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Now we're ready to use the physics definition of WORK to develop ideas about -- and formulas
for -- some important FORMS of energy.
By "forms of energy," I don't mean anything anthropocentric (like enthusiasm, or spirit).
And I don't mean anything BIOcentric (like a mysterious life energy, or the Chinese Qi).
It turns out that nature's deepest interpretation of energy TYPES -- that is, FORMS of energy
-- seems to come from applying the definition of work in supremely simple situations.
By the way, I'm now indulging in what philosophers call reductionism: reducing issues to their
SIMPLEST elements. This is often the way of science. I don't know if reductionism is the
road to ULTIMATE truth. But it certainly has given us PRICELESS insights.
Consider THE SIMPLEST case of a force doing work: a CONSTANT force acting on a FREE -- i.e.,
unattached -- object that is initially at rest.
For example, let gravity act on a mass over a short distance near Earth's surface (pretend
Earth is down there), and ignore the tiny effect of air resistance: [drop ... thud]
The force of gravity, F, does work, W, on mass, m, through a distance, d. [Draw it!]
Another way of drawing this situation would focus on the constant acceleration, a, and
the resulting speed, v, acquired at time, t, just before impact. [Draw it!]
The definition of work tells us that work equals force times distance.
N2, Newton's second law, allows us to rewrite this single force F as m times a, so that
W = Force times distance = mad
And simple linear kinematics tells us how "distance traveled" is related to the constant
acceleration: d = one-half a t^2. So...
W = ma-times-one-halfat^2 ... which equals... one-half times m times the quantity (at)-squared
And, finally, from the very definition of acceleration, we know that constant acceleration
times the elapsed time gives the speed acquired. That is,
a is defined as: delta-v over delta-t, which becomes, when initial time and speed are zero,
simply
v = at.
So, W = one-half m times v-squared
This equation says that the work done on a free mass is manifested as SPEED. This "one-half
m v-squared" is so useful in physics that we give it a special name and its own symbol:
one-half mv^2 is called KINETIC ENERGY, capital K, the energy of MOTION. This is our first
FORM of energy. A 1-kilogram object moving at 1 meter-per-second (ordinary walking speed)
has one-half joule of kinetic energy.
Notice that the v in the formula for kinetic energy is speed, not velocity. Direction is
not involved. This will remind us that kinetic energy -- like ALL energies -- is NOT a vector
quantity.
Our 1-kilogram object has exactly the same energy whether it moves to the right at 1
meter per second, or to the left at 1 meter per second.
And notice that the speed v is SQUARED in the formula. This means that speed determines
kinetic energy in a BIG way. If, for example, we double a BULLET's speed, then we quadruple
its energy. This is an important -- and perhaps scary -- insight.
Capital K is defined as one-half mv^2. Remember that.
The work done on the mass has given it kinetic energy. If the mass had already been moving,
then the work would just have CHANGED the kinetic energy. In other words, we can generalize
our result to become a VERY useful law of physics:
capital W-sub-net = delta-(capital-K)
The work done by the net force on a mass equals the mass's CHANGE in kinetic energy. If the
work is a POSITIVE number of joules, then the kinetic energy increases by that many
joules, and the mass speeds up. If the work is NEGATIVE, then capital-K decreases, and
the mass slows down.
This very general law of physics has acquired a strange name. It's called the work--kinetic-energy
THEOREM, as if we had proven it mathematically.
We sure DID use math to help derive this relationship, but, in the end, we BELIEVE the so-called
"Work--Kinetic-Energy Theorem" because EXPERIMENT -- and every-day life -- confirm it again
and again and again.
Perhaps you noticed that little n-e-t subscript that I slipped-in alongside the W in the Work--kinetic-energy
theorem.
As I said, W-sub-net is the work done by the NET force acting on the mass. Or, equivalently,
by the SUM of the works (positive and negative) done by EACH of the forces acting on the mass.
We'll come back to the Work--Kinetic-Energy Theorem in another Radical Physics episode.
But, for now, let's remember what we originally set out to do: to learn about different FORMS
of energy by seeing work done in simple situations.
When episode RP11 first introduced the concept of work, I said, "WORK is the TRANSFER of
ENERGY from one object to another by means of a FORCE." Let's take that seriously.
Look, again, at the mass that simply falls through a distance d. The work done by a force
(gravity) is transferring energy (kinetic energy) to THIS object.
But as this kilogram strikes the ground, is a DIFFERENT force doing work to transfer the
KINETIC energy into ANOTHER form of energy? After all, the mass ends up NOT MOVING. It's
sitting on the ground, so it has zero KINETIC energy.
Evidently, the force of impact DOES do work, but at the MOLECULAR level. Take a look at
this animation that depicts what's happening during a slightly bouncy impact. Those gray
circles represent MOLECULES within the falling object.
As the impact occurs, the kinetic energy of the entire falling object is transformed into
the kinetic energy of the object's MOLECULES. In a SOLID, those molecules can't go very
far, so they just VIBRATE more than usual.
This molecular vibrational energy is commonly called "heat." But let's be a little more
formal and call it THERMAL ENERGY, capital T.
The KINETIC energy of the falling object has become THERMAL energy: vibrational kinetic
energy of that object's molecules (AND of molecules in the object hit, AND even of the
surrounding air molecules).
Notice that this heat, or thermal energy, is NOT created by the motions of the molecules.
It *IS* the motions -- the kinetic energies -- of the molecules. No mysterious stuff called
"heat" is created when molecules collide.
Collisions tend to turn the kinetic energy of MACROscopic objects into the RANDOM kinetic
energy of MICROscopic objects (the molecules).
This miniaturization and randomization of energy happens virtually EVERYWHERE and at
almost every instant. We will come back to look at this steady disorganization of energy
in later RP episodes.
The force of impact did work on the falling object, and transformed its kinetic energy
into thermal energy. But where did the original kinetic energy of the falling object come
from?
Gravity did work to transfer WHAT form of energy into kinetic energy?
Scientist's have found a very useful way of speaking about the work done by gravity, whether
the mass is rising or falling. We say that the mass's POTENTIAL ENERGY is changing.
Unfortunately, potential energy is symbolized with a capital U. (Think of it as a big bucket
-- so big, in fact, that it holds most of the ordinary energy of the universe!)
delta-U (the change in potential energy) is defined as negative W (work done by any force
of a special class of forces)
One such special force is gravity. So let's use the formula for work done by gravity to
find the change in potential energy that resulted:
∆U-sub-g = -- W-sub-g
Remember what the delta means...
Ulater - Uearlier = -- W-sub-g
U-at-the-bottom - U-at-the-top = -- m times g times d Let's rearrange this:
U-at-the-TOP - U-at-the-bottom = mgd
This says that the gravitational potential energy at the top, minus the gravitational
potential energy at the bottom, equals mgd.
But if we recognize that this distance d is just the height h above the ground, then U-at-the-bottom
is the potential energy of the object when it can't fall any farther.
So, if we DEFINE U-at-the-bottom to be the local zero of potential energy, then
U-at-the-top = mgh
where h, the height, is just the distance above our DEFINED starting point -- the zero
point -- for gravitational potential energy. If we want to say that U-sub-g is zero not
on the ground, but actually deeper down, that's fine.
We can even define U-sub-g to be zero at the center of Earth. However, then our earlier
formula F-sub-g = mg wouldn't apply all the way down, cuz g (the acceleration due to gravity)
changes underground. So our formula U-at-the-top = mgh would need to be amended.
But, for objects near Earth's surface, U-sub-g = mgh is just fine. In fact,
U-sub-g is defined as mgh is our second form of energy: local gravitational potential energy.
Other types of potential energy will be considered later, but THIS type is incredibly useful
-- especially for you humans trapped there on Earth!
The mgh formula tells us that every elevated object has potential energy. The greater the
elevation, the more the joules of potential energy. But wait, ... elevation ...as measured
from WHERE?! From WHEREVER we have DEFINED the elevation to be zero!
This may seem fishy, but we did something quite similar for kinetic energy. We said
that K = one-half m v-squared, but we didn't say "relative to what" that v must be measured!
This means that the kinetic energy of an object, as with its speed, is a REALTIVE quantity.
Relative to these stars, I have no kinetic energy. But relative to a passing comet, I
may have a great deal!
Likewise, this kilogram, at a height of 1 meter on Earth, has ... m times g times h
... it has about 10 joules of gravitational potential energy RELATIVE to the ground. But
relative to the center of Earth, it has many millions of joules.
This kind of RELATIVITY of potential and kinetic energy does not diminish their usefulness.
That's because the incredibly deep law of physics that employs them deals only with
CHANGES in energy, not with absolute values. That law is the law of ENERGY CONSERVATION,
and it is the subject of our next taproot episode.