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The Law of Universal Gravitation

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Title: The Law of Universal Gravitation


1
The Law of Universal Gravitation
  • Physics
  • Montwood High School
  • R. Casao

2
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.

3
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

4
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.

5
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.

6
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.

7
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.

8
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)

9
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

10
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.

11
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

12
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.

13
Keplers Laws
  • First law
  • Second law

14
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.

15
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

16
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.

17
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.

18
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

19
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

20
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.

21
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

22
Satellite Orbits
  • The period of a satellite can be determined by

23
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.

24
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

25
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.

26
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

27
Web Sites
  • Kepler's Laws (with animations)
  • Kepler's Three Laws
  • Kepler's Three Laws of Planetary Motion
  • Kepler's Laws of Planetary Motion
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