Chapter 12: Gravitation

1 22

-11 2 2

G=gravitational constant = 6.673(10) 10 /

g

Gm mF

r

Nm kg

E

The weight of a body of mass m on the earth's surface with

radius

Note:

R is

2 2 or E E

E E

Gm m Gmmg g

R R

Newton’s Law of Gravitation

Gravitational attraction

Note: Two particles of different mass exert equally strong gravitational force on each other

Gravitational Forces (I)

‘‘Attractive Force”

MM ME

FG = G r 2

ME

MM

Cavendish balance

Cavendish(1798) announced that he has weighted the earth

g = 9.80 m/s2

RE = 6.37 x 106 m

ME = 5.96 x 1024 kg E = 5.50 x 103 kg/m3

= 5.50 g/cm3 ~ 2 x Rock

???

Average Density of the Earth

The gravitational force is conservative

Note:

Work done is

Path independent!

Gravitational potential energy

Note: The gravitational potential is zero at infinity

Projectile

1 1 2 2

21

1

1( ) 0 ( )

2 2

/

E E

E E

E E

K U K U

Gm m Gm mmv

R R

v Gm R

Projectile

Circular orbit

2

2

/

E

E

G m m mv

r r

v Gm r

Kepler’s Laws

1. Each planet moves in an elliptical orbit, with the sun at one focus of the ellipse

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

3. The periods of the planets are proportional to the 3/2 powers of the major axis lengths of their orbits.

Geometry of an ellipse

Elliptical orbit

0d L

r Fdt

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