Universal Gravitation
Lecturer:
Professor Stephen T. Thornton
Reading Quiz
The International Space Station is at an altitude
of 200 km above the surface of the Earth.
What is the net force on an astronaut at rest
inside the Space Station?
A)
B)
C)
D)
E)
Equal to her weight on Earth.
A little less than her weight on Earth.
Less than half her weight on Earth.
Zero (she is weightless).
Somewhat larger than her weight on Earth.
Answer: B
The astronaut is falling
around the Earth. The
gravitational force is
keeping her from going
in a straight line. The
acceleration of gravity
is a little less than g at
200 km above the
Earth.
200 km
6380 km
Last Time
Non-uniform circular motion
Drag
Terminal velocity
Fundamental forces
Today
History of gravitation
Newton’s law of universal gravitation
Kepler’s laws
History of Gravitation
Greeks used the geocentric frame in
which the Earth was at the center.
Ptolemy, 2nd A.D., prepared a
detailed formulation of heavenly
body motion.
epicycles
retrograde motion
 Nicolaus Copernicus – (1473-1543 )
introduced heliocentric frame with the
Sun at the center of the solar system.
 Catholic church thought this was heresy.
 Danish astronomer Tycho Brahe (15461601) made huge number of observations.
Johannes Kepler continued his work and
did analysis.
Galileo (1564-1642) is said to have invented
the telescope. He made many observations.
Issac Newton (1642-1727) was one of the
smartest persons to ever live.
He invented calculus so he could solve the
problem of how the moon rotates around
Earth.
Newton looked at Kepler’s results and figured
everything out in a manner of days!
Newton’s Law of Universal Gravitation
If the force of gravity is being exerted on objects
on Earth, what is the origin of that force?
Newton’s realization was
that the force must come
from the Earth.
He further realized that
this force must be what
keeps the Moon in its
orbit.
conic sections
Newton showed
that planetary
motion and other
similar motion
had to be of conic
sections.
Newton’s Law of Universal Gravitation
The force of gravity between any two
point objects is attractive and of
magnitude:
m1m2
F G 2
r
G is the universal gravitational constant
G  6.67 10
11
N  m / kg
2
2
Gravitational Force
Between Point Masses
m1m2
F G 2
r
Important points about gravity
1) Gravitational force is a vector and
always attractive.
2) Gravity is difficult to measure, except
for large bodies.
3) Gravity has an exact 1/r2 dependence.
4) For several masses, just add forces.
Called superposition.
Gravitational Force Between a Point Mass
and a Sphere (uniform mass density)
Use symmetry. For
this case, force acts
at center.
FMm
FmM
Gravitational Force Between
the Earth and the Moon
FEm
FmE
FEm
FmE
Conceptual Quiz
Which is
stronger,
Earth’s pull on
the Moon, or
the Moon’s
pull on Earth?
A)
B)
C)
D)
the Earth pulls harder on the Moon
the Moon pulls harder on the Earth
they pull on each other equally
there is no force between the Earth
and the Moon
E) it depends upon where the Moon is
in its orbit at that time
Conceptual Quiz
Which is
stronger,
Earth’s pull on
the Moon, or
the Moon’s
pull on Earth?
A)
B)
C)
D)
the Earth pulls harder on the Moon
the Moon pulls harder on the Earth
they pull on each other equally
there is no force between the Earth
and the Moon
E) it depends upon where the Moon is
in its orbit at that time
By Newton’s 3rd Law,
the forces are equal and
opposite.
Conceptual Quiz
If the distance to the
Moon were doubled,
then the force of
attraction between
Earth and the Moon
would be:
A)
B)
C)
D)
E)
one quarter
one half
the same
two times
four times
Conceptual Quiz
If the distance to the
Moon were doubled,
then the force of
attraction between
Earth and the Moon
would be:
A)
B)
C)
D)
E)
one quarter
one half
the same
two times
four times
The gravitational force depends
inversely on the distance squared.
So if you increase the distance by a
factor of 2, the force will decrease by
a factor of 4.
Mm
F G 2
R
Follow-up: What distance would increase the force by a factor of 2?
Force of gravity on a mass m on the
surface of the Earth is mg.
Let’s use Newton’s universal law.
mM E
F  mg  G 2
RE
GM E
g 2 
RE
(6.67 10
11 N  m
kg
2
2
)(5.98 1024 kg)
(6.38 106 m) 2
 9.80 m/s 2
Assume we know g , but not M E . Solve for M E :
2
E
gR
ME 
 5.98  1024 kg
G
Kepler’s three laws follow naturally from
Newton’s Law of Universal Gravitation.
First law occurs because of 1/r2. Orbits
must be ellipses.
We will derive Kepler’s laws later after we
study angular momentum.
Kepler’s 1st
and 2nd laws
http://physics.bu.edu/~duffy/
semester1/semester1.html
2nd law: Radius
vector sweeps out
equal areas in same
time interval.
Kepler’s 3rd law follows directly from the
form of the gravitational force law.
4 3
T 
r
GM S
2
2
 2  3/2
T 
r

 GM 
S 

Kepler’s Third Law and Some Near Misses
Free Fall
Look at
http://galileoandeinstein.physi
cs.virginia.edu/more_stuff/Ap
plets/newt/newtmtn.html
Gravitational Attraction. Two
objects attract each other
gravitationally with a force of
10
2.5 10 N when they are 0.25
m apart. Their total mass is 4.00
kg. Find their individual masses.
Sun’s Mass Determination.
Determine the mass of the Sun using
the known value for the period of the
Earth and its distance from the Sun.
[Hint: The force on the Earth due to the
Sun is related to the centripetal
acceleration of the Earth.] Compare
your answer to that given in your
textbook.
Conceptual Quiz
The gravitational constant G is
A) equal to g at the surface of Earth.
B) different on the Moon than on
Earth.
C) obtained by measuring the speed of
falling objects having different
masses.
D) all of the above.
E) none of the above
Answer: E
None of them determine G.
Newton’s Law of Universal Gravitation
Using calculus, we can show:
Particle outside a thin spherical shell:
gravitational force is the same as if all mass
were at center of shell.
Particle inside a thin spherical shell: gravitational
force is zero. See next slide.
Can model a sphere as a series of thin shells;
outside any spherically symmetric mass,
gravitational force acts as though all mass is at
center of sphere.