Learning to Think About Gravity

Concepts of Gravity from Aristotle to Newton

Esther Zirbel

Claudine Kavanagh

Tufts University

Overview of Class1) Tell students the objective and tell them they will get confused2) Elicit Student’s Ideas (let them fill out the motions in the universe sheet)3) Historic Overview – Constructivist Approach4) Talk about Aristotle’s Ideas

a) Have a discussion where you challenge students to disprove Aristotle5) The Birth of Scientific Methodology – “How”

a) How does the apple fall off the tree– Galileo – the Experimentalist

b) How do planets orbit the Sun?– Brahe – the Observer– Kepler – the Analyst

c) How does scientific methodology deepens the understanding of how things work?d) What is “modeling” in science?

6) Talk about Newton’s Theory – “Why”a) Mention Newton’s three laws but focus on conceptually deriving the Universal

Law of Gravitationb) Do the Cannon Ball thought experiment with the studentsc) Discuss what makes a theory a theory

7) What did you learn?8) Concluding Remarks and Summary

1) Goal of Today’s Class

• Elicit your own Ideas about Gravity• Confront you with your own Understanding

of Gravity!• Learn Scientific Methodology• Differentiate between Empirical Models

and Theories• Discover how Discoveries are made• Which Theory should you chose?

A Comparison of Aristotle and Newton

1. How & why do objects fall?Terrestrial motion

2. How & why does Earth orbit Sun? Celestial motion

3. What is Gravity?

Cause of motion

Learning to Think About Gravity

2) Motions in the Universe

Your interpretationsHow – 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?

3) A (very) short history:

• Aristotle philosophizes on matter and motion• logic but no experiment• differentiates between earthly and heavenly motions

• Copernicus suggests alternate model of solar system• Galileo provides “scientific methodology”

• 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?”

philosophy

How?EmpiricalModeling

why

what

Timeline

A) Overview on Free-fallHow and why does the apple fall off the tree

Aristotle – uses “common sense”The motions of earthly things are determined by their natural tendencies to move towards their proper place in the cosmos, i.e. towards the center of the Earth.

How does the Apple fall? Galileo – Experiment and Analysis

Observe and Experiment (e.g. Tower of Pizza)Deduce (& mathematical description; e.g. s= ½ g/t)

Behavior of Forces & Laws of MotionNewton – Discover and Formulate three laws

Prove using scientific methodology(not covered in this class!)

Why does the Apple fall?Newton – Discover Theory

A force acts on the appleThis force makes the apple accelerate toward Earth==> Gravity acts on the body

Philosophy

Scientific Theory

Scientific Methodology

Section 5a

Section 4

Section 6

B) Overview on Orbital MotionHow and why do the planets orbit the Sun?

Aristotle – uses “Religion and Plato”Heavenly things move in perfect circles Plato’s principle

How do the planets go around the Sun? Brahe – ObserveKepler – Deduce (mathematical description of three

laws)

Why do the planets go around the Sun?Newton -- Theory

A force acts on the planetsForce is counterbalanced by centrifugal forceEffect is that the planets accelerate toward Sun==> Gravity acts on the bodies

Philosophy

Scientific Theory

Scientific MethodologyEmpirical Modeling

Section 5b

Section 6

Section 4

We have been Aristotelian for a long time, indeed…..

Aristotle: ~380-320 BC

Newton: 1678 / 1728

Einstein: 1905 (published)

Today: 2007 schools are (still!) teaching 300 yr old theory

4) Aristotle and MotionTwo 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.

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!!

Discussion

• How to disprove Aristotle’s Ideas about Gravity

• Experience that this is not easy to do!

5)The Birth of Scientific Methodology

Galileo Galilei

Father of Scientific Methodology:1) Studies Free Fall & Pendulum

2) Motion & Mechanics

Points Telescope at Celestial Objects:1) “New” stars (many more to be seen)

2) Pitfalls on the Moon (anti Plato)3) Moons of Jupiter (not everything revolves

around Earth)4) Phases of Venus (the Sun is in the center)

5a) How do objects fall? – the Story

Experiment 1

How does the apple fall from the tree?

Drop ObjectsWhat does the Fall depend on?

a) Constant Speed?

Galileo found that 'gravity' caused objects to fall at a changing rate. This is called acceleration..

b) Dependence on shape?c) Dependence on weight?

Test! Experiment! Look at Results! Then Philosophy!

Experimental Results

• Quote Result

Speed increases as bodies fall; they accelerate!

Mathematical Description(based on Empirical Results)free-fall s= ½ at2

• Prove Result

All bodies “fall” at the same rate– NO dependence on Weight– NO dependence on Shape

Experiment 2

Introduction – elaborate on Galileo’s experiment of rolling balls down an incline and measuring s= ½at2

Explain experimentWhich of these cans rolls fastest?Make a Prediction! Discuss “reasons” for the predictionsThen do the experiment

• Empty can• Can filled with liquid• Can filled with solid food

Discussion after the experiment

• Discuss “correct” interpretation but DO NOT tell students the final answer

• Discuss purpose of the exercise– Why did you do this exercise?– What do you want your students to get out of this exercise?

Comment to Students• So far have discussed how the apple falls from the tree –

have not yet discussed why the apple falls from the tree. Note that Galileo does not provide answer for “why”.

5b) How does Earth go around the Sun?

— Aristotle: Philosophy

Based on Plato’s Ideas of the “Perfect Heavens”

Motion of celestial bodies is circular

— Galileo: Scientific Methodology – no answer (only supplies proof that Earth goes around Sun, not vise versa)

— Brahe observes

— Kepler analyzes Brahe’s Observations:

— Invent Laws empirically

— describes orbits mathematically no cause!

Tycho Brahe (1546 – 1601)

Use of high-precision instruments for precise astronomical observations, meticulously reported in tables.

Later used by Kepler to develop laws of planetary motion.

Johannes Kepler (1571 – 1630)

Used the precise observational tables of Tycho Brahe to study

planetary motion mathematically.

1.Circular motion and

• Planets move around the sun on elliptical paths, with non-uniform velocities.

Found a consistent description by

abandoning both:

2.Uniform motion.

Kepler’s 3 Laws:

1. Orbits of planets are ellipses with the sun at one focus

2. A line from a planet to the sun sweeps out equal areas in equal times

3. A planet’s orbital period squared is proportional to its average distance from the Sun cubed.

mathematical description -- no cause!

32 aP

Scientific Methodology Summary

So far:

Models only

no explanation

• Galileo “Introduces Methodology”

• Brahe "Observer"

• Kepler "Analyst”

• Newton “Provides Model” Why

Discussion

• How does scientific methodology lead to a deeper understanding of how things work?

• What is the role of empirical modeling?

6) 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 motion2. Discovered the three laws of motion

3. Discovered the universal law of mutual gravitation

6a) Newton—Laws of Motion

lst law: Inertiaobject at rest remains at rest object in motion remains in motion unless a force acts on it Absence of force

2nd law: F=maGalileo—free fall accelerationAcceleration is the Result of a force acting on the mass

3rd Law: Actio = ReactioFor every action there is an equal and opposite reactionBalance of forces

It’s all about Forces

What does it mean to be smart in society?

Was Newton were really smart?

Discussion

How and why does the apple fall to the ground?

• Apply Newton’s laws

• Why do heavy and light objects fall at the same rate?

• Aside: How would apples fall on the Moon?

m

Fa

6b) Is 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 the Earth?

Discussion

Why does the Moon not fall towards the Earth?

Why does the moon not travel in a straight line?

Recall Newton's first law

--> A force must act on the moon

What is this force?

Effect of this force?

Centrifugal force

--> change in direction

The Resulting Path of the Planets…

Gravitational pull (blue)

Centrifugal force (green)

Planet’s velocity (red)

Resulting Path (black)

The Theory of Gravity

221

distance

MassMassF

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:

Is this hypothesis correct? — How do you test this? (a) Cannon Ball Thought Experiment

(b) Mathematical Proof

How would you test whether this hypothesis is correct?

Discussion

Is this hypothesis correct? — How do you test this? (a) Cannon Ball Thought Experiment

(b) Mathematical Proof

How would you test whether this hypothesis is correct?

Discussion for students in class

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?

Speed of the Moon? For orbital motion have:

Does this agree to the observed speed?

r

GMv

month1

2

month1time

distance rncecircumferev

Yep!

r

mvF

r

GMmF CG

2

2

Discussion for students

How would you calculate the speed of the moon from theory? Does this agree with observations?

The Universal Law of Gravity

Gravity decreases with the inverse square law

221

distance

MassMassF

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.

“Derive” Kepler’s Third Law

32

2

2

2

4

2

2

rGM

P

P

r

r

GM

P

rvuse

r

GMv

r

mvF

r

GMmF CG

6c) Theory

• What is a Scientific Theory?

• Distinguish this from an Empirical Model?

Discussion

7) What did you learn?

• Compare Aristotle to Newton• Look at your initial answers from section 2• Update those answers• Convince yourself and your class mates that your updated

answers make more sense• How does your everyday experience of gravity contribute

to your understanding of gravity?

Discussion

If appropriate mention student’s common misconceptions about gravity

8) Concluding Remarks

• Science– Learn principles of gravity: How, Why, What– Make the transition from Aristotelian to Newtonian thinking

• Learn scientific methodology– Discover how discoveries are made– Understand the purpose of a model– Understand the meaning of a theory

• Sociology– Learn that making sense of difficult concepts took millennia and a

scientific revolution– Making mistakes is part of learning