gravitational interaction

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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.

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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.

Therefore

or

Next page.

�������� �� #������ ������ � $������� �� %������� ������

$������� �� #������ ������ � �������� �� %������� ������

The System.

(TWO interacting

persons; Dr. Yamada

and the student)

Increase in student money and

Decrease in Dr. Yamada’s money.

But NET change is ZERO!

22

These are consequences of “no energy input” and “no energy output” in

������ ����� � ������ ������ � ������ � �����

0 0

This is satisfied if and only if

������ ! ����� � �

So, go back to the I/O energy diagram I drew on page 16. In this diagram, we

have

$������� �� #������ ������ � �������� �� %������� ������

This is the Law of Conservation of Energy for a non-contact interaction in which

TWO objects interact with each other without physical contact.

The above result clearly tells us that

When the kinetic energy increases, the potential energy decreases.

When the kinetic energy decreases, the potential energy increases.

The amount of “increment” is always equal to the amount of “decrement” so

that the TOTAL NET change becomes ZERO. This is the meaning of

������ ! ����� � �

Just like the money exchange between Dr.Yamada and the student, the fact

that the NET change in energy of the system is zero means that the total NET

change in energy of the system is ZERO. This is what it means

������ ! ����� � �

Since the TOTAL energy of the system does NOT change, the TOTAL energy of

the system will remain the SAME. We say that “the TOTAL ENERGY IS

CONSERVED”. (CF; The total amount of money of the system that consists of Dr.

"�� � � � � �

� � � � �

does NOT make sense!

makes sense.

23

Yamada and the student will remain the same, even though each person’s

money changes. The TOTAL money is conserved.)

So, “CONSERVED” means “REMAINS THE SAME”. OK?

Since we are considering ONLY two types of energy, namely “kinetic energy”

and “potential energy”, the TOTAL energy of the system is simply the sum of the

kinetic energy and the potential energy”

& � &��� ������ � #������ ������ � %������� ������

The total energy remains unchanged even though each type of energy (kinetic

and potential) changes.

See the next page for the CONCLUSION.

24

Conclusion:

Start with the general statement of the Law of Conservation of Energy, which is

������ ����� � ������ ������ � ������ ! �����

Then since we have (for a non-contact interaction between two objects)

������ ����� � � and ������ ������ � �,

the law of conservation of energy reduces to

������ ! ����� � �

Here “energy” means TWO different kinds of energy. They are, “kinetic energy”

and “potential energy”.

Therefore

������ ! ����� � �

means that the NET change in energy of the system is zero or over all change is

zero. The NET energy does NOT change at all!. Therefore, if kinetic energy

increases, then the potential energy must decreases by same amount so that the

NET energy-change is zero, and vice versa.

Since total net change in energy of the system is zero (The total energy does

NOT change), the TOAL amount of energy cannot change. The total energy is

conserved.

Total energy for the system consisting of TWO interacting

objects is

&��� ������ � #������ ������ � %������� ������

This will remain unchanged

But

or

On the next page you will see some information on Chapter 3 test.

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25

Chapter 3 Test (one of MANY questions)

Using the law of conservation of energy, which says

Energy Input = Energy Output + Energy Changes,

explain why the “amount of increase in one type of energy” is the

SAME as the amount of decrease in other type of energy”.

Please do NOT give a lengthy answer. Please!!!!!!

Please do NOT ask me any question regarding this matter. If you ask

me ANY question, I simply shut you down by saying “Refer to the

lecture note in my web site”. COMPRENDE?

By the way, I still ask you “balanced force” and “unbalanced force”,

which you are supposed to know and understand VE~~~~~~~~RY well

by now.

Just one more on the next page.

26

One more shot on the Gravitational Interaction

Iron ball

Gravitational force

In any interaction between TWO objects, always some kind of energy is

transferred from one object to the other object.

In the gravitational interaction between the iron ball and the earth, as shown

above, what type of energy is transferred?

The answer is: The Mechanical Energy. Why mechanical energy?

Recall that “under what circumstances, Mechanical Energy is transferred? “

The answer is: Whenever at least one the interacting objects changes its speed

(either speeding up or slowing down). While the iron ball and the interact with

each other, as shown above, the speed of the iron ball increases as it falls,

getting faster, faster, and faster as it falls. Thus, the kinetic energy of the falling

ball increases as it falls. The ball is receiving energy from the gravitational

potential energy. That’s why its energy increases. Therefore, the falling ball is

the energy receiver. Mechanical Energy is being transferred to the ball while

it is falling because its speed is increasing (changing).

Earth