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Gravitational Potential Energy near the surface of the EARTH.
Written By Katsuya Yamada
13 pages
Look at the following two pictures;
Case (A)
A chunk of iron
The position of the chunk of iron is
reasonably near the surface of the
EARTH; much less than the height of
Mt. Everest or much lower than the
summit of the Rocky Mountain.
Although unrealistic, ASSUME that
there is no air.
Case (B)
8 feet
A chunk of iron
1 foot
A relatively large ice cube
A relatively large ice cube
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Case (A): A chunk of iron is placed 8 feet right above the large ice cube.
Case (B): A chunk of iron is placed 1 foot right above the large ice cube.
The temperature of the ice cubes may be MINUS 30 degrees in Celsius.
Move on to the next page.
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We assume that the both size and weight (mass) of the two chunks of iron are
exactly the same and further the both the size and shape of the two ice-cubes
are also exactly the same.
Now, the chunk of iron in each case is released from rest (dropped) and hits the
ice cube underneath. What do you think would happen to each ice cube?
After hitting the ice cube, the ice cube in Case (A) would be broken apart
(shattered). But the ice cube in Case (B) may not be broken at all.
Can you explain why?
In case (A), the iron is released from a high position (8 feet above the ice cube)
and in Case (B), the iron is released from a low position (only 1 foot above the
ice cube).
The fact that the ice cube is broken in Case (A) OBVIOUSLY means that the ice
cube in Case (A) receives much more energy than the ice cube does in Case (B).
But why so? Because the iron is released from the position in Case (A) much
higher than the iron released from the position in Case (B). This further means
that there is much more (gravitational) Potential Energy stored at the position
of the chunk of iron in Case (A) than in Case (B). After the iron is released, the
potential energy stored at the position of the chunk of iron is released and it is
changing into the kinetic energy of the chunk of iron as it falls. Thus, as the iron
falls, the stored potential energy decreases and the motion energy (the kinetic
energy) of the falling chunk of iron increase. Let me REPEAT! The potential
energy is being changing into the kinetic energy as the iron falls. The higher the
initial position of the chunk of iron, the more kinetic energy the iron gains right
before hitting the ice cube.
The initial (Gravitational) Potential Energy stored depends on the initial position
(the initial height) of the chunk of iron. The higher the initial position (the initial
height) of the chunk of iron, the more potential energy stored at that initial
position. Likewise, the lower the initial position of the chunk of iron, the less
potential energy stored at that initial position.
The initially stored potential energy stored at the initial position of the chunk of
iron is being converted into the kinetic energy of the chunk of iron as it falls.
Continues the next page.
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The initial potential energy stored at the initial position (initial height) is 100%
converted into the kinetic energy (motion energy) of the falling chunk of iron at
the instant right before hitting the ice cube.
In Case (A), since the iron is released from a higher position than in Case (B), the
iron starts to fall with a larger amount of potential energy stored at the initial
position. Then, the iron hits the ice cube with a larger amount of kinetic energy.
That’s why the ice cube would be broken a apart when the iron hits the ice cube
in Case (A) and the ice cube may not be broken at all in Case (B) because the
kinetic energy of the falling iron right before hitting the ice cube is much less.
Right before hitting
Right before hitting
A large
Case (A)
kinetic-energy
A small kinetic-
Case (B)
energy
Invisible gap
Why do we have to consider this? b
e
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f
o
In Case (A), since the chunk of iron is released from a higher
r position (8 feet), it
will hit the ice cube with greater kinetic energy. The result is that the iron gives
e
a greater DAMAGE to the ice cube.
h position (1 foot), it
In Case (B), since the chunk of iron is released from a lower
i the iron gives a
will hit the ice cube with less kinetic energy. The result is that
t
less DAMAGE to the ice cube.
t
Yah! This can (may) be used as the very “definition of the iKinetic Energy of an
object”
n
To the next page, please.
g
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Therefore, the Kinetic Energy of an object (iron) is directly related to the
amount of DAMAGE the OTHER object (the ice, in this case) receives when the
two objects COLLIDE with each other. The greater the kinetic energy of the
“bombarding object”, the greater is the damage the other object (the ice)
receives. But when a car moving at high speed hits the rigid hard wall, the car
will experience a large DAMAGE. The greater the kinetic energy of the car, the
greater is the damage (of the car) also.
Kinetic Energy
Cause of DAMAGE.
BUT, more detailed discussion(s) is (are) given from page 7 THROUGH
page 11. Anyway, please keep reading.
After hitting
After hitting
Case (A)
Case (B)
NOW completely
touched! No gap
and “hits” with
b
e
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f be broken at all!
The ice cube breaks apart
The ice cube may NOT
heavily damaged.
Small Kinetic Energy.
o Small DAMAGE.
Large Kinetic Energy.
But the temperature of the ice increases
r
Large DAMAGE.
and It may melt to some
extent.
e
“THUD” sound.
Please note that in Case (B), although the ice cube is NOT broken at all, the
temperature of the ice cube (and the iron too) MUST increase!
h They become
warmer (Because the molecules of the ice are agitated byi the “the hitting THUD”
of the chunk of iron; the detail will be discussed later.) Otherwise, the energy
t
t
i
n
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will not be conserved. But soon the heat energy will be transferred to the
ground (and the air) and dispersed.
Anyway, the potential energy is proportional to the height of the position of
the chunk of iron. The higher the height, the more potential energy stored at
that height and the lower the height, the less potential energy stored at that
height.
For this reason, the potential energy is interpreted as “Energy of Position”.
Note that it is very wrong to say that the potential energy is stored inside the
chunk of iron. The potential energy is stored at the position of the chunk of iron.
It is also wrong to say that the chunk of iron alone possesses the potential
energy. The potential energy comes into play because of the gravitational
interaction between the chunk of iron and the EARTH. No isolated object can
have potential energy. The potential energy is the interaction energy between
the chunk of iron and the earth.
On the other hand, we can say that an isolated object can have kinetic energy.
Remark
The gravitational potential energy of an object depends on its height. The higher
the position of the object, the greater is the potential energy. The higher the
position, the more is the energy stored at that height.
It is extremely important, however, the gravitational potential energy
(stored energy) comes into play thanks to the existence of the EARTH.
If the earth does not exist, there would be NO gravitational potential
energy. That’s why the reference position for the gravitational
potential energy is chose at the ground level (the surface of the earth).
In other words, the gravitational potential energy is the interaction
energy between the object in question and the EARTH. There is an
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attractive force between the object and the earth. The object interacts
with the earth through this attractive gravitational FORCE.
So, the gravitational potential energy is stored at the height of the
object, the height above the ground (the surface of the earth), is the
result of the interaction between the object and the EARTH.
I repeat that there would be no gravitational potential energy if the
EARTH does not exist! Therefore, it is meaningless to say that an
isolated single object has a potential energy. Any kind of
potential energy comes into play between TWO objects that attract
(repel) each other. (non-contact interaction). An interaction requires
at least TWO objects! TWO objects interact with each other through
attractive or repulsive force that acts on the two objects. One object
exerts the force on the other object. MUTUAL force. Only under this
situation, POTENTIAL energy comes into play. In our present case, one
object is the chunk of iron and the other object is a HUGE object called
the EARTH (a large mass)!
Please move on to the next page. Thank you.
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The TOTAL ENERGY (KE + PE), [which is 100 in this “EXAMPLE”], is conserved
(remains the same). It is always 100 at EVERY moment while the object is falling.
𝑬𝒕𝒐𝒕𝒂𝒍 = 𝑲𝑬 +
𝑷𝑬
100
=
0
+ 100
100
=
10
+ 90
100
=
20
+ 80
100
=
30
+ 70
100
=
40
+ 60
100
=
50
+ 50
100
=
60
+ 40
100
=
70
+ 30
Getting faster and faster
Falling
Invisibly tiny GAP
100
=
80
+ 20
100
=
90
+ 10
100
=
100
+ 0
Right BEFORE hitting the ground
(It is NOT YET touching the ground).
Continues on the next page.
ONLY the POTENTIALENERGY. PE = 100
and KE = 0
Kinetic Energy (KE) is
the Energy associated
with the MOTION of
the object (Energy of
Motion). The FASTER
it moves, the greater
is the Kinetic Energy
and vice versa.
Potential Energy (PE)
is the energy STORED
at the height of the
object; the higher the
height the more
energy is stored (the
greater PE) and vice
versa.
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For your information;
𝑲𝑬 =
𝟏
𝟐
𝒎𝒗𝟐
𝒎 and 𝒗 represent
the mass and speed
of the falling object,
respectively.
ONLY the KINETIC- ENERGY.
KE = 100 and PE = 0
Ground Level
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As the chunk of iron falls, it’s KE (the Kinetic Energy (Energy of Motion)
increases as shown in the picture on the preceding page, like, 0, 10, 20, 30, 40,
50, 60, 70, 80, 100. This means that the as the chunk of iron falls, its speed
increases, getting faster, faster, and faster. On the other hand, the (The
GRAVITATIONAL) potential energy decreases like 100, 90, 80, 70, 60, 50, 40, 30,
20, 10, 0.
As CLEARLY SHOWN in the picture on the preceding page, the total energy,
which is (KE + PE) is A~~~~~~~~~LWAYS, at ANY moment while the iron FALLS,
100. Yes, 100, 100, 100, 100, ・・・・・, 100.
At ANY MOMENT (Yes, at any moment) WHILE THE CHUNK OF IRON IS FALING,
the TOTAL ENERGY (sum of Kinetic Energy and Potential Energy) is always 100,
which is a constant value, always remains the SAME. In the very beginning
before the chunk of iron is released, the iron has NOT YET started the motion.
So, its KE is zero. But the MAXIMUM amount of potential energy is STORED at
the initial height of the chunk of iron. This maximum potential energy is 100.
Why 100? Because I said SO! I can say 500, I can say 8906. I can say ANY
number! But the number 100 is VE~~~RY convenient and easy to handle (in
number calculation). That’s why I chose 100.
From the picture on the preceding page, it is easy to see that the initially
STORED potential energy 100, stored at the initial height of the iron chunk, is
being converted into the Kinetic Energy (KE) of the falling iron. Thereby, the
potential energy decreases and the kinetic energy increase.
At the very INSTANT right before the chunk of iron hits the ICE CUBE (Not yet
contact, but almost contact; the chunk of iron is STILL moving!), ALL THE INITIAL
potential energy (Initial PE), which is 100, is 100% converted into the kinetic
energy (No PE is left) and at that very special instant, the kinetic energy of the
chunk of iron becomes 100, which is the MAXIMUM KINETIC ENERGY (the chunk
of iron moves with the maximum speed just before hitting the ice cube). At this
very instant (right before hitting the ice cube, not yet contact occurs) NO
potential energy left because ALL the potential energy is completely 100%
converted into the kinetic energy. Only the kinetic energy at this instant.
Continues on the next page.
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This means that, the chunk of iron HITS the ice cube with the maximum kinetic
energy 100.
Once the chunk of iron HITS (touches, contact) the
ground, the situation DRASTICALLY changes! HOW changes?
1. The ice completely melts (becoming pure liquid water) and the chunk of
iron comes to REST (Complete Stop). So, its kinetic energy becomes
ZERO.
2. (The ice cube already COMPLETELY melts to become liquid water), AFTER
hitting the ground, the height of the chunk of iron is ZERO (it is touching
the ground). Then, since the POTENTIAL ENERGY strictly depends on the
height (The higher the height, the more is the potential energy and vice
versa), AT the very moment the chunk of iron hits the ground (its height is
zero), the POTENTIAL ENERGY becomes ZERO also.
Thus, the instant the chunk of iron hits the ground; BOTH the Kinetic-Energy
AND the Potential Energy become ZERO simultaneously. So, at this instant
(hitting the ground); TOTAL energy (KE + PE) becomes ZERO!
KE (Kinetic Energy) + PE (Potential Energy) = 0
But, before the chunk of iron hits the ground, the total energy (KE + PE) was 100.
The total energy which was 100 before the chunk of iron hits the ground
SUDDENLY becomes ZERO (DISAPPEARS or VANICHES!).
100 suddenly becomes ZERO! 100
0 This DEFINITELY VIOLATES
“The Law of Conservation of Energy”; because the WHOLE energy vanishes!
This does NOT make sense! This is a SERIOUS problem.
Solution:
Right before hitting the ground, the chunk of iron possesses the Kinetic energy
ONLY whose value is 100. This Kinetic energy of amount 100 is now converted
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into the kinetic energy of each of many debris of the broken ice. In addition,
some of 100 kinetic energy of the chunk of iron warms up the ice cube and the
debris of the broken ice cube (the ice debris melts and become liquid water,
which is warmed up). So, eventually, all initial kinetic energy (100) right before
hitting the ground is used for WARMING UP everything in the “hit area” of the
ground and chunk of iron itself. As a result, the very last kinetic energy
(100) of the chunk of iron right before hitting the ground is, after
touching the ground, transformed into the thermal energy of the soil
and the chunk of iron even though the chunk of iron NEVER move any more
after hitting the ground. Because the soil particles in the hit region in the
ground and the iron particles in the chunk of iron are AGITATED by this
“HITTING”, BOTH the soil particles AND the chunk of iron (made of iron
particles) gain EXTRA motions or extra vibrations. In other words, the AVERAGE
KINETIC ENERGY of these soil particles and iron particles INCREASE. But the
AVERAGE KINETIC ENERGY represents the TEMPERATURE! Thus, “The
Temperature of the soil and the chunk of iron” increase also. So, the
temperature of everything warmed up by the final kinetic energy of the
chunk of iron RISES (Becomes HIGHER TEMPERATURE) when the chunk of iron
hits the ground. Because, ONLY because that the TOTAL energy is conserved
(will REMAIN THE SAME), The (KE + PE), which is 100, is 100% converted into the
THERMAL-ENERGY (increase in temperature) of the soil of the hit area on the
ground and chunk of iron. Macroscopically “TEMPERATURE” is the SAME as
“THERMAL ENERGY”. Therefore, the above result means that the numerical
value of the total final thermal
energy is 100 also. It started with
100 and ended up with 100 also. This is what it means that the total energy is
A~~LWAYS conserved even AFTER the chunk of iron hits the ground.
In other words, the VE~~~~~RY initial energy at the initial height is 100
(before released). This much energy is the TOTAL energy given to the
system. So, the TOTAL energy is 100. This TOTAL energy 100 neither
vanishes nor destroyed. It WILL survive forever! Yes, forever!! Forever!
A~~~~LWAYS 100! What happens is SIMPLY one type of energy changes
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into a different type of energy WITHOUT changing its NUMERICAL value
(in this very particular example, its value is 100. I am the one who said
100, NOT GOD! Remember???).
INFRARED RADIATION
The chunk of iron itself is warmed up
FORCE!
Thud!
These regions of the soil are warmed up by the
“HITTING ACTION (THUD)” of the chunk of iron
(The soil particles are agitated; extra motions
of soil particles and extra vibrations of iron
particles of which the iron is made are caused
by this “THUD” by the chunk of iron and the
ground.) “Thud” is accompanied by some
amount of sound energy also.
An extra thermal energy appears.
The amount of this extra thermal energy is 100, which is the same as the
VE~~~~RY initial POTENTIAL energy at the very initial height of the
chunk of iron before it is released (KE was zero). Look at the picture on
page 1. Since there occurs higher temperature region (THUD) and
lower temperature regions outside the “THUD region, the heat energy
escapes from the THUD region to its surrounding region. Also some heat
energy radiates into the air (Infrared Radiation). Eventually, the
temperature becomes the SAME everywhere! Continues on the next page.
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As time goes on, the HEAT ENERGY is transferred from the heated soil
and heated chunk of iron into the surroundings (mostly the surrounding
AIR): That is the heat energy SCATTERED and SPREAD out into the air.
This does NOT mean that the energy disappears: the energy whose
numerical value is 100 in this example went out far from the heated
chunk of iron and the heated soil. The energy (100) is STILL somewhere
else in the world! Because the energy (100) dispersed into the all over
the world, the temperatures of the soil and the chunk of iron go DOWN
and eventually becomes the same temperature as the temperature of the
surroundings.
The NUMERICAL value of the TOTAL energy, which is 100, is always 100.
If we collected all the dispersed energies, it becomes equal to 100.
This is what it means that energy NEVER vanishes and the TOTAL energy
is conserved.
But, in Quantum Mechanic, that handles the particles smaller than
molecules, the TOTAL energy will not be conserved. If you are interested
in Quantum Mechanics, I STRONGLY suggest you to major in Physics. In
the past, one GIRL who took Physical Science from me was moved by my
lectures and she transferred to UCLA as Physics major. She earned Ph. D
in physics. She is now an associate professor of physics! How about that?
The FINAL EXAM question is shown on the next page. Please go to the
next page.
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OK. The question in FINAL EXAM.
Suppose you are in a room. The floor is carpeted with VE~~Y fuzzy-carpet. Suppose
that you hold a heavy book with hand. You release the book from rest (Don’t push it!)
from a certain height in the room. The book naturally falls down and hits the fuzzy
carpet and makes a COPMLETE stop on the FUZZY carpet. The carpet level becomes
the reference ZERO level for the gravitational POTENTIAL ENERGY. Initially, the
book has a certain amount of gravitational POTENTIAL ENERGY corresponding to the
initial height of the book (above the floor level). As it falls, since its speed increases,
it gets faster, faster, and faster. Therefore, its kinetic energy increases as it falls. But
after hitting the VERY fuzzy carpet, it will stop completely. Then, with respect to the
BOOK, there will be NO kinetic energies and there will be NO gravitational
POTENTIALE ENERGY either. That is, after (Yes, AFTER!) the BOOK hitting the fuzzy
carpet and stops, it seems that all energies associated with the book are GONE and
com~~~letly disappeared. This definitely VIOLATES the Law of Conservation of
Energy. The total energy CANNOT disappear! Explain this AS SHORT AS POSSIBLE
(two LINES at most! OK?). Where the HELL is the total energy GONE ????? Your
answer MUST be consistent with THIS lecture note just given from page 1 THROUGH
page 12 (I DO know that you do NOT want to read. But this is your problem!)
ESPECIALLY page 10 and page 11. The “THERMAL ENERGY arising from temperature
increase” is the KEY word!
NOTE that in this collision between the book and the fuzzy carpet, the Kinetic Energy
of the book does not give any DAMAGE to anything but AGITATES the atoms or
molecules of the book and the carpet to give them extra vibrations. This is a sort of
DAMAGE??