The Law of Universal Gravitation

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The Law of Universal Gravitation

• Physics
• Montwood High School
• R. Casao

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Newtons Universal Law of Gravity

• Legend has it that Newton was struck on the head
  by a falling apple while napping under a tree.
  This prompted Newton to imagine that all bodies
  in the universe are attracted to each other in
  the same way that the apple was attracted to the
  Earth.
• Newton analyzed astronomical data on the motion
  of the Moon around the Earth and stated that the
  law of force governing the motion of the planets
  has the same mathematical form as the force law
  that attracts the falling apple to the Earth.

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Newtons Universal Law of Gravity

• Newtons law of gravitation every particle in
  the universe attracts every other particle with a
  force that is directly proportional to the
  product of their masses and inversely
  proportional to the square of the distance
  between them.
• If the particles have masses m1 and m2 and are
  separated by a distance r, the magnitude of the
  gravitational force is

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Newtons Universal Law of Gravity

• G is the universal gravitational constant, which
  has been measured experimentally as 6.672 x
  10-11 .
• The distance r between m1 and m2 is measured from
  the center of m1 to the center of m2.

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Newtons Universal Law of Gravity

• By Newtons third law, the magnitude of the force
  exerted by m1 on m2 is equal to the force exerted
  by m2 on m1, but opposite in direction. These
  gravitational forces form an action-reaction pair.

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Properties of the Gravitational Force

• The gravitational force acts as an
  action-at-a-distance force, which also exists
  between two particles, regardless of the medium
  that separates them.
• The force varies as the inverse square of the
  distance between the particles and therefore
  decreases rapidly with increasing distance
  between the particles.
• The gravitational force is proportional to the
  mass of each particle.

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Properties of the Gravitational Force

• The force on a particle of mass m at the Earths
  surface has the magnitude
• ME is the Earths mass (5.98 x 1024 kg) RE is
  the radius of the Earth (6.37 x 106 m).
• The net force is directed toward the center of
  the Earth both masses accelerate, but the
  Earths acceleration is not noticeable due to its
  extremely large mass. The smaller mass
  accelerates towards the Earth.

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Weight and Gravitational Force

• Weight was previously defined as FW mg, where
  g is the magnitude of the acceleration due to
  gravity. With the new perspective related to the
  attractive forces existing between any two
  objects in the universe,
• The mass m cancels out, giving us (also called
  surface gravity)

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Bodies Above the Surface of a Mass

• Consider a body of mass m at a distance h above
  the Earths surface, or a distance r from the
  Earths center, where r Re h. The magnitude
  of the gravitational force acting on the mass is
  given by

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Gravity/Radius Ratio

• If the body is in free fall, then the
  acceleration of gravity at the altitude h is
  given by
• Thus, it follows that g decreases with increasing
  altitude.

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Gravity/Radius Ratio

• The value of g at any given location can be
  determined using the following proportional
  relationship
• This proportional relationship can also be
  applied to the weight of an object

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Keplers Laws

• Kepler formulated three kinematic laws to
  describe the motion of planets about the Sun
• Keplers First Law All planets move in
  elliptical orbits with the sun at one of the
  focal points.
• Keplers Second Law The radius vector drawn
  from the sun to any planet sweeps out equal areas
  in equal time intervals.
• Keplers Third Law The square of the orbital
  period of any planet is proportional to the cube
  of the semi-major axis of the elliptical orbit.

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Keplers Laws

• First law

• Second law

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Keplers Laws

• Third law equation
• where k is a constant 3.35 x 1018 m3/s2
• r is the radius of rotation
• T is the period of rotation (the time necessary
  to complete one revolution)
• Keplers laws apply to any body that orbits the
  Sun, manmade spaceship as well as planets,
  comets, and other natural objects. The mass of
  the orbiting body does not enter into the
  calculation.

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Keplers Laws

• The ratio of the squares of the periods (T) of
  any two planets revolving about the Sun is equal
  to the ratio of the cubes of their average
  distances r from the Sun

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Period of a Satellite

• The period of a satellite or planet orbiting
  about a central body is given by
• Mbody is the mass of the central body being
  orbited.

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The Gravitational Field

• The gravitational field concept revolves around
  the general idea that an object modifies the
  space surrounding it by establishing a
  gravitational field which extends outward in all
  directions, falling to zero at infinity.
• Any other mass located within this field
  experiences a force because of its location. So,
  it is the strength of the gravitational field at
  that location that produces the force - not the
  distant object.
• The situation is symmetrical - each object
  experiences a gravitational force because of the
  field set up by any other object.

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The Gravitational Field

• The gravitational field is a vector quantity
  equal to the gravitational force acting on a
  particle divided by the mass of the particle
• The gravitational field equation can be used to
  determine the value of g at any location by

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Escape Velocity

• Suppose you want to launch a rocket vertically
  upward and give it just enough kinetic energy
  (energy of motion) to escape the Earths
  gravitational pull.
• The minimum initial velocity of an object at the
  Earths surface that would allow the object to
  escape the Earth, never to return, is the escape
  velocity.
• Escape velocity from Earth

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Satellite Orbits

• A satellite is held in a circular orbit because
  the force of gravity supplies the necessary
  centripetal force to keep the object moving in a
  circular path about the central body.
• In order for a satellite to orbit around a
  central body, such as the Earth, there must be a
  net force on the object directed toward the
  center of the circular orbit, a centripetal
  force.
• For a satellite in orbit around Earth, the
  centripetal force is equal to the gravitational
  force exerted by the Earth on the satellite.

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Satellite Orbits

• A satellite does not fall because it is moving,
  being given a tangential velocity by the rocket
  that launched it. It does not travel off in a
  straight line because Earths gravity pulls it
  toward the Earth.
• The tangential speed of an object in a circular
  orbit is given by
• If the period of the orbit is known, the velocity
  may be determined using

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Satellite Orbits

• The period of a satellite can be determined by

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Satellite Orbits

• The Goldilocks principle can be used to explain
  the relationship between the speed of a satellite
  and its orbit. The velocity of the satellite is
  critical, and the velocity described by the
  equation
• Vmin describes the minimum velocity necessary for
  the satellite to maintain its proper circular
  orbit (JUST RIGHT). If the satellite velocity is
  TOO HOT (greater than the vmin), it will not
  maintain the proper circular orbit and fly into
  space. If the satellite velocity is TOO COLD
  (less than vmin), it will be pulled into the
  Earths atmosphere by the Earths gravitational
  force, where it will either burn up in the
  atmosphere or slam into the Earths surface.

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Gravitational Potential Energy Revisited

• Gravitational potential energy near the surface
  of the Earth is given by the equation Ug mgh,
  where h is the height of the object above or
  below a reference level. This equation is only
  valid for an object near the Earths surface.
• For objects high above the Earths surface, the
  equation for potential energy is

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Gravitational Potential Energy

• The negative sign comes from the work done
  against the gravity force in bringing a mass in
  from infinity where the potential energy is
  assigned the value zero, towards the Earth. This
  work is stored in the mass as potential energy.
• As r gets larger, the potential energy gets
  smaller the gravitational force approaches zero
  as r approaches infinity.

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Gravitational Potential Energy

• Only changes in gravitational potential energy
  are important.
• For an object that moves from point B to point A,
  the expression for the change in potential energy
  is

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Web Sites

• Kepler's Laws (with animations)
• Kepler's Three Laws
• Kepler's Three Laws of Planetary Motion
• Kepler's Laws of Planetary Motion