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Learning to Think About
Gravity
Concepts of Gravity from Aristotle to Einstein
Proving and Disproving Newton
Esther Zirbel
Claudine Kavanagh
Tufts University
Learning to Think About
Gravity
A Comparison of
Aristotle, Newton, and Einstein
1.
2.
3.
How & why do objects fall?
Terrestrial motion
How & why does Earth orbit Sun?
Celestial motion
What is Gravity?
Cause of motion
Overview about the Class
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Tell students the objective and tell them they will get confused
Elicit Student’s Ideas (let them fill out the motions in the universe sheet)
Historic Overview
Talk about Aristotle’s Ideas
– Have a discussion where you challenge students to disprove Aristotle
•
Talk about Newton’s Theory
– First Galileo, Brahe and Kepler and about how scientific methodology leads to a
deeper understanding of how things work
– Mention notion of empirical modeling
– Mention Newton’s three laws but focus on conceptually deriving the Universal Law
of Gravitation
– Do the Cannon Ball thought experiment with the students
– Discuss what makes a theory a theory
•
Talk about Einstein
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–
–
–
•
Elicit what students know about Einstein
Explain Equivalence Principle
Present Evidence of pictures students might be familiar with
Mention objective of General Relativity and how space-time gets distorted by
masses
Pose concluding questions comparing Newton to Einstein
Goal of Class
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Elicit your Ideas about Gravity
Confront you with your Understanding of Gravity
Confuse you
Learn scientific methodology and thinking
Differentiate between empirical and mathematical
models and scientific theories
Experience how discoveries are made
Learn two different theories of gravity
Which theory should you chose? When? Why?
Differentiate between science fiction and fact
Motions in the Universe
Your interpretations
How – Orbits:
How and why does
the Moon orbit the
Earth?
How – Free Fall:
How and why does
the apple fall off
the tree?
What?
What is the nature
of Gravity?
A (very) short history:
• Aristotle philosophizes on matter and motion
philosophy
• logic but no experiment
• differentiates between earthly and heavenly motions
• Copernicus suggests alternate model of solar system
• Galileo provides “scientific methodology”
How?
Empirical
Modeling
• studies free fall – how (no why)
• no answer to orbital motion (but proof of heliocentric system)
• Brahe observes the stars and planets
• Kepler analyzes Brahe’s records
• describes laws empirically & orbits mathematically (no cause!)
• Newton reasons
• 3 laws and “universal” law of gravity
• suggests a conceptual and mathematical theory of gravity
• answers question “What is gravity?”
why
what
Timeline
An Important Side Point
• Aristotle's Physics was written in the fourth century
BC. For more than two thousand years this book
served as the basis of natural philosophy up to the
sixteenth century, the time of Galileo.
• Humanity believed Aristotle for 2000 years
• Einstein invented his new theory 100 years ago and
we do not yet teach it in schools
Changing your mind about theories takes a very
long time (society takes generations)
It is okay to take your time to understand new
theories – and it is okay to make mistakes!!
Aristotle and Motion
Two types of motion: natural motion and violent motion.
Natural motion is motion arising from the nature of an object.
• This motion does NOT require an external cause to occur.
• Four elements: earth, water, air, fire
Motion of things are determined by their natural tendencies to
move towards their proper place in the cosmos:
• Earthly things towards the center of the Earth.
• Heavenly things …
Violent motion is contrary to the nature of an object.
• This motion does require a FORCE to cause motion.
• e.g. a stone thrown into air moves in a violent motion.
Isaac Newton (1643 - 1727)
• Building on the results of Galileo
and Kepler
• Adding physics interpretations to the
mathematical descriptions of
astronomy by Copernicus, Galileo
and Kepler
Major achievements:
1. Invented calculus as a necessary tool to solve
mathematical problems related to motion
2. Discovered the three laws of motion
3. Discovered the universal law of mutual gravitation
Newton—Laws of Motion
It’s all about Forces
lst law:
Inertia
object at rest remains at rest
object in motion remains in motion unless a force acts on it
Absence of force
2nd law:F=ma
Galileo—free fall  acceleration
Acceleration is the Result of a force acting on the mass
3rd Law: Actio = Reactio
For every action there is an equal and opposite reaction
Balance of forces
Discussion
How and why does the apple fall to the ground?
• Explain this using Newton’s laws
• Why do heavy and light objects fall at the same rate?
• How would apples fall on the Moon?
For more details see Gravity I Module
In this Theory of Gravity Universal?
• Does this theory also apply to celestial bodies?
• Recall: Aristotle distinguished between the motions
of earthly and heavenly bodies
• So then, why does the Moon not fall towards Earth?
Discussion
The Resulting Path of the Planets…
Gravitational pull (blue)
Centrifugal force (green)
Planet’s velocity (red)
Resulting Path (black)
The Theory of Gravity
Any two bodies are attracting each other through
gravitation, with a force proportional to the product of
their masses and inversely proportional to the square of
their distance:
Mass1  Mass 2
F
distance 2
In this Theory of Gravity Universal?
How would you test whether this
hypothesis is correct?
Is this hypothesis correct? — How do you test this?
 (a) Cannon Ball Thought Experiment
 (b) Mathematical Proof
Discussion
Cannon Ball Thought Experiment
• How fast would the cannon
ball have to travel to go around
the earth in a circle?
• If the cannon ball was at the
moon's distance how fast
would it have to travel then?
• What is the velocity of the
Moon?
Discussion
How would you calculate the speed of the moon from theory?
Does this agree with observations?
Speed of the Moon? For orbital motion have:
GMm
FG  2
r
v

mv2
FC 
r
GM
r
Does this agree to the observed speed?
v
distance circumference
2 r


time
1 month
1 month
Yep!
The Universal Law of Gravity
Gravity decreases with the inverse square law
Mass1  Mass 2
F
distance 2
This applies to all Objects, Apples, Cannon Balls, the Moon, and other Planets.
 UNIVERSAL LAW
Newton provided an explanation of WHY the Moon goes around the Earth.
Optional:
“Derive” Kepler’s Third Law
GMm
mv 2
FG  2
FC 
r
r
GM
v
r
2r
use v 
P
GM 2r


r
P
2
4

 P2 
r3
GM
Theory?
• What is a Scientific Theory?
• Distinguish this from an Empirical Model?
Discussion
Who is this guy?
Near universal recognition –
Every body knows who he is, yet not many know his theories
What do you know about Einstein?
Discussion
• Elicit Student Ideas
• Did not start talking until he was 3 yrs old
• Worked at a Patent office before he got famous
• When did he live?
• How old are his theories?
• Why don’t we teach them in schools?
Einstein’s Theories
• E=mc2
• Special Relativity
Motions in the universe – “Relative” motions
• [General Theory of Relativity]  discovered later
A new “Theory of Gravity”
Describe structure of Space-Time and describe
how objects move through Space-Time
• Photoelectric Effect
Got the Nobel Price for this!
Einstein
Principle of Equivalence
 No Force!!
 ONLY acceleration
Need New Theory of Gravity "General Relativity"
Test of General Relativity – 1917 Eclipse
• Prediction of GTR: Bending of light
• Curvature strongest in vicinity of dense and massive objects
Perihelion of Mercury
• Newtonian physics could not explain Mercury’s orbit around the Sun
• General Relativity does a more accurate job predicting Mercury’s orbit
Castle
Second image: A black hole with the mass of Saturn is placed over the
middle of the Mall.
(McLeod CfA)
Image of Abell 2218, A Galaxy Cluster Lens
– taken from HST archives
Traveling with relativistic speeds
Downloaded from http://www.spacetimetravel.org/
Difference between
Newtonian Theory of Gravity and GTR
The Way of Newton:
Mass tells Gravity how to exert a Force
Force tells Mass how to accelerate
F 
GM g mg
r2
F  mi a
a
GM g
r2
The Way of Einstein:
Mass-Energy tells Space-Time how to curve
Curved Space-Times tells Mass-Energy how to move
(if mg  mi )
Earth- Sun Motion
• Newtonian: The Sun creates a gravitational field that exerts a force upon
the Earth, which, in turn, causes it to orbit around the Sun rather than
move in a straight line.
• General Relativity: The Mass-Energy Distribution of the Sun changes the
geometry of space-time. The Earth is in free motion (no forces acting on
it!) and travels on a geodesic of space-time. But because space-time is
curved around the Sun, the Earth orbits the Sun.
Explain this in words and with next two slides
The “Revised” Theory
Newtonian Mechanics – 3 space coordinates – no time coordinate
 no relation between event 1 and event 2
 need the Special Theory of Relativity
 need frames of reference
 need Lorentz Transformation
However: Galaxy is accelerating due to
- other galaxies around it
- expansion of the Universe
 Include accelerations into the Theory of Special Relativity
 Describe geometry of space-time surrounding massive objects
 Link geometry to motions in the universe
 General Theory of Relativity
Task of General Relativity
Couple Geometry to the Mass distributions and motions
1) How does matter affect the Geometry of Space-Time
2) How do particles move in this Geometry
Procedure:
1) Describe Geometry
2) Incorporate Kinematics
3) Link the two
NO Forces!
link
Geometry
Mass Distribution
Discussion
Space-Time travel? Warp Speed? Black Holes? Worm Holes?
What do you know about Space-Time Distortions?
What is science fact and what is science fiction?
Elicit Student Ideas: What do you know?
Summative Activity or Review
How and why does the apple fall off the tree?
How and why does the Earth go around the Sun?
Provide an explanation using
a) Aristotle’s Theory
b) Newtonian Theory of Gravity
c) Einstein’s General Theory of Relativity
Aristotle/Newton/Einstein Matrix
Aristotle
Orbits
Free Fall
What is
Gravity
Newton
Einstein
How & why does Earth orbit Sun?
Aristotle: The natural movement of the celestial bodies is circular
rather than a movement toward the center of the earth – free
motion
No Force
Newton: The Sun creates a gravitational field that exerts a force
upon the Earth, which, in turn, causes it to orbit around the Sun
rather than move in a straight line. (Revised Newtonian: N-body
problems and motion around the Center of Mass)
Gravity is a Force
Einstein: The Mass-Energy Distribution of the Sun changes the
geometry of space-time. The Earth is in free motion (no forces
acting on it!) and travels on a geodesic of space-time. But
because space-time is curved around the Sun, the Earth orbits the
Sun.
No Force
How & why does the apple fall off the tree?
Aristotle: The natural movement of the earthly bodies is toward the
center of the earth – free motion
No Force
Newton: The Earth has a gravitational field that exerts a force upon
the apple. This force accelerates the apple and causes it to fall
towards the earth. (Revised Newtonian: N-body problems and
motion around the Center of Mass)
Gravity is a Force
Einstein: The Mass-Energy Distribution of the Earth changes the
geometry of space-time. This causes the apple to accelerate
towards the Earth.
No Force
Final Questions for Students to ponder
• Which theory is more accurate? Which theory to use when?
• Aristotle and Newton both support the notion of “no forces”. So then,
what is gravity?
• Aristotle and Newton both support the notion of “free motion”. What
is free motion?
• Which theory is more intuitive to you? Why?
Discussion