Universal Gravitation

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Universal Gravitation Satellite Motion

• What is gravity?
• What is its role in our solar system?

2
The Law of Universal Gravitation
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What do we know about gravity?

• Force between the Earth and objects on or near
  it.
• Causes a ball tossed upward to slow down on the
  way up and speed up on the way down.
• Acceleration due to gravity on or near Earth's
  surfaceg 9.8 m/s2 (downward)
• Same acceleration value for all objects,
  regardless of mass.

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The Law of Universal Gravitation

• Gravity doesnt just affect things on the Earth.
• Gravity is an interaction between any two masses.
  (particles, planet, people, etc.)
• It exists between all objects simultaneously.
• The force of gravity can be determined the same
  way, regardless of the objects.
• In this sense, it is universal.

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The Law of Universal Gravitation

• Developed published by Isaac Newton in 1687.
• The gravitational force between any two masses is
  directly proportional to the product of the two
  masses and inversely proportional to the square
  of the distance between their centers.
• Translation
• As the masses increase, the gravitational force
  increases.
• As the distance increases, the gravitational
  force decreases.

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The Law of Universal Gravitation
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What it means
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The Law of Universal Gravitation

• In 1798, Lord Henry Cavendish determined the
  value of the universal gravitation constant using
  a torsion balance.
• Thus, the law could be expressed as an equation

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Gee, thats small!

• The value of G is an extremely small numerical
  value.
• Its smallness accounts for the fact that the
  force of gravitational attraction is only
  appreciable for objects with large mass.
• Knowing the value of G allows us to calculate the
  force of gravitational attraction between any two
  objects of known mass and known separation
  distance.

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Calculating Fgrav

• Determine the force of gravitational attraction
  between the earth (m 5.98 x 1024 kg) and a 70
  kg physics student if the student is standing at
  sea level, a distance of 6.37 x 106 m from
  earth's center.

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The solution is as follows
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Fgrav vs. Fgrav

• We have often calculated the force of gravity
  (Fgrav) with which an object of mass m was
  attracted to the earth using Fgrav mg (We
  called this weight.)
• Now a second equation has been introduced for
  calculating the force of gravity with which an
  object is attracted to the earth.
• Does it matter which equation we use?

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Fgrav vs. Fgrav

• For an object on or near the surface of the
  Earth, it doesnt matter.
• The answer is the same regardless of the equation
  used.
• If the object is not on or near the surface of
  the Earth, then we must use the equation given by
  the Law of Universal Gravitation (the big
  equation).
• Reason the distance has changed!
• When we consider gravitational attraction with
  other planets, its pretty clear that we cant
  use Fgrav mg anymore, unless we know g for
  that planet.

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g vs. G

• Which g do I use? Little g or big G?
• Always use G when using the Law of Universal
  Gravitation. That number never changes and can
  be used to determine the gravitational force
  between any two objects, even between Mr.
  Dellibovi and the cell phone in your hand. Yes,
  he sees you. Youve been, how do you say,
  sniped? Yes. Sniped.
• You can only use little g for a given planet when
  an object is on or near the surface of a planet.
  Remember, its 9.8 m/s2 for Earth ONLY. Its
  different on other planets.

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Where are you, little g?

• So, how do we determine the value of g on another
  planet?
• Set Fgrav equal to Fgrav
• We just need to knowthe mass of the planetand
  its radius
• Easier said than done!

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Planet Radius (m) Mass (kg) g (m/s2)
Mercury 2.43 x 106 3.20 x 1023 3.61
Venus 6.07 x 106 4.88 x1024 8.83
Earth 6.40 x 106 5.98 x 1024 9.81
Mars 3.38 x 106 6.42 x 1023 3.75
Jupiter 6.98 x 107 1.90 x 1027 26.0
Saturn 5.82 x 107 5.68 x 1026 11.2
Uranus 2.35 x 107 8.68 x 1025 10.5
Neptune 2.27 x 107 1.03 x 1026 13.3
Pluto 1.15 x 106 1.20 x 1022 0.61
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Determining the Radius of a Planet

• Determining the radius of a planet can be done
  using a good telescope and some geometry. Not
  too difficult.
• The mass of a planet is a little more difficult
  and involves the planets orbit around the Sun.

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Determining the Mass of a Central Body

• For any case involving the orbit of an
  astronomical object, the object that is orbiting
  is called a satellite and the object that is
  being orbited is called the central body
• Lets assume that satellites orbit a the central
  body in circular paths
• Gravity causes the circular motion, so gravity is
  the centripetal force in this case.
• So, lets set Fgrav equal to Fcent

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Determining the Mass of a Central Body
This is an interesting result! The speed of a
satellite in its orbit does not depend on the
mass of the satellite, just its orbital
radius. The farther away from the central body,
the slower thesatellite moves, because there is
less gravitational pull.
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Determining the Mass of a Central Body
So, to determine the mass of a central body, we
need to know the orbital radius of the satellite
and its period of revolution around the central
body.
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Gravity and the Motion of Planets

• In the early 1600s, German mathematician and
  astronomer Johannes Kepler mathematically
  analyzed known astronomical data from his mentor,
  Tycho Brahe.
• Through inductive reasoning, Kepler developed
  three laws to describe the motion of planets
  about the sun.

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Formation of Ellipses

• There are two fixed points which a string would
  go around. These are the focal points.

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Law 1 Planets travel in elliptical orbits

• One of the focal points is the location of the
  sun

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Law 1 Planets travel in elliptical orbits

• There is ABSOLUTELY NOTHING at the other focal
  point

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Law 2 Equal area law
1 month
I
II
1 month

• Planets sweep out equal areas in equal amounts of
  time.
• The green and the blue areas are equal to each
  other.
• Planets move faster the closer they are to the
  sun.

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Law 3 Law of Harmonies

• The period of a planet (time it takes a planet to
  travel an orbit) squared is proportional to the
  cube of its average distance from the sun

What the !?_at_!? Im not solving this!
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Law 3 Law of Harmonies

• Its easier to understand in this form
• Where k is a constant for a given star (or
  central body)
• T is measured in Earth years. (TEarth 1 Year)
• r is measured in AU (astronomical units) (rEarth
  1 AU)

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Law 3 Law of Harmonies

• Any satellite around a central body has the same
  T2 / r3 ratio.
• This ratio changes as we change the central
  body.
• All the planets (satellites) around the sun
  (central body) have the same T2 / r3 ratio k
  1.
• However, the ratio for the moon (satellite)
  around the earth (central body) is a different
  number.

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Why do these motions happen?

• Kepler couldnt explain why the planets orbited
  the sun. He said they were somehow
  magnetically driven by the sun.
• Explanation of Keplers Laws was not provided
  until Isaac Newton proposed the Law of Universal
  Gravitation.
• The gravitational interaction between planets and
  the sun is the reason for Keplers laws.