Developing a Theory of Gravity

Does the Sun go around the Earth or Earth around Sun?Why does this happen?

Plato }Artistotle } PhilosophyPtolemy } & ModelsCopernicus }

Galileo } ExperimentKepler } & Mathematical Description

Newton } Physics Einstein??

Timeline

How do objects fall to the ground?

• Aristotle philosophies only[without looking for deeper explanations]

• Galileo performs experiment– Observe– Deduce– Then philosophize

• experiment – free-fall (S= ½ at 2) fig. 5-4

• rule out philosophies through observations

--> “scientific proof”

Galileo performs experiments to proof his philosophical views -- he observes and deduces.

Scientific Methodology

Experiment

Drop ObjectsWhat does the Fall depend on?

a) Constant Speed?b) Dependence on shape?c) Dependence on weight?

Test! Experiment! Look at Results! Then Philosophy!

Experimental Results

• Speed increases as bodies fall; they accelerate!Mathematical Description(based on Empirical Results)free-fall s= ½ at 2

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

Intermission

Which of these cans rolls fastest?Make a Prediction!

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

How does Earth go around the Sun?

— Brahe: Observations— Kepler analyzes Brahe’s Observations:

— Invent Laws empirically— describe orbits mathematically

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

Why does the Earth go around the Sun?

So far:

Models only

no explanation

• Galileo “Introduces Methodology”• Brahe "Observer"• Kepler "Analyst”• Newton “Provides Model” Why

Newton—Laws of Motionlst 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

Why does the apple fall off the tree?

Aristotlemotion 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.

NewtonA force acts on the appleThis force makes the apple accelerate==> Gravity acts on the body

…and one day the apple fell off the tree

Aristotle – uses common senseThe 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)

Why does the Apple fall?Newton -- Theory

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

Philosophy

Scientific Methodologyand

Scientific Thinking

Is Gravity a “Universal" Force?

— Why does the Apple fall off the Tree?— Why does the Moon not fall towards the Earth?

Hypothesis:— Gravity acts on Moon— Gravity is counterbalanced by Centrifugal force(remember: Action = Re-actio)

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

Is Gravity a “Universal" Force?

— Why does the Apple fall off the Tree?— Why does the Moon not fall towards the Earth?

Why doesn’t the Moon 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 Moon…

Gravitational pull (blue)Centrifugal force (red) Resulting Path (black)

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?

How would you test whether this hypothesis is correct?

Discussion

Speed of the Moon? For orbital motion have:

Does this agree to the observed speed?

rGMv

month12

month1timedistance rncecircumferev

Yep!

The Universal Law of Gravity

Gravity decreases with the inverse square law

221

distanceMassMassF

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.

Quiz Questions

• Deriving Kepler's Third Law

• Calculating Earth’s Mass(By measuring earth’s acceleration on its surface and distance from center of Earth)

Einstein

Acceleration pulls you down

No Gravity!!

ONLY acceleration

Principle of Equivalence

Need New Theory of Gravity "General Relativity"

Test of General RelativityEclipse in 1917

• Curvature strongest in vicinity of dense and massive objects• (black holes are theoretical playgrounds for relativity people)

Einstein Ring

Difference between Newtonian Theory of Gravity and General Theory of Relativity

• 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.

From the Special Theory of Relativity to the General Theory of Relativity

Newtonian Mechanics – 3 space coordinates -- no time coordinate no relation between event 1 and event 1

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

Acceleration is due to forces Include “forces” into the Theory of Special Relativity General Theory of Relativity

Task of General Relativity

Couple Geometry to the Mass distributions and motions

How does matter affect the Geometry of Space-Time

How do particles move in this Geometry

(no forces!)

ijij TcGG 2

8

Geometry described by Robertson

Walker Metric

Stress Energy Tensor

A constant

22222

22222 sin

1)( ddr

krdrtRdtcdS

Last two equations

(only if coordinates are

chosen cleverly)