Law of Universal Gravitation

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Law of Universal Gravitation

• Gravitational Force
• Continued from Walker Chapter 12

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Its everywhere

• Every particle in the universe attracts every
  other particle with a force that is proportional
  to the mass of the particles and inversely
  proportional to the square of the distance
  between them.
• Like all other forces, the force of gravity is
  measured in Newtons

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Heaven and Earth

• Until Newtons brilliant Universal law of
  Gravitation it was truly thought that the rules
  were different for the heavens and than for the
  earth.
• Newton proved that to be wrong. There are rules,
  nature plays by those rules, we can understand
  those rules and the rules apply EVERYWHERE!!.

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12.1 12.2

• What is important to recall is that gravity is
  only an atractive force, mass 1 pulls on mass 2
  and mass 2 pulls on mass 1.
• The net gravitational force on any given mass due
  to 2 or more masses in a system of interest, is
  the vector sum of the gravitational forces due to
  each of the other masses individually.

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Acceleration due to gravity

• Measurements of distances between two masses of
  interest are measured from the center of gravity
  for each mass.
• So objects on or near the surface of the Earth
  are measured as Mass of object times Mass of the
  Earth divided by the distance squared between
  them. That distance is the radius of the Earth.
  The center of the Earth is taken to be the center
  of gravity of the planet.
• This produces acceleration of 9.8 m/s/s.
• The farther you are from the center of the Earth,
  the greater the radial distance that would
  produce LESS gravitational acceleration, less
  gravitational force.
• Further, smaller planets in mass would produce
  smaller gravitational forces.

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The first and least

• There are only four fundamentally identifiable
  forces in the universe.how simple nature
  continues to beonly 4 forces in the whole
  universe
• Strong force and weak forces are found in the
  nucleus of the atomnot of interest to us in this
  course.
• Electromagnetic forcelight and electricity and
  magnetism and friction and all of chemistry are
  examples of the E-M force.
• Force number 4 is gravity. Of the 4 it is the
  weakest! You can actually overcome gravity for a
  momentjump! What makes it SOOOO important is its
  infinite influence. The force of gravity of
  planet Earth NEVER goes to zero no matter how far
  you fly away from home planet, the pull of
  gravity, though weaker, never goes to zero.

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There was confusion

• The connection between a falling object, an apple
  falling to the ground, and a moon orbiting a
  planet or a planet orbiting a sun was not
  understood to be the SAME force until Newton!
• Newton began to realize that what the Moon was
  doing was constantly falling toward the center of
  the earth.
• Just like astronauts are constantly falling
  around the earth bedcause of the force of
  gravity, so too the moon falls toward earth due
  to gravity just like and apple falls due to
  gravity.
• Recall that in circular motion, the acceleration
  is always pointed toward the center of the orbit.
  The force, gravity, is always directed toward the
  center of the orbit as well.

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Action-Reaction

• When a bug hits the windshield of a moving truck,
  the force is the same for the bug and for the
  truck. It is simply the EFFECT of the force that
  differs.
• So too with the force of gravity
• F G mM / r2.
• Betrween the Earth and the Moon the force is the
  same for each mass.
• The moon pulls on the earth with the same force
  that the earth pulls on the moon. The equation
  above would produce the same results.
• It is the EFFECT that gravitational force has
  that differs and causes the moon to orbit the
  earth since its mass is less and so that force
  would produce a different acceleration on the
  moon than on the earth.
• Recall F MA

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G

• The gravitational constant is the conversion
  factor that relates the force of gravity to the
  masses and the distances between the masses.
• G 6.67 x 10 -11 Nmm / kgkg
• It is a very small value.
• Thus, we cannot detect easily the effect of
  gravity between you and your pencils or between
  you and the chair across the roombut it IS
  there! Every object attracts every other
  object!!! It is just tough for our senses to
  detect since our 5 senses are more attuned to the
  macroscopic worldour senses are limited!!

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Calculations

• The tough part in calculating the true F of
  Gravity between masses is finding the distances
  between the CENTERS of the massesthe
  gravitational centers of irregularly shaped
  objects can be tricky to find.
• In addition one might want to consider all the
  masses that are influencing the mass in question
  which proves to be REALLY difficult
• SOwe will consider only uniform, spherical
  masses and only two at a time
• It was in solving such gravitational problems
  that Newton invented calculus to solve such.
• Thus, we will simply use F G mM / d2
• So the force of gravity, weight, mg, equals the
  mass of one object times the mass of the second
  object divided by the distance squared between
  their centers of mass.

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The Earth is SOOOO BiG

• The force of gravity causes us to have weight.
  Our mass is being pulled down by the earth.
• When we solve for the gravity pulling on us, we
  can really ignore our mass! It is truly
  negligible when compared to our planet..so we can
  drop our mass out of the equation.
• For example to solve for acceleration due to
  gravity, g we can simply write
• g G M / d2
• That is 9.8 m/s/s on our planet Earth Mwhich
  is the same as 9.8 N /kg9.8 Newtons of force
  pulling on every kg of mass.

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Who weighed the Earth?

• A British fellow by the name of Henry Cavendish
  conducted an elegant experimentsee Walker text
  Fig. 12-6
• In that experiment Cavendish was able to quantify
  the value of big G
• Big G is a VERY small value!
• That is why it took so long to come up with an
  experiment that would measure such a small value.
• But he did it and so

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And the Earth weighs

• Well, he didnt really find the weight of the
  earth
• He DID find a way to calculate the mass of the
  earth.
• Gravity is a force that causes weightso we can
  write
• Mg G mM / d2
• Canceling the m values and rearranging the
  equation just a bit, we get
• M g d2 / G
• And if you work it all out you should get
  something like 5.97 x 10 24 kg
• Other scientists used that info to calculate
  volume and density
• The density of the Earth, by the way, is about
  5.5 times the density of waterthe planet earth
  will not float in an ocean of water.
• Is there a planet that WOULD float in water like
  a cork???

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Just to restate..

• Big G is a constant.
• It is the conversion factor that relates the
  force exerted due to gravity and the masses in
  question and the distances between them.
• That BIG G value of 6.67 x 10 -11 Nm2/kg2is
  true throughout the universe..
• Big G is called the universal gravitational
  constant. Its the same everywhere!
• As a result of that fact, we can determine the
  mass and densities of all the other planets.

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Johannes Kepler

• Kepler is a true scientific hero
• Kepler had to turn his back on everything he had
  been taught, everything he thought to be true in
  order to accept the truth of his calculations.
• Trained to be a minister, Kepler had always been
  taught that the Heavens were perfect and Earth
  was corrupt.
• And everyone knew that because the Heavens were
  perfect then the planets had to move in perfect
  orbits of perfect circles.the circle being
  perfect.

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Opposites dont always Attract

• Kepler wanted to know how the planets moved in
  their orbits across the night sky.
• He loved astronomy.
• Newton told us WHY they moved predictablygravity!
  
• Kepler wanted to know How they moved around the
  sun.
• The religious, serious,no-nonsense Kepler, met
  the party animal, Tycho Brahe.
• No two men could have been more different and yet
  together they turned the world on its ear.

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Tycho

• Tycho was so wild that somewhere along the line,
  he got into a fight, had his nose sliced off and
  spent the rest of his life wearing a silver
  (gold) nose.
• Not a very serious fellow, he loved to eat and
  drink and party.
• BUT
• He had one passion. Every night, before the
  invention of the telescope, he would plot the
  positions of the planets in the night sky.
• He had a huge grid in his backyard and every
  night he plotted the objects in the night sky in
  degrees north, south, east, west of the
  gridlines.
• ALL of his observations were jotted down in inky
  notebookslots of themrepresenting YEARS of
  work.

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Kepler needed Tycho

• In order to understand HOW the planets moved
  around the sun, Kepler came to realize he needed
  the data Tycho had been collecting all of his
  adult life.
• He went to meet with Tycho but Tycho had little
  interest in the serious minded Kepler.
• Tycho diedeven that is a legendary story..his
  death was probably a result of wild partying
• Kepler hung around and pleaded with the family
  for those notebooks of Tycho.
• FINALLY, he had all that data.

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The Best Laid Plans

• Kepler settled down to prove what everyone
  knew.the planets traveled at various distances
  from the sun in orbits that were perfect
  circles.
• But the numbers didnt produce perfect circles.
  He tried and tried but no matter what, Kepler
  came to understand that Tychos plotted planetary
  positions would not produce perfect circles.
• So, either Tycho, the drunken party animal was
  wrong, or everything Kepler had ever been taught
  by all of his teachers was wrong.
• Malley

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A Profile in Courage

• Kepler trusted the data. It represented years,
  decades of observations. He trusted Tychos
  numbers. It was proof. It was quantifiable
  values.
• Furthermore, when plotted, the orbits drawn using
  Tychos data, answered a lot of unanswered
  questions, like Mars apparent retrograde motion.
• Kepler trusted numbers, he trusted geometry.
• He believed that if one understand geometry, one
  could understand the mind of God.
• Kepler turned his back on all he had been taught.
  Consider such a dilemma in your life
• Thanks to his courage, we know how planets move
• Keplers three laws of planetary motion are

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Keplers First Law of Planetary Motion

• Once Kepler plotted the data and came to the
  conclusion that Mars, and all other planets as
  well, orbit the sun not in circular orbits but in
  more oval shaped paths, in elliptical paths.

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Keplers Second Law of Planetary Motion

• The text talks a lot about the fact that in the
  second law, planets sweep out equal areas in
  equal amounts of time
• More to the point, this is true because it is
  gravity that keeps planets in orbit around the
  sun.
• Because planets orbit not in circles but in
  ellipses, that means that sometimes the planets
  are closer to the sun as they orbit and sometimes
  they are farther away from the sun as they go
  around.
• So the second law, shown clearly on pg 356 of
  the Walker text, is true due to law number one
  along with Newtons law of gravity which says
  that as distance increases, force decreases
• Thus, when planets are farther away from the sun
  they move more slowly.
• When planets are closer they move faster. Look
  carefully at figure 12-9.

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Keplers Third Law of Planetary Motion

• The math of the third law is established on
  page 357 of Walker text.
• Basically, it says simply, the farther out a
  planet orbits the sun, the slower it moves and
  thus the longer its period of revolution, the
  longer its year.
• So Mercury, the closest to the sun, takes just a
  handful of days but if you lived on Pluto you
  would not live to see your first birthday.
• So the time for the orbit, the period of orbit,
  T
• The mean distance of the planet from the sun is
  r
• Study the derivation of the equation which solves
  for the period of orbit of each planet on page
  357.
• T2 4 pi2 r3 / Gravitational constant times
  the Mass of sun
• Malley

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Geosynchronous Orbit

• A satellite that orbits around the earth with a
  period equal to 24hours, one day, so that it
  always appears to be in the same place overhead .
  
• Read Walker pg 359 for details.
• For read only value READ pg 360-361.

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12.4 Gravitational PE

• Gravitational PE approaches zero as distance from
  object of interest approaches infinity since
  there is NOT much force of gravity acting since
  its influence decreases as distance increases.
• Keep in mind that distances between two objects
  of interest, r, are measured from the centers
  of the masses in question.
• As one object approaches another, the
  gravitational PE increases since the force of
  gravity increases as the distances between the
  centers of masses get closer.

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12.5

• We will take 12.5 and 12.6 as optional for
  reading only.
• Basically, what you need to know is that even in
  astronomical terms, energy is still conserved and
  can be quantified as PE and/or KE when measuring
  two or more bodies acting as a system.
• Further, you should know that to attain orbital
  velocity around planet Earth, one must achieve at
  least 17,500 miles / hr
• If one wishes to escape Earth orbit, one must
  achieve closer to 25,000 miles / hr.
• The equation one can use to solve for escape
  velocity for any astronomical body is equation
  12-13
• Escape velocity depends on the mass of the
  planet, the radius of the planet and the
  gravitational constant.