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Physics 218
Lecture 21
Dr. David Toback
Physics 218, Lecture XXI
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Schedule for the Rest of the Semester
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Today: Chapter 12
Tuesday Nov 21st: Exam 3
Thursday Nov 23rd: No class, Thanksgiving
Tuesday November 28th: Chapter 13, Part 1
Thursday November 30th: Chapter 13, Part 2
Tuesday December 5th: Final Exam Review
No lecture on December 7th (Reading day)
Final exam is Monday, December 11th, 1-3PM
We will skip Chapter 15
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Chapter 12: Overview
• Newton’s gravitational law
• Dynamics and Gravity
• Gravity and Uniform
Circular Motion
• Escape Velocity
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Physics 218, Lecture XXI
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Gravitation
Newton’s law of Universal
Gravitation
“Every particle in the
universe attracts every
other particle”
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Large number of scales
Kinda amazing!
Gravity covers the attraction between
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An apple near the earth
The earth and the moon
The earth and the sun
The sun and our galaxy
Our galaxy and the universe
Every particle in the universe and an apple
The Earth and you
Bevo and Reveille
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Newton’s Law
“Every particle in the universe attracts every
other particle with a force that is
proportional to the product of their masses
and inversely proportional to the square of
the distance between them. This force acts
along the line joining the particles”
• Gravity has a magnitude and direction
 Gravity is a force
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The Force of Gravity

m1m 2
Fgravity  G 2 (rˆ)
r
Direction of the
Distance between
force
the masses
G  6.67 10
11
N  m /kg
2
2
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The motion of the moon and the planets
• One of the great achievements of physics is
that it explains the motion of planets
• It took awhile, but they eventually figured out
that the motion of the planets made much
more sense if one assumed that the Sun was
the center of motion rather than the Earth
• Newton was able to use his gravitational
law and Uniform Circular Motion to
“Predict” these observations
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Gravity and Circular Motion
• Use the force of gravity
along with other forces in
force diagrams
• Circular motion is motion
with the acceleration pointed
towards the center of the
circle
• The Earth is a good “center”
acceleration for Satellites
and Moons
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Binary System
Two equal mass stars maintain a constant
distance RS apart and rotate about a point,
midway between them, at a revolution rate
of once per time T
What must be the mass of each star?
RS
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Gravitational Potential Energy
How much potential
energy does a ball of
mass m have in outer
space?
Assuming you know ME
and G, calculate how
much work you would
have to do move a ball
from the surface of the
earth to some distance R
Physics 218, Lecture XXI
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Escape Velocity
• What happens if I throw a ball up in
the air? Will it fall down?
• What if I throw it up really fast?
• What if I throw it up REALLY fast?
• Can I throw it up so fast that it will
never come down? How fast would
that be?
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Example: Geosynchronous Satellite
A satellite is in orbit around the Earth
and its speed is such that it always stays
above the same point on the earth
throughout the day.
Assuming a spherical Earth with mass
ME, determine the height of the satellite
(from the center of the Earth) in terms
of the period, ME and G
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Next Week
• Monday: Chapter 11 HW due
• Tuesday: Exam 3
– Covers Chapter 8 through 11
• Only sections listed on the syllabus
– Mini-practice exam is open and available for
people who are caught up. Usual 5 points
• Thursday: Thanksgiving, no class
• Following Tuesday: Lecture on Chapter 13
– Periodic Motion
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Note:
Do not sell your textbook!
You will need the same
book for 208
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Three Bodies
Three bodies of identical and known mass M form
the vertices of an equilateral triangle of side L. They
rotate in circular orbits around the center of the
triangle and are held in place by their mutual
gravitation.
What is the speed of each?
L
L
L
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Density of a Planet
A satellite orbits very near a planet’s
surface (I.e., R=RPlanet, but we don’t
know R) with period T.
What is the density of the planet in
terms of the measured period T?
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Kepler’s
st
1
Law
Kepler: The path of
each planet about the
Sun is an ellipse with
the Sun at one focus.
Newton: Gravity gives
a center seeking
force! Centripetal
forces can lead to
both elliptical motion
or circular motion
(circular motion is
just a special case
which we happen to
have studied already!)
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Kepler’s
nd
2
Law
Kepler: Each planet
moves so that an
imaginary line drawn
from the Sun to the
planet sweeps out area
in equal periods of time
Newton: Can show this,
but it’ll be easier when
we get to angular
momentum. We’ll come
back to this later
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Kepler’s
rd
3
Law
Kepler: The ratio of the squares of the periods of
any two planets revolving about the Sun is equal to
the ratio of the cubes of their semi-major axes.
2
 T1   S1 
    
T
S
 2  2
3
For a Circle S = Radius
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Derivation of
rd
3
Law for a Circle
Let’s say there are two small planets revolving
around the sun in circular motion with periods
T1 and T2 and radii R1 and R2 respectively.
Use Newton’s laws to show the following:
2
 T1   R 1 
    
T
R
 2  2
3
Physics 218, Lecture XXI
Assume the masses of
the planets are so
small or that they are
so far apart that we
may neglect the
gravitational
attraction between
the two.
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Understanding the motion of planets
• One of the great achievements of physics is
that it explains the motion of planets
• It took awhile, but they eventually figured out
that the motion of the planets made much
more sense if one assumed that the Sun was
the center of motion rather than the Earth
• Then Kepler made some important
observations WAY before Newton
• Newton was able to “Predict” these
observations (and now so are we…)
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A way simplify problems
It takes some fancy integration, but
one can show that we can “model”
the Earth as if all the mass were
concentrated at its center
• One can model any sphere as a
point
• This is why we like to model things
as spheres in the first place.
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Calculate the Magnitude of g
For a person on the earth what is
g. Use
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– MEarth = 5.97*10 kg
-11
2
2
– G = 6.67*10 N*m /kg
– R = REarth = 6.38*106 m
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Ok… So what?
A single formula can show that:
1. the acceleration of an apple
towards the Earth has magnitude g
and is pointing down
2. as you may have noticed, Bevo and
Reveille aren’t attracted to each
other very much
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Bevo and Reveille
Bevo and Reveille have masses mB and mR and
are standing D meters apart.
Despite what you might like to believe, what is
the attraction between them? More
specifically, find Reveille’s acceleration.
Hint: Assume a spherical cow
Perhaps this explains why we’ve never
observed any attraction…
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How to solve these types of problems
• Some thoughts
– What keeps a satellite up? Its speed
– |Accel| = v2/r
– |Force| = ma = mv2/r
• The trick is going to be to ask the question
– What are the forces?
Is it in uniform circular motion? If so, we
can use Newton’s law:
FGravity = FUniform Circular Motion
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Space craft in Orbit
A space craft, with mass m, is circling
the earth at radius R = 2rEarth.
What is the force on the space craft in
terms of g and m?
Model the space craft
as a single point near
the Earth which we
model as a point at its
center
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Satellites/Orbiting Problems
A satellite problem is a good example of the
substantive problems we need to be able to
solve. Predict the outcome of the experiment
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Weightlessness
• What is the weight of
the person in the
figure?
• What is the difference
between being in “free
fall” and being “out of
the reach of large
gravitational forces?”
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Throwing a Baseball
A person throws a baseball at 100 km/hr, but
it is attracted back towards him because of
gravity.
• Estimate the force 1 Meter away?
• Assuming constant acceleration (Bad
assumption), how long does it take to turn
around?
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Vector Form
Force ON mass 2 due to mass 1

m1m 2
F21  G 2 rˆ12
r21
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More on Vector Form
r̂21  r̂12
By Newtons Laws :


F21  F12
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