• Historically very interesting, Geocentric vs. heliocentric universe

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

Chapter 13: The Law of Gravity

Reading assignment: Chapter 13.1 to 13.5

Homework : (due Monday, Nov. 12):

Problems: 3, 5, 9, 11, 17, 20

- 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

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

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

- 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 Gravitation

Every 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

Newton’s Law of Universal Gravitation

1 212 122

ˆm m

F G rr

- Particle 1 is attracted by particle 2 (and vice versa).

- F12 is force exerted by mass 1 on mass 2 (and vice versa).

- r12 goes from mass 1 to mass 2.

- F12 and F21 form an action-reaction pair

- Force drops off as 1/r2 as distance r between particles increases

- Can treat spherical, symmetric mass distributions as if the mass were concentrated in center of mass.

2112 FF

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 = 30° for both?

Black board example 13.1

Measuring the gravitational constant – Cavendish apparatus (1789)

Free-Fall Acceleration and the Gravitational Force

mgF

R

mMGF

g

E

Eg

2

Gravitational force:

Thus: 2E

E

R

MGg

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

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

Earth’s gravitational field

Close to the surfaceFar away from the surface

Gravitational force acts from a distance through a “field”

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

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

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

.3

2

consta

T

Kepler’s laws about planetary motion

Most planets, except Mercury and Pluto, are on almost a circular orbit

Earth:

Ratio of minor to major axis b/a = 0.99986.

For planets around sun:

3

219

3

2

1097.2m

s

a

T

Black board example 13.3

The solar system

If the Mars year is 1.88 earth years, what is Mars’ distance from the sun?

Calculate the mass of the sun using the fact that the period of the earth’s orbit is 3.157·107 s and it’s distance from the sun is 1.496·1011 m.

Inner planets

Further out: Saturn, Uranus, Neptun, Pluto

All nine eight planets of the solar system