1 Gravitational Interaction (26 pages) Written by Katsuya Yamada Falling object A basket ball? The earth does NOT move by any appreciable amount. WHY? Earth Gravitational Force exerted on the EARTH by the falling object. Gravitational Force exerted on the falling object by the earth.
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
1
Gravitational Interaction (26 pages)
Written by Katsuya Yamada
Falling object
A basket ball?
The earth does NOT move by any appreciable amount. WHY?
Earth
Gravitational Force
exerted on the
EARTH by the falling
object.
Gravitational Force
exerted on the falling
object by the earth.
2
The two objects are interacting with each other through the “gravitational force”
or just “gravity”. This is the “gravitational Interaction”. The gravitational
interaction is another example of “non-contact interaction”. Then, you should
expect “the gravitational potential energy”.
What are the two objects that are involved in the gravitational interaction?
They are:
1. A falling object (a basket ball, a tennis ball, a base ball, anything!)
2. The Earth (Gravity is NOT an object! OK?)
But, according to Isaac Newton, there occurs the gravitational interaction
between any TWO objects. There is the gravitational interaction between a
baseball and a pencil. There is the gravitational interaction between a girl and a
stone. The VE~~~~RY origin of the gravitational interaction is the MASSES of the
two objects. Since just every object possesses a mass (no exceptions!), there is
the gravitational interaction between any two objects. For this reason, the
gravitational interaction is called “the universal interaction”.
A~~LWAYS “gravitational force” is involved in the gravitational interaction.
Look at the beautiful picture I drew on the preceding page one more time.
Isaac Newton discovered that the gravitational interaction is a MUTUAL
interaction. The EARTH exerts the gravitational force on the BASKET BALL. This
may not be too difficult to understand. But, the problem is, the BASKET BALL
ALSO exerts the gravitational force on the EARTH simultaneously. Further, the
strength of the force exerted on the BASKET BALL is EXACTLY the same as the
strength of the force exerted on the EARTH by the BASKET BALL. In the
gravitational interaction between the BASKET BALL and the EARTH, each
experiences exactly the SAME force from the other side object. This situation is
indicated by ARROWS in the figure on the preceding page. Each arrow
represents the gravitational force. The directions of the arrows are opposite but
the LENGTHS of the arrows are the same (the same strength).
The, the EARTH should be attracted to the BASKET BALL and the EARTH should
approach the BASKET BALL. But ONLY the BASKET BALL moves towards the
EARTH and the EARTH dose not move towards the BASKET BALL even though the
force is acting on the EARTH. Why? The answer is: Wait. Keep reading.
3
The gravitational force exerted on any object by the EARTH is called “the
gravity”. The gravitational force between two objects is primarily ATTRACIVE.
One object is attracted toward the other object by the gravitational force.
The gravitational force (or gravitational interaction) between two ORDINARY
objects (between two tennis balls, between two basket balls, between a pencil
and a book, between you and me, etc) is so weak that we cannot observe it. To
observe the gravitational interaction directly with our naked eyes, the mass of
one of the two objects must be HUGE, as huge as the mass of the earth.
For your information, let me tell you the mass of the earth: It is
�. �� � �� �� ������
There are 24 zeros after 1.
�� � �������������������������
24 zeros
This is the mass of the earth. It is indeed HUGE! Then we can easily observe the
gravitational interaction between an ordinary object (e.g., a basket ball) and
the earth. The ordinary object approaches to the earth due to the attractive
gravitational force. The ordinary object “FALLS” toward the earth, as shown in
the figure on the very first page of this note.
Because of this HUGE mass, the EARTH cannot move towards the BASKET BALL.
The earth pulls the basket ball and the basket ball pulls the earth with the same
strength of force simultaneously. But what moves (falls) is the basket ball. The
earth doses not move by any appreciable amount even though the basket ball is
continuously pulling the earth. Why? Because the mass of the earth is too
much! Any object possessing a very large mass is very difficult to change its
speed [Recall a heavily loaded truck (large mass). Such a truck cannot change its
speed easily, VE~~~~RY difficult to move! Ve~~~ry heavy!]
Continues on the next page.
So, we consider the gravitational interaction between the BASKET BALL and
the EARTGH.
In the class experiment, you released a basket ball from underneath the motion
detector. This motion is monitored on the computer screen through the interface.
The distance of FALL is something like 1.5 meters (4 feet?). The air resistance
(viscous force) is negligibly small for such a small vertical distance. Therefore we
can safely assume that THERE IS
influence of the gravity only. Such a fall is called
tried THE FREE FALL in the Magic Mountain?
On the screen, you saw the straight line on the graph of “Spee
See the graph on the next page.
4
So, we consider the gravitational interaction between the BASKET BALL and
In the class experiment, you released a basket ball from underneath the motion
detector. This motion is monitored on the computer screen through the interface.
The distance of FALL is something like 1.5 meters (4 feet?). The air resistance
(viscous force) is negligibly small for such a small vertical distance. Therefore we
y assume that THERE IS NO AIR! The basket ball simply falls under the
influence of the gravity only. Such a fall is called “THE FREE FALL”. Have you
tried THE FREE FALL in the Magic Mountain?
On the screen, you saw the straight line on the graph of “Speed VS Time”
See the graph on the next page.
Motion Sensor
(Detector)
A basket ball
Gravitational
force
So, we consider the gravitational interaction between the BASKET BALL and
In the class experiment, you released a basket ball from underneath the motion-
detector. This motion is monitored on the computer screen through the interface.
The distance of FALL is something like 1.5 meters (4 feet?). The air resistance
(viscous force) is negligibly small for such a small vertical distance. Therefore we
NO AIR! The basket ball simply falls under the
. Have you
d VS Time”
Motion Sensor
(Detector)
A basket ball
Gravitational -
5
Speed of the falling ball
Time
Waiting time
The graph clearly shows that the speed of the basket ball increases as it falls. In
other words, the speed of the basket ball CHANGES. The only CAUSE of change
of speed is FORCE. Therefore, some force MUST be acting on the basket ball
continuously as it falls. This force is the earth-gravity or “the gravitational force”.
Further, the graph is almost perfectly a STRAIGHT line. This implies that the
speed-increasing RATE is constant and therefore the gravitational force acting
on the falling basket ball is CONSTANT also. Yes, the gravitational force is
constant. In other words, the strength of the gravity acting on the ball does
NOT change while it is falling.
Strictly speaking, this is NOT true. The strength of the earth-gravity changes
as the height of the ball changes. The higher the height, the weaker is the
gravitational force and vice versa. That is, the strength of the gravitational force
between the basket ball and the earth is greater (stronger) when the separation
(height) between the ball and the earth is short and vice versa (closer). You
observed that when two magnets are closer, the magnetic force is stronger and
vice versa. When the two charged scotch tapes are closer, the electric force is
stronger and vice versa.
(continues the next page)
6
However, when the height of the basket ball is placed at an ordinary height,
less than, say, 100 miles, and released, the variation of the strength of the
gravitational force with the height is negligible (it varies VE~~~~~~RY little).
Therefore, when the basket ball is released from a height less than 100 miles,
the strength of the gravity is essentially constant while the ball is falling. But
this is true if the effect of the air is completely ignored. If the effect of the air is
NOT ignored, then we have to worry about “the terminal velocity”. The COURSE
“Physical Science” will not touch the terminal velocity. Don’t worry about it!
Like any other non-contact interaction (magnetic and electrical), the
gravitational interaction is a mutual interaction. Thus, when we consider the
gravitational interaction between an ordinary object and the earth, since only
the ordinary object moves (falls), we only consider the motion of the ordinary
object. We do NOT pay attention to the earth simply because the earth is
essentially at rest. Here “at rest” means that the earth does not change its
position due to the gravitational force coming from the ordinary object (falling
object). As far as we only consider the mutual gravitational interaction between
the earth and the ordinary object (like a basket ball), we ignore the motion of
the earth around the sun and the spinning motion of the earth. OK?
A Clear Evidence of Interaction When an ordinary object and the earth is interacting through the gravitational
force, what is the evidence of the interaction?
The evidence is “Change in Speed”. Surely, when the ordinary object is
interacting with the earth through the gravitational force, the speed of the
object is increasing. The object is getting faster, faster, and faster as it falls (as it
interacts with the earth “gravitationally”.) You SAW this in the class experiment
the other day. Didn’t you? Definitely the speed of the ordinary object is
changing. “Change in Speed” indeed becomes the evidence of interaction.
Why does the speed of the falling object increases as it falls? What is the
VE~~~~~~RY cause of change in speed???? A FORCE! ONLY FORCE can change
the speed of an object (NOTHING ELSE!). Since the speed of the falling object
7
increases as it falls, there must be some force acting on it. What is that force?
Yes “the gravitational force” or just “gravity”. This gravitational force acting on
the falling object comes from the earth. In other words, the gravitational force is
exerted on the falling object by the earth.
Again, like the magnetic potential energy, the gravitational potential energy
depends on the configuration of the TWO objects. This means that the
gravitational potential energy depends on the separation distance of the two
objects. In our present case, the TWO objects are
1. A falling object (A basket ball, a tennis ball, base ball, whatever!).
2. The Earth.
When a tennis ball is released from the position very far from the surface of the
earth (a very HIGH position), the ball falls with increasing speed (getting, faster,
faster, and faster) as it falls (as it approaches the earth.) This “speed increase”
continues until the ball hits the surface of the earth (So called “ground”.)
Therefore, the kinetic energy of the falling ball continuously increases until it
hits the ground. Since this “Speed increasing period” (or the accelerating period)
is very long because it was released from a very high position. Thus, the falling
ball has gained a large speed by the time it hits the ground.
Therefore, when the ball is released from a very high position, it will gain a large
amount of kinetic energy just before hitting the ground. This is the MAXIMUM
kinetic energy of the ball. The energy supplier to the falling ball is the
Gravitational Potential energy that was originally stored in the initial
configuration of the TWO objects (The initial separation between the ball and
the Earth, before the ball is released.) So, the initial gravitational potential
energy must be very large so that the falling ball receives the large kinetic
energy at the end of her journey.
On the other hand, when the ball is released from a much lower position
above the surface of the earth, then since the vertical distance covered by the
falling ball is shorter now, the ball cannot gain a very high speed by the time it
hits the ground. Therefore, in this case (a lower releasing position), the kinetic
energy obtained by the falling ball by the end of its journey is small. Since the
energy supplier is the gravitational potential energy, the initial gravitational
8
potential energy that is stored in the initial configuration of the two objects
(before the object is released) must be small also.
Object (a basket ball, a base ball, a tennis ball,
any kind of an ordinary object)
Object
The higher the height of the ball, the greater is the gravitational potential
energy.
The lower the height of the ball, the less is the gravitational potential energy.
This means that as the ball falls and approaches the earth, since its height
becomes lower, lower, and lower (the separation between the ball and the earth
decreases), the gravitational potential energy decreases (less, less, and less.).
Earth
Earth
Large Potential-
Energy is stored in
this configuration
Small Potential-
Energy is stored in
this configuration
9
But, the kinetic energy of the ball increases and it obtains the MAXIMUM kinetic
energy right before hitting the ground.
Now you understand that, as the ball falls and approaches the earth, since
initially (before letting go) stored gravitational POTENTIAL energy is being
converted into the kinetic energy of the falling ball, the gravitational potential
energy decreases and the kinetic energy of the object increases, as it falls.
This situation is exactly the same as the situation of two magnets that attract
each other. The magnetic potential energy depends on the separation distance
between the two magnets (configuration of the two magnets.) One of the
magnets is fixed so that it cannot move and the other magnet is free to move.
As long as the two magnets attract each other, the stored magnetic potential
energy is greater for a greater separation distance and less for a less separation
distance. This means that when the free magnet is released from a certain
separation, it approaches towards the fixed magnet. As it approaches the fixed
magnet, it speed increases, getting faster, faster, and faster. That is, its kinetic
energy increases continuously as it approaches the fixed magnet. It will gain the
maximum kinetic energy right before it hits the fixed magnet.
NOTE: The gravitational force is A~~~~~LWAYS acting on just every object 24
hours a day and every year. Further, the direction of the gravitational force is
A~~~~~LWAYS downward. The gravitational force will NEVER disappear, NEVER
vanish!
Move onto the next page.
10
Tossing an object vertically upward.
Stops momentarily (ZERO speed)
Remember that the gravitational force (or the gravity) acting on any object will
NEVER disappear from its object under any circumstances. Whether the object is
moving upward or falling, the gravitational force is A~~~~~~~LWAYS acting on
the object and the direction of the gravitational force is A~~~~~~~LWAYS
downward (even the object is rising!)
Move onto the next page.
Earth
11
1. The force diagram for the object during the tossing period.
Force of hand
Direction of motion
(Upward) A tennis ball
Gravitational force
(NOTE: the gravitational force
is A~~~~~~LWAYS downward)
You hold the tennis ball and toss it. Your hand exerts an upward force on the
ball. Both force of hand and the gravitational force act on the same object,
which is the tennis ball. You MUST understand that during the tossing period,
the force of hand has to be greater (stronger) than the force of gravity (the
gravitational force). Otherwise, the ball would NEVER move up due to the
downward gravitational force. In other words, the upward force of HAND must
overcome the downward gravity. That is, the forces must be UNBALANCED! The
unbalanced force (or the NET force) must be upward so that the object moves
upward. Then, the equivalent force diagram for the ball will be
Unbalanced force (the NET force)
Direction of Motion
(Starts to rise)
Note that the unbalanced force (or the NET force) is replaced by a SINGLE force
as shown above. A single force means ONE force! OK?
12
Then since the direction of this single force (upward) and the direction of motion
(also upward) are the same, the object will be sped up; it moves upward, faster,
faster, and faster, during the tossing period during which your hand is in contact
with the object. But this tossing period can be very short, as short as 1 second.
Anyway, during the tossing period, the speed of the object (moving upward)
INCREASES. This is what “tossing” means.
2. The force diagram after tossing (is over)
After tossing, the hand and the object are not in contact any more. Then, the
force of hand disappears now. But since the gravitational force will NEVER
disappear, the only force acting on the object after tossing is the gravitational
force (or the gravity)
No force of hand (Why?)
Direction of motion
(Still rising!)
Gravitational Force
Now (After tossing), since the direction of force (the gravitational force only)
and the direction of force are OPPOSITE from each other, as clearly seen in the
diagram right above, the speed of the object (still rising!) will decrease, getting
slower, slower, and slower, as it rises. The object will slow down while moving
upward (Yes, upward!). Can you picture this in your mind? OK?
“Slowing down” means that the object MUST stop completely at a certain height.
“Stops” means that the speed of the object is exactly ZERO. But even the object
stops completely at a certain height, the gravitational force will NEVER
disappear! Why? Because ….. Whether the object is moving or stops, the object
and the earth are A~~~~~LWAYS interacting with each other. Therefore, the
13
gravitational force will NEVER disappear even at the instant the object makes a
complete stop at the highest point. The stopping period should be extremely
short, as short as one-thousands of a second (But not zero second,)
3. During the stopping period (no motion) The only force acting on the object is the downward gravitational force.
No direction of motion Stops momentarily.(Has just reached the
(because STOPS!) highest point)
Gravitational force
Because NOBODY is touching the object, it will start to fall downward due to
the downward gravitational force, which NEVER disappears under any
circumstances. The downward gravitational force NEVER, NEVER, disappear regardless of
the state of the object, moving or not moving! Why????? There is a reason! What is it?
4. During falling period Recall the effect of force on an object. In our present case, only the
gravitational FORCE is acting on the object. Then, this force increase (change)
the speed of the object, from zero speed to non-zero speed. But since the
downward gravitational force will never disappear, the speed of the object
continuously increases due to the effect of the downward gravitational force
(look at the figure above.). That is, it is getting faster, faster, and faster as it
falls. Then, during the falling period, the force diagram for the object is
Direction of Motion
(Falling) Gravitational force
Next page please!
After “falling” starts, direction of motion
and the direction of force are the same.
Thus, the speed increases.
14
Since the direction of Motion is the same as that of the gravitational force,
the object will be sped up as it falls.
5, Catching period To catch the falling object, you MUST hold the object by hand to stop it. But
remember that nothing will change instantly. “Changing” takes a certain
amount of time no matter how short it is. The falling object cannot stop
instantly. Before it makes a complete stop, it must slow down first and then
stops. To do this, the hand must exert a force on the falling object against the
direction of motion of the object. Since the object is moving downward (falling),
the hand must exert a force upward. But this upward hand force must be
greater than the downward gravitational force so that the direction of the
unbalanced force is against the downward motion
So, the force diagram for the object during catching period is
Upward force of HAND
Direction of Motion
(Falling)
Gravitational Force
The upward force of hand must be greater than the downward gravitational
force so that the unbalanced force (the NET force) becomes upward.
15
The unbalanced force or the NET force can be replaced by a SINGLE force, as
shown below
Unbalanced force (Net force)
Direction of Motion
(Falling)
Catching period
Now, since the direction of motion (downward, falling) is opposite to the
direction of the gravitational force (always downward), the object will SLOW
DOWN as it falls. “Slowing down” implies that it must stop eventually. When it
stops, you adjust the upward force of HAND so that the upward force of hand
becomes exactly equal to the downward gravitational force.
6. Stopped momentarily! No motion! Speed is ZERO.
Force of HAND (upward)
No arrow of Motion
Gravitational force
The two forces are balanced out to zero.
Next page.
16
Now go back to the business of “AFTER tossing”. After tossing (no more force
of hand!), the object is STILL rising (the direction of motion is upward). But, the
only force acting on the object is the downward gravitational force; the force
diagram for the rising object is given by the one shown on page 9 of this note.
Let me reproduce it here.
Direction of Motion
(Rising) Gravitational force
Since the direction of Motion is opposite to that of the gravitational force, the
object slows down while rising. That is, while the object is rising, it slows down
and its kinetic energy decreases.
Rising
Gravitational Force
Next page.
Earth
The gravitational potential
energy is increasing as the
objet rises. (Yes INCREASE!)
17
While rising with decreasing speed (slowing down), the object is moving away
from the earth. As I explained on pages 5 and 6, when the separation between
the two interacting objects (one is an ordinary object like a basket ball and the
other is the earth) is increasing (moving away from each other), then the
gravitational potential energy increases also. Yes, while the ordinary object is
moving upward (after tossing), the gravitational potential energy increases. But,
since the object is slowing down while rising, its kinetic energy decreases.
Then, after tossing and the object is rising with decreasing speed (slowing down),
the kinetic energy increases but the gravitational energy decreases. This can be
interpreted in the following way.
While the object is rising, the kinetic energy of the rising object is being
converted into the gravitational potential energy. In other words, the kinetic
energy of the rising object is being taken away by the gravitational potential
energy.
This is what it means that the kinetic energy is decreasing and the potential
energy is increasing.
Although I mentioned nothing about the existence of the air, in physical science,
unless otherwise stated clearly, it is always understood that we completely
ignore the air. NO AIR! OK?
We consider the system that consists of the ordinary object and the earth. The
gravitational force acting on both the object and the earth simultaneously is the
internal force. No external forces are needed to move the ordinary object.
No external energy supplies are needed to the two interacting objects. Therefore,
no energy input is needed to the system consisting of the ordinary object and
the earth. The energy is already contained within the system in the form of
(gravitational) POTENTIAL ENRGY. The same thing can be said to the system
consisting of two magnets or consisting of two charged scotch tapes.
Since we ALWAYS assume that there is no air, no energies are dissipated in the
interaction (gravitational interaction, magnetic interaction, or electric
interaction). This means that when we consider two interacting objects as a
single system, there is neither energy input nor energy output.
Then . . . Next page please.
18
Input/Output Energy diagram for the system consisting of
two interacting object is
No energy input No energy output
The system contains self supporting energy supply in it (potential energy)
Let me apply this idea to the system consisting of the rising object and the earth
(the situation AFTER tossing). See page 13. Draw I/O energy diagram for this
system.
On the next page.
System (consisting of
TWO interacting
objects)
Energy
Changes (NET change)
ZERO!
I/O energy diagram
19
I/O energy diagram for the gravitational interacting
system after tossing (the object is rising with decreasing
speed)
We apply the Law of Conservation of Energy to the system, which is,
������ ����� � ������ ������ � ������ � �����
But since we are considering a self supporting system that it
contains energy source (energy supply) and there are no
dissipated energy in it, there is neither energy input nor energy
output.
The system
consisting of the
rising object and
the earth.
Decrease
in Kinetic Energy
and
Increase in Gravitational-
potential Energy
20
Therefore, the law of conservation of energy becomes
� � � � ������ � �����
Therefore we must have
������ ! ����� � �
The right hand side is ZERO. What does this mean? “ZERO” in this
case means that there is NO net change in energy
Consider a SINGLE SYSTEM that consists of Dr. Yamada and one of my
favorite students. Dr. Yamada and the student interact with each other by
exchanging money. Suppose Dr. Yamada has $500 in cash in his wallet and this
student has $300 in her/his wallet. Then the TOTAL money in this SYSTEM is
$800. Because I like this student so much I give her/him $100. So, my money
decreases by $100 and your money increases by $100. But TOTAL amount of
money in this SYSTEM is still $800 ($400 + $400 = $800). Therefore, the change in
money in this SYSTEM is zero. But, as we noted already, the TOAL amount of
money in the system, which is $800, will remain the same. No matter how much
money I give the student (but less than $500), the TOTAL amount of money
($800) in the system will NOT change even though my money decreases and the
student’s money increases, because the amount of decrease in my money is
exactly equal to the amount of increase in the student’s money. That’s why the
NET change in money of the entire system is zero. “ZERO” means no change in
the total money of the system
No changes at all! Nothing has changed in the system even though my money
decreased and your money increased. But, “change in amount of money of the
system is zero” is guaranteed under the following condition
The amount of decrease in money in Dr. Yamada’s wallet
is equal to the amount of increase in the student’s wallet.
In the interaction between Dr. Yamada and the student, “money” is transferred
from one to the other. There is nobody else except these twp people. There is
"�� � � � � �
� � � � �
does NOT make sense!
makes sense.
21
NEITHER money input into this system NOR money output (No money escapes
from this system. Therefore, I/O money diagram is constructed like the one
shown below.
No money input No money output
In our present case (Two objects interacting with each other), there are only
TWO types of energy involved; Kinetic Energy and Potential Energy. So, in
the I/O money diagram above, simply replace MONEY by ENERGY.
Therefore, the above result says that over all change in the energy of the system
is zero. Means that the NET change in all energies is zero.
More explicitly, this means that when the kinetic energy increases, then the
potential energy MUST decrease by the same amount to compensate so that the
total change is zero.
Conversely, when the potential energy increases, the kinetic energy MUST
decrease by the same amount to compensate so that the NET change is zero.