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Newton’s Law of Gravity
Newton’s Law of Gravity
Key Concepts
1) Gravity is an attractive force between
all pairs of massive objects.
2) Gravity is an inverse-square-law force.
3) Newton’s law of gravity can be used
to compute the shape of an orbit.
Newton’s insight:
Gravity is an attractive force between
all pairs of massive objects.
How big is the force? That’s given
by a (fairly) simple formula.
Newton’s Law of Gravity
FG
F = force
mM
r2
m = mass of one object
M = mass of other object
r = distance between centers of objects
G = “gravitational constant”
(G = 6.7 × 10-11 Newton meter2 / kg2)
What is gravitational force between
Earth and cookies?
FG
mM
r2
M = mass of Earth = 6.0 × 1024 kg
m = mass of cookies = 0.454 kg
r = radius of Earth = 6.4 × 106 meters
G = 6.7 × 10-11 Newton meter2 / kg2
F = 4.4 Newtons = 1 pound
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What’s the acceleration of the cookies?
Combining the two equations:
Newton’s 2nd law of motion:
aF/m
a
GmM 1 GM
  2
r2
m
r
For the Earth,
Newton’s law of gravity:
mM
FG 2
r
Gravitational force varies
directly with mass and
inversely with square of distance.
Double the distance between objects:
Force 1/4 as large.
Triple the distance between objects:
Force 1/9 as large.
G  6.7 1011 Newtons m2 / kg2 (very small! )
Newton Says:
•Objects move in straight lines at constant
speed unless a force acts on them.
•The Moon moves on a curved path
at changing speed.
•Therefore a force is acting on the Moon:
that force is gravity.
a
GM
 9.8 meters/sec2
2
r
INDEPENDENT OF THE MASS OF THE COOKIES!
Gravitational acceleration decreases with
distance from the Earth’s center.
Top of CN Tower:
weight = 180 pounds
minus ½ ounce.
560 m
Base of CN Tower:
weight = 180 pounds.
Gravity makes apples fall; it also keeps
the Moon on its orbit around the Earth, &
the Earth on its orbit around the Sun.
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HALFTIME QUIZ:
You tie a string to a ball and whirl it in a circle. If
the string suddenly breaks, in which direction
will the ball move? (Ignore gravity.)
a) radially outward
b) radially inward
c
c) at a tangent
a
b
As Sun pulls on Earth,
Earth pulls on Sun.
Both Sun and Earth orbit the center
of mass of the Sun – Earth system.
Newton modified and expanded
Kepler’s Laws of Planetary Motion.
Kepler’s First Law: The orbits of
planets around the Sun are ellipses
with the Sun at one focus.
Newton’s revision: The orbits of
any pair of objects are conic sections
with the center of mass at one focus.
Center of mass is
closer to the more
massive body.
Center of mass = balance point.
Center of mass of the Earth – Moon
system is inside the Earth.
Circles, ellipses, parabolas, and
hyperbolas are called conic sections.
Why? Slice a cone and see!
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Newton: shape of orbit depends on
speed of satellite at launch.
A satellite will have a circular orbit if its initial
speed = circular speed ( vcirc )
v circ 
GM
r
Presented without proof (life is too short).
Low speed = closed orbit (circle, ellipse).
High speed = open orbit (parabola, hyperbola).
To stay in low Earth orbit, a satellite must have
v = vcirc= 7.9 km/sec (18,000 mph).
r = radius of circular orbit
M = mass of object being orbited
Escape Speed
• Minimum speed needed to break free of the
gravity of a massive body starting from a
given distance
vE  2GM
r
• From the Earth’s surface:
– vC = 7.9 km/sec (28,400 km/hr)
– vE = 11.2 km/sec (40,300 km/hr)
Hyperbola
v>vE
Parabola
v = vE
Kepler’s Second Law: A line from a
planet to the Sun sweeps out equal
areas in equal time intervals.
Ellipse
Ellipse
vC<v<vE
Circle
v = vC
v<vC
Closer the planet to the Sun,
faster it moves
Newton’s revision: Angular momentum
is conserved.
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The product of orbital speed (v) and
distance from center of mass (r) is constant.
Second Law of Orbital Motion
• Orbital motions conserve angular
momentum.
• Angular Momentum:
• L = mvr = constant
• m=mass, v=speed, r = distance from the
center of mass.
• At constant mass:
As r decreases, v must increase.
Kepler’s Third Law:
P2  a3
– Increase r, v must decrease proportionally
– Decrease r, v must increase proportionally
D
Kepler’s Third Law (P2=a3) applies
only to objects orbiting the Sun.
Newton’s revision applies to
all orbiting pairs of objects.
Newton’s revision:
 4 2
 3
P 
a


G
m

m
1
2 

2
Mass of one object
Example:
Binary stars
Mass of other object
Kepler described how planets move;
Newton explained why they move
that way.
Kepler’s Laws result naturally from
Newton’s Laws of Motion & Law of Gravity.
Kepler’s Laws of Planetary Motion, as
modified by Newton, are now universal.
The Mathematics of Change
• The full statement of the laws required the
invention of a new mathematical language:
the Calculus
– Independently invented by Newton & Leibnitz
• Calculus is the mathematics of change:
– Gives us a way to describe the change in the
velocity of a moving object with time.
– Calculus sets geometry into motion.
• Provides a framework for exploring
motion.
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The works of the giant
• Newton
–Invented the integral & differential calculus.
–Developed the binomial theorem.
–Started fundamental work on optics.
- invented a reflecting telescope
–Formulated his laws of motion & gravitation.
– experimented in alchemy.
–Was always unprepared for classes & hated to
teach.
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