• Historically very interesting, heliocentric vs. geocentric universe

• The main cast: Copernicus, Brahe, Galileo, Kepler, Newton

• Satellite motion

Chapter 13: The Law of GravityReading assignment: Chapter 13.1 to 13.5

Homework (due Wednesday, Nov. 14): OQ1, OQ2, OQ5, OQ8, 2, 3, 6, 10, 13, 14, 18, 28, 33

Copernicus1473 – 1543

Brahe1546 – 1601

Galileo1564 – 1642

Kepler1571 – 1630

Newton1643 – 161727

- Earth is round! Slowly accepted, starting in ancient Greece, Magellan + Elcano circumnavigation (1519 – 1522).

- Ptolemy (100 –170 A.D.) geocentric model: Sun revolves around earth (Wrong!)

From astronomical observations: - Copernicus (1473-1543) heliocentric model: Earth & planets revolve around sun

- Brahe (1546 - 1601) Accurate observation of planetary motion

- Galileo (1564 - 1642) (1610) supports the heliocentric model

- Kepler (1571-1630), 1609: Laws I, II of planetary motion,

- Kepler 1619: Law III of planetary motion

- Aristotle (384-322 B.C.) Heavier objects fall faster than light objects (Wrong!)

- Galileo (1564 - 1642) Neglecting air resistance, all objects fall at same acceleration

Geocentric vs. heliocentric model of earth

About falling objects

Newton’s Law of Universal GravitationEvery particle in the Universe attracts every other particle with a force of:

1 212 122

ˆm m

F G rr

G… Gravitational constant G = 6.673·10-11 N·m2/kg2

m1, m2 …masses of particles 1 and 2

r… distance separating these particles

… unit vector in r direction12r̂

Experimentally confirmed.

Measuring the gravitational constant – Cavendish apparatus (1789)

What is the attractive force you (m1 = 100 kg) experience from the two people (m2 = m3 = 70 kg) sitting in front of you. Assume a distance r = 0.5 m and an angle q = 30° for both?

Black board example 13.1

q

Free-Fall Acceleration and the Gravitational Force

mgF

R

mMGF

g

E

Eg

2

Gravitational force:

Thus: 2E

E

Mg G

R

g is not constant as we move up from the surface of the earth!

G is a universal constant (does not change at all).

a. What is the value of g in the ISS space station that is at an altitude of 400 km. Assume ME = 5.960·1024 kg and RE = 6.370·106 m.

b. Why does it feel like g = 0?

Black board example 13.2

Variation of g with altitude

The ISS photographed from shuttle Discovery in 2006.

From http://www.daviddarling.info/encyclopedia/I/ISS.html

Gravitational potential energy(similar equation for charge-charge interaction)

r

mmGrU 21)(

• Notice the – sign

• U = 0 at infinity

• U will get smaller (more negative) as r gets smaller.

• “Falling down” means loosing gravit. potential energy.

• Use only when far away from earth; otherwise use approximation U = mgh.

(a) What is the escape speed of a particle on earth (ignore air resistance)?

A) ~10,000 m/s B) ~11,000 m/s C) ~20,000 m/s D) ~22,000 m/s

First Rocket Launch from Cape Canaveral (NASA); July 1950

Black board example 13.3

Kepler’s first two laws (1609):

I. Planets move in elliptical paths around the sun. The sun is in one of the focal points (foci) of the ellipse.

II. The radius vector drawn from the sun to a planet sweeps out equal areas in equal time intervals (Law of equal areas).

Kepler’s laws about planetary motion

These laws hold true for any object in orbit around a central mass

Area S-A-B equals area S-D-C

Kepler’s third law (1619):

III. The square of the orbital period, T, of any planet is proportional to the cube of the semimajor axis of the elliptical orbit, a.

Kepler’s laws about planetary motion

32 aT

Thus, for any two planets:

3

2

1

2

2

1

a

a

T

T

2 2

3

4.

S

Tconst

a G M

G… Gravitational constant G = 6.673·10-11 N·m2/kg2

MS … central mass, (i.e. mass of sun for planet motion)

All nine eight planets of the solar system

Inner solar system:

Mercury

Venus

Earth

Mars

Outer solar system:

Jupiter

Saturn

Uranus

Neptune

(Pluto)

My Very Educated Mother Just Served Us Nine (Plums)