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
