Newtons Law of Universal Gravitation

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Gravitation

• Newtons Law of Universal Gravitation
• Every mass in the universe attracts every other
  mass in the universe according to
• F GMm
• r2
• where
• G 6.67 x 10-11 Nm2kg-2

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Gravitation

• Newtons Law of Universal Gravitation
• The law deals only with point masses. (A sphere
  can be considered a point mass located at the
  center of the sphere.)
• When dealing with two objects, the gravitational
  force acting on each is equal and opposite.

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Gravitation

• Newtons Law of Universal Gravitation
• Gravitational forces are always attractive.
  (Neither the mass nor the distance between the
  objects can be negative.)
• Although gravitational force acts between all
  objects, the value of G shows the force will be
  significant only for very massive objects.

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Gravitation

• Newtons Law of Universal Gravitation
• Note The law applies only to two masses at a
  time, but the gravitational force vectors may be
  added to the force vectors coming from a third
  mass in the system. This may be done iteratively
  for additional masses.
• The resulting mathematical complexity limits
  solutions for 3 or more masses to approximate
  solutions, (the three body problem).

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Gravitation

• Gravitational Field Strength (g)
• g F
• m
• Units are N?kg-1
• Gravitational
• Field Lines ?
• (sphere and point)

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Gravitation

• Gravitational Field Strength
• Since F GMm then F GM
• r2 m r2
• Since F mg then F g
• m

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Gravitation

• Gravitational Field Strength
• and therefore g GM
• r2
• is the gravitational field at the surface of a
  sphere of radius r. Use vectors to get the
  resulting field of two objects on a third.

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Gravitation

• Gravitational Field Strength
• g GM
• r2
• G 6.67 x 10-11 Nm2kg-2
• Mearth 5.97 1024 kilograms
• rearth 6.4 x 106 kilometers

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Gravitation

• LaGrangian Points (Earth Moon)
• L1 - Between the Earth and Moon
• L2 - On the other side of the Moon
• L3 - Opposite the Moon
• L4 - 60 ahead of the Moon
• L5 - 60 behind the Moon

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Gravitation

• LaGrangian Points (Earth Moon)
• L1, L2 and L3 are quasistable, station-keeping is
  needed to keep an object on location at these
  points.
• L4 and L5 are stable. Trash will collect at L4
  and L5. (Think Jupiter and asteroids.)

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Gravitation

• Gravitational Potential Energy
• When a mass moves between two different heights
  (h) near the earths surface
• difference in energy mgh PE
• works only if (h) is not very large

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Gravitation

• Gravitational Potential Energy
• The everyday solution for mgh treats the PE at
  the surface of the earth as 0.
• The true solution for gravitational PE is 0 for
  PE taken at infinity. Therefore, an object at
  infinity loses potential energy as it moves
  towards a mass M.

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Gravitation

• Gravitational Potential Energy
• Therefore the PE is negative for any object at
  point P, where P is not infinity.
• This means that the work done in moving from
  infinity to point P is also negative.
• Gravitational PE is a scalar.

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Gravitation

• Gravitational Potential Energy
• By math not covered in IB...
• PEG - GMm
• r

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Gravitation

• Gravitational Potential Energy
• STOP!
• PROVE
• ?PE mgh

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Gravitation

• Gravitational Potential Energy
• Therefore, for r h, the change in PE
• ?PE - GMm (- GMm)
• r r h

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Gravitation

• Gravitational Potential Energy
• ?PE - GMm GMm
• r r h
• Since F GMm then Fr GMm
• r2 r

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Gravitation

• Gravitational Potential Energy
• and F GMm then F(r h) GMm
• (rh)2 (r h)
• ?PE - Fr F(r h)
• ?PE - Fr Fr Fh

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Gravitation

• Gravitational Potential Energy
• ?PE - Fr Fr Fh Fh
• Since F mg
• ?PE Fh mgh

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Gravitation

• Gravitational Potential
• A scalar that measures the energy per unit test
  mass.
• VG W - GM
• m RP
• where W Work m Test Mass RP Planet
  Radius

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Gravitation

• Escape Speed
• Speed needed for a rocket to go from the planet
  surface to infinity
• VG - GM since VG W
• RP m
• and the energy difference between the surface and
  infinity is

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Gravitation

• Escape Speed
• Speed needed for a rocket to go from the planet
  surface to infinity
• VG (m) 0(m) - (- GM)(m) since W VG (m)
• RP
• W GM(m) and therefore W GMm
• RP RP

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Gravitation

• Escape Speed
• By the Work-Kinetic Energy Theorem
• W ?KE ½mv2 GMm
• RP
• therefore v2 2GM and vescape
  (2GM/r)½ RP

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Gravitation

• Keplers Third Law
• When one object is in orbit around another, the
  gravitational and centripetal forces must be
  exactly balanced.
• FG GMm and FC mv2
• r2 r
• and therefore GMm mv2
• r2 r

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Gravitation

• Keplers Third Law
• Therefore the orbital speed is
• vorbital (GM/r)½
• Now rearranging this equation, we note that
• GM v2r

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Gravitation

• Keplers Third Law
• We can use geometry to figure the speed
• v 2pr and v2 4p2r2
• T T2
• Substituting into the previous equation
• GM 4p2r3 and therefore GM r3
• T2 4p2 T2

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Gravitation

• Keplers Third Law
• Then since G, p, and 4 are all constant, then any
  object rotating around M will be subject to the
  following constraint, usually known as Keplers
  Third Law
• r3 constant
• T2

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Gravitation

• Total Energy
• Total energy is the sum of potential and kinetic
  energies (or TE PE KE)
• PEG - GMm and KE ½mv2
• r
• therefore TE - GMm ½mv2
• r

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Gravitation

• Total Energy
• Using the equation for orbital velocity
• vorbit (GM/r)½ and vo2 GM/r
• therefore TE - GMm ½mGM
• r r

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Gravitation

• Total Energy
• Finally
• TE - ½GMm
• r

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Gravitation

• Gravitational Interactions
• In essence, a small object in orbit around a
  large object is falling around the large
  object.
• Satellites, the shuttle and the International
  Space Station are examples.

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Gravitation

• Gravitational Interactions
• Two large objects rotating around each other
  actually rotate around a common center of gravity
  between them.
• The earth and moon, or the sun and Jupiter, or
  double stars are examples.

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Gravitation

• Gravitational Interactions
• The gravitational drag of the moon create the
  ocean tides on earth.
• Tides in high latitudes can be very significant.
• The moon creates 254,000,000,000 watts of tidal
  energy daily.

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Gravitation

• Gravitational Interactions
• In order of increasing gravitational attraction
  are
• Sun
• Blue Giant Star
• Neutron Star
• Black Hole
• (of Sun Mass diameter 1 inch)

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Gravitation

• Satellite Motion
• We usually think of the orbits of satellites and
  the planets as circular.
• In actuality, most orbits are elliptical, with
  the larger body at the focus of the ellipse.
• The earth is a satellite of the sun with an
  elliptical orbit around the sun.

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Gravitation

• Keplers Three Laws
• 1st Any object bound by a force that varies as
  1/r2 travels in an ellipse.
• 2nd A line drawn from the sun to any planet
  will sweep out equal areas in equal times.
• 3rd The ratio r3 /T2 is a constant for any
  planet.

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Gravitation

• END OF GRAVITATION

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GRAVITATION

• Notes