Chapter 6 - Gravitation

• Newton’s Law of Gravitation (1687)

• Kepler’s Laws• Implications of Newton’s Law

of Gravitation– Gravitation near the Earth’s

surface– Superposition of forces– Justification for Kepler’s second

and third law

• Gravitational Potential Energy

Newton’s law of (universal) gravitation

GF acts along the line

joining the two objects

G 1 2F m m

G 2

1F

r

1 2G 2

Gm mF

r

Cavendish Experiment (1798)

211

2

N mG 6.67x10

kg

Vector Form of Newton’s Law of Gravitation

1 212 122

21

Gm mˆF r

r

21 2 1r distance from mass m to mass m

12 1 2r̂ unit vector oriented from mass m to mass m

m1m2

12r̂12F

21r

12 2 1F force on mass m due to mass m

by on

Superposition of Forcesn

0 10 20 30 i0i 1

F F F F .... F

0 1 0 2 0 30 10 20 302 2 2

10 20 30

Gm m Gm m Gm mˆ ˆ ˆF r r r ....

r r r

n31 2 i

0 0 10 20 30 0 i02 2 2 2i 110 20 30 i0

mm m mˆ ˆ ˆ ˆF Gm r r r .... Gm r

r r r r

m3

m2

m1

m0

10F

30F20F

10r

20r

30r

Problem 1

• Three masses are each at a vertex of an isosceles right triangle as shown. Write an expression for the force on mass three due to the other two.

r

r

m3

m1

m2

Gravity near the earth’s surface

1 2G 2

Gm mF

r

E2

Gm mmg

r

E2

Gmg

r

Kepler’s Laws

• The Law of Orbits– All Planets move in elliptical

orbits, with the sun at one focus.

• The Law of Areas– A line that connects a planet

to the sun sweeps out equal areas in equal times.

• The Law of Periods– The square of the period of

any planet is proportional to the cube of the semi major axis of its orbit

2 3T R

Kepler’s 2nd Law

• The Law of Areas– A line that connects a

planet to the sun sweeps out equal areas in equal times.

1dA rvdt

2

2 vL I mr mrv Constant

r

1 LdA dt

2 m

Justification of Kepler’s third law(for circular orbits around the sun)

Ra

v

2

R R

vF ma m

r

2S E

E2

GM m vm

r r

32

2

S

4T r

GM

Problem 2

• Verify Kepler’s third law for the earth revolving around the sun.

• distance from sun to earth = 1.496 x 1011 m• mass of sun = 1.99 x 1030 kg.

r

Ms

Problem 3

• What would be the height of a satellite with a period of one day

• mass of earth = 5.98 x 1024 kg.

r

me

hRe

Gravitational potential energy again

E2

Gm mˆF r

r

2

1W F dl

W U

EGM mU r

r

Escape velocity

2 Eesc

E

GM m1mv 0

2 r

Eesc

E

2GMv

r