelength are so fundamental that simple questions about
them can bump us up against the limits of what is known. Science does not ignore these
questions. But to keep from stalling on these imponderables, we often fall back to what we
can observe, or measure. Definitions that do this differ in important ways from Dictionary
Definitions. Definitions of this class focus on how we measure something rather than what
it is or means. Oh, the meaning of the idea is not ignored in these definitions, it is just not
the main focus. Let us construct one of these definitions.
A College Sophomore is a term that behaves this way nicely. For the purposes of this
example, we define a college sophomore as someone who meets the following conditions. 1)
They are a matriculated student (meaning that they have enrolled in the university for the
purpose of getting a degree), and 2) they have completed more than 32 semester hours, but
less than 64 semester hours of credit.
Notice that we have listed the steps or operations necessary to observe or measure whether
someone is a sophomore. For this reason, these are called Operational Definitions. In this
case the steps are 1) Observe if they are enrolled to get a degree, 2) Count how many credits
they have completed, and 3) See if the number of credits is within the boundaries of 32 and 64
credits. Occasionally we include math operations, such as divide, in our definitions.
Operational definitions state or imply the steps needed to observe or measure a property.
Length, time, mass, and charge
We are going to study motion in the first part of this class. As an example, let's take a simple
formula
tdv.
where v is the speed of the object, d is a distance (a length) and t is time. We are all familiar
with these quantities. We use this equation (speed) to tell an officer how fast we were going.
"Honestly, I was only going 50mi/h. Miles are distance, hours are time. The word per
often implies that we divide.
In science, we must use careful definitions. Lets look at three quantities as our initial
building blocks: Time, Length, and Mass.
Time
What is time? It turns out that time is hard to define. We usually use the idea that time is how
long we wait,2 but that can be tricky to measure. Let's start with something simple. How
much time will you spend in this class today? That is a time we can wait, about an hour. But
it is harder to answer questions like "how long does it take for light to travel a foot. The
answer is about a nanosecond-but what is a nanosecond? We cannot perceive times this
small. Likewise, we cannot wait for a million years (well we could, but it might be less
interesting after the first 70 or so).
2Feynman, Richard, Robert Leighton, Matthew Sands, The Feynman Lectures on Physics,
Vol. I, Addison-Wesley, Reading Massachusetts, 1963
To measure time we use events that are periodic, that is, they occur at regular intervals.
Nature itself is rich in things that regularly repeat. We probably began noting time against
some of them. The apparent motion of the sun in the sky marks a day. Tides, seasons, and
months are all names attached to natural markers of time. The time required for the moon to
repeat its motion is approximately a Moonth. An early device using this idea is the
pendulum of a clock. It periodically swings back and forth. From a fundamental periodic
phenomena, we can build up larger or smaller units of time (say, half a swing, or 20 swings).
The current basic unit of time is the second, abbreviated "s. Our standard for this unit is
given in terms one specific, much more fundamental, aspect of nature. It is times the period
of oscillation of radiation from the cesium atom. This definition is difficult to use in practice.
Fortunately in this class, we can just use a clock or watch to measure seconds. But if we need
a very accurate time measurement, the standard for time measurement is the atomic clock.
Atomic Clock
Length
To begin our discussion of length, we will start with the concept of space. Our goal is to have
some idea of how far away things are or how long or tall things are. If you wish to know how
far away you sit from the door of our class room, it helps to have a mental picture of the size
of our room. In this class, our view of space will be that it is like a large room, a container in
which things happen. For example, the solar system exists in space. The Earth orbits around
the Sun in that space. Space is the very large container in which the action occurs. If you
were to study Einstein's Relativity, you would find that this is not perfectly true, but for our
purposes, space is a big container, and length is a measure of how far away in this container
something is.
In ancient Egypt, the standard of measuring length was when Pharaoh took his ceremonial
reed and measured the length of the foundation for the temple. This might sound strange, but
in essence this is what we all did until 1960. Prior to this, a meter, our unit of length, was
defined as one ten millionth of the distance from the North Pole to the equator. Since this was
not a very practical day-to-day measuring definition, a standard "reed" (this time made of
platinum) was kept in France, and meter sticks were made to match this standard. There are
terrible problems with this! Each stick of a different materials changes length with
temperature. So in 1983 the meter was defined as the distance light travels in vacuum during
a time interval of
1299792458s
The abbreviation for meter is "m. You might be more familiar with a yard stick. A meter
is about the same length as a yard (three feet). The yard is more difficult to use in science with
its divisions of feet and inches. And a yard stick suffers from the same difficulties as a meter
stick. Like most scientists, we will usually use the metric system. Luckily, our measurements
are not too demanding in this class. Usually rulers will work fine for us. We wont have to use
the precise definition.
Mass
Mass is an amount of matter. Exactly what matter is, is still somewhat of a question (Job
security for Physicists!). Einstein equated mass and energy. For this class we shall take mass
as just the amount of matter and trust our intuition on what that means. A good way to see
what our intuition tells us is to pick up two objects, one in each hand (a pen and pencil, a book
and folder, or whatever you have handy) and feel which has more mass. Usually to do this we
move the objects up and down. What we are doing is feeling how hard the objects are to get
moving and to stop moving. The amount of matter, or mass, tells us how hard something is to
move. This will be the central role that mass will play in our study of motion.
Standard Kilogram
The standard for mass is still a little like Pharaoh's reed. The standard unit of mass is the
kilogram, abbreviated "kg." It is the mass of a standard piece of platinum alloy, again kept in
France (see the picture above). An operational definition for mass would be to compare the
yx
amount of material you have with the standard kilogram. Note that mass and weight are very
different quantities. You can see this if we use a bathroom scale. On earth it gives a reading
that is proportional to the amount of matter in our bodies, but if we took it on a space craft in
orbit, it would not measure any weight at all. Yet the amount of in matter our bodies has not
changed! You might ask how we measure mass if it is not the same as weight. We use a
balance to do this, but to understand how it works, we will need to know about forces, and
that is a subject we will take up next lecture.
Charge
Let's start with something we all know. Let's rub a balloon in someone's hair. If we do
this we will find that the balloon sticks to the wall. Why? We say the balloon has become
charged. Charge is some property that provides this phenomena we have observed with the
balloon (i.e. it sticks to the wall).
You may remember from you high school science that there are small particles in atoms
that are charged. These little particles (electrons and protons) have charge as part of their
nature. Our balloon must have collected more of one type or the other of these small particles.
These particles and their sticking power plays a great part in our study of light and sound.
For example, in your MP3-player, electrons form the electrical current that drives the
earphones to make sound.
Derived quantities
Derived quantities are combinations of our four fundamental quantities (time, length, mass,
and charge). One of the simplest derived quantities is displacement. Let's start there.
Reference Systems
We will start talking about motion by making a setting for our motion. We will use a
coordinate system.
In the picture, we see a person standing by an arrow marked x and an arrow marked y. By
measuring distances along these two arrows, we can tell you how far our person will go. In
choosing a set of axes and an origin, we have chosen a reference system. We will measure our
motion with respect to the origin (zero point) of this system. Note that for measuring lengths,
there is really nothing special about where we choose the origin. We place it where it is most
convenient. So picture our reference system being centered on a lamp post.
The man is standing at the lamp post. We will call his position (length from the origin), 0. If
he moves away from the origin, we can use a number line along the ground (like a tape
measure) to tell how far away he goes.
Displacement
Displacement is the difference between two lengths. Think of measuring how far away the
Snow building is from the Romney building. That measurement is a displacement. Often we
will see a graph of a quantity measured from an origin. Displacement is easiest to understand
on this type of graph, so let's take a moment to make sure we all understand how to read such
a graph now.
Suppose our person is standing 2m from the lamp post. He moves to 5m from the lamp post.
We may wish to describe the change in his position. We do that by writing
12xxx...
We often use the letter x to stand for position in math shorthand. The strange symbol that
looks like a Doritos chip is called delta. We use it to say in shorthand the change in the
next quantity, in this case, the change in the position or x. We call this change in position
displacement.
In our example, the displacement is
mmmxxx32512
.
..
...
Notice that with displacement, direction matters, if we went from 5m to 2m, we would have
mmmxxx35221
..
..
...
The negative sign tells us which direction we are going. In our coordinate system, negative
directions tell us we are going left. We always take the position we end up at, and subtract the
position we started from in that order.
Now that we know about displacement and we have a coordinate system, lets try interpreting
a graph.
012345024t(sec)
x(m)
What does this graph mean? Notice that on the horizontal axis we are measuring time. On the
vertical axis, we are measuring position or length. This graph says our person starts 2m
from our lamp post (the origin). Then, he moves away from the lamp post. We know this
because the position gets larger the longer we wait. Five seconds later, the person is at about
5m from the origin. His displacement would be 5m - 2m or 3m.
You will need to be familiar with reading graphs, so if this is not clear, you need to ask
questions from class mates or your professor.
Speed and Velocity
Our second derived quantity is speed--how fast something goes. In physics, velocity and
speed are different but closely related. Velocity is a speed and a direction, like saying 50mi/h
due South. Because velocity and speed are so closely related, we will write the letter v for
speed. This may seem strange, "vspeed" would be hard to pronounce. But both velocity and
speed tell you how much position changes in a given time.
We can write speed as
timetotaldistance totalspeed average.
or in math shorthand
tdvave.
Lets go back to our guy walking by the light pole. Suppose our fellow walks three meters to
the right in one second. We will define the velocity as
timetotalntdisplaceme total
velocityaverage.
txvave
.
.
Then this velocity is
s/m3s0s1m0m3
.
.
.
.avev
Now, let's have the guy walk to the left three meters in one second.
012345024t(sec)
x(m)
012345024t(sec)
x(m)
s/m3s0s1m0m3
..
.
..
.avev
Notice this is negative. That is the direction part of velocity showing up. It is okay to have a
negative velocity. It just means he is going to the left.
Graphical Interpretation of Velocity
Let's go back to the definition of average velocity. This time, lets write it
ififavettxxv
.
.
.
Notice at the top of the fraction we have a change in position and on the bottom of the
equation, we have a change in time. If the velocity is constant, we will get a straight line on a
plot of position vs. time.
This is just what we had in our first graph, but now suppose our guy were to run instead of
walk. The xf - xi part of the equation would get bigger. This would make the graph steeper.
You may have thought as you have been reading that average velocity and average speed are
nice, but generally when you think of speed, you are looking at a speedometer. The
speedometer does not seem to be measuring average speed. It gives the speed you are
traveling right now, an instantaneous speed. Of course that is exactly what a speedometer
does. Both average and instantaneous speeds are useful. We will use both average and
instantaneous speeds in our studies.
Acceleration
Thinking about driving a car, we must get the car going. When we come to our destination,
we must slow down. This is a change in our velocity. We call a change in velocity
acceleration. Let's define average acceleration as change in velocity .v divided by the
av
av
amount of time it took to make the change, .t. In math shorthand we may write
ififttvvtva
.
.
.
.
.
.
Where vf is the final velocity, and vi is our shorthand for the initial velocity. Since we are
dealing with velocity, we would expect acceleration to have directions attached to it. This is
true. In one dimension, we can indicate direction with a plus or minus sign. So the equation
above could be positive or negative. Let's investigate what a negative acceleration means. To
do this, lets first start with positive accelerations.
Start with a positive velocity and a positive acceleration. In this case, we have a positive
initial velocity, and we would expect the velocity to get larger, so vf - vi is positive.
The ball is speeding up. Now suppose we have a positive velocity and a negative
acceleration. This means that vf - vi is negative tf - ti is never negative, that would be going
back in time. For this to be true vf must be smaller than vi. The object is slowing down!
If we use average velocity as an example, we can guess that if the acceleration is constant,
then the acceleration is the slope of the line in a velocity vs. time graph. We can summarize
what we have learned about acceleration in the following statement.
When the object's velocity and acceleration are in opposite directions, the object slows down.
Questions to ponder for Lecture 3
1. Name the basic quantities.
2. Which of the following is considered a basic unit?
a. Displacement
b. Charge
c. Velocity
d. Speed
3. Density is the mass divided by the volume. Discuss: Is this a good example of an
operational definition?
4. I have a ball on a football field. The ball starts at the endzone (the 0-yard line) and is
moved to the 20-yard line. How far did the ball move? Is this a displacement?
5. If I go to Idaho Falls (a distance of about 30 miles) in an hour, what is my speed? What is
my velocity?
6. Which of the following lines represents the fastest of the two objects, A or B? Why?
7. You are going south and but your acceleration direction is north. Describe what is
happening.
8. Name a derived quantity and list the basic quantities from which it is made.