Gravity, Planetary Orbits, and

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Chapter 11

• Gravity, Planetary Orbits, and
• the Hydrogen Atom

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11.1 Newtons Law of Universal 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
• G is the universal gravitational constant and
  equals 6.673 x 10-11 N?m2 / kg2

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More About Forces

• The forces form a Newtons Third Law
  action-reaction pair
• Gravitation is a field force that always exists
  between two particles, regardless of the medium
  between them
• The force decreases rapidly as distance increases
• A consequence of the inverse square law

Fig 11.1
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Notation

• is the force exerted by particle 1 on
  particle 2
• The negative sign in the vector form of the
  equation indicates that particle 2 is attracted
  toward particle 1
• is the force exerted by particle 2 on
  particle 1

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Law of Gravitation, cont

• This is an example of an inverse square law
• The magnitude of the force varies as the inverse
  square of the separation of the particles
• The law can also be expressed in vector form

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G vs. g

• Always distinguish between G and g
• G is the universal gravitational constant
• It is the same everywhere
• g is the acceleration due to gravity
• g 9.80 m/s2 at the surface of the Earth
• g will vary by location

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Gravitational Force Due to a Distribution of Mass

• The gravitational force exerted by a
  finite-sized, spherically symmetric mass
  distribution on a particle outside the
  distribution is the same as if the entire mass of
  the distribution were concentrated at the center
• For the Earth, this means

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Measuring G

• G was first measured by Henry Cavendish in 1798
• The apparatus shown here allowed the attractive
  force between two spheres to cause the rod to
  rotate
• The mirror amplifies the motion
• It was repeated for various masses

Fig 11.2
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Gravitational Field

• Use the mental representation of a field
• A source mass creates a gravitational field
  throughout the space around it
• A test mass located in the field experiences a
  gravitational force
• The gravitational field is defined as

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Gravitational Field of the Earth

• Consider an object of mass m near the earths
  surface
• The gravitational field at some point has the
  value of the free fall acceleration
• At the surface of the earth, r RE and g 9.80
  m/s2

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Representations of the Gravitational Field
Fig 11.3

• The gravitational field vectors in the vicinity
  of a uniform spherical mass
• fig. a the vectors vary in magnitude and
  direction
• The gravitational field vectors in a small region
  near the earths surface
• fig. b the vectors are uniform

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Fig 11.4
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11.2 Structural Models

• In a structural model, we propose theoretical
  structures in an attempt to understand the
  behavior of a system with which we cannot
  interact directly
• The system may be either much larger or much
  smaller than our macroscopic world
• One early structural model was the Earths place
  in the Universe
• The geocentric model and the heliocentric models
  are both structural models

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Features of a Structural Model

• A description of the physical components of the
  system
• A description of where the components are located
  relative to one another and how they interact
• A description of the time evolution of the system
• A description of the agreement between
  predictions of the model and actual observations
• Possibly predictions of new effects, as well

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11.3 Keplers Laws, Introduction

• Johannes Kepler was a German astronomer
• He was Tycho Brahes assistant
• Brahe was the last of the naked eye astronomers
• Kepler analyzed Brahes data and formulated three
  laws of planetary motion

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Keplers Laws

• Keplers First Law
• Each planet in the Solar System moves in an
  elliptical orbit with the Sun at one focus
• Keplers Second Law
• The radius vector drawn from the Sun to a 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 semimajor axis of
  the elliptical orbit

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Notes About Ellipses

• F1 and F2 are each a focus of the ellipse
• They are located a distance c from the center
• The longest distance through the center is the
  major axis
• a is the semimajor axis

Fig 11.5
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Notes About Ellipses, cont

• The shortest distance through the center is the
  minor axis
• b is the semiminor axis
• The eccentricity of the ellipse is defined as e
  c /a
• For a circle, e 0
• The range of values of the eccentricity for
  ellipses is 0 lt e lt 1

Fig 11.5
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Notes About Ellipses, Planet Orbits

• The Sun is at one focus
• Nothing is located at the other focus
• Aphelion is the point farthest away from the Sun
• The distance for aphelion is a c
• For an orbit around the Earth, this point is
  called the apogee
• Perihelion is the point nearest the Sun
• The distance for perihelion is a c
• For an orbit around the Earth, this point is
  called the perigee

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Keplers First Law

• A circular orbit is a special case of the general
  elliptical orbits
• Is a direct result of the inverse square nature
  of the gravitational force
• Elliptical (and circular) orbits are allowed for
  bound objects
• A bound object repeatedly orbits the center
• An unbound object would pass by and not return
• These objects could have paths that are parabolas
  
• and hyperbolas

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Orbit Examples

• Pluto has the highest eccentricity of any planet
  (a)
• ePluto 0.25
• Halleys comet has an orbit with high
  eccentricity (b)
• eHalleys comet 0.97

Fig 11.6
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Keplers Second Law

• Is a consequence of conservation of angular
  momentum
• The force produces no torque, so angular momentum
  is conserved

Fig 11.7(a)
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Keplers Second Law, cont.

• Geometrically, in a time dt, the radius vector r
  sweeps out the area dA, which is half the area of
  the parallelogram
• Its displacement is given by

Fig 11.7(b)
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Keplers Second Law, final

• Mathematically, we can say
• The radius vector from the Sun to any planet
  sweeps out equal areas in equal times
• The law applies to any central force, whether
  inverse-square or not

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Keplers Third Law

• Can be predicted from the inverse square law
• Start by assuming a circular orbit
• The gravitational force supplies a centripetal
  force
• Ks is a constant

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Keplers Third Law, cont

• This can be extended to an elliptical orbit
• Replace r with a
• Remember a is the semimajor axis
• Ks is independent of the mass of the planet, and
  so is valid for any planet

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Keplers Third Law, final

• If an object is orbiting another object, the
  value of K will depend on the object being
  orbited
• For example, for the Moon around the Earth, KSun
  is replaced with KEarth

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Fig 11.8
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11.4 Energy in Satellite Motion

• Consider an object of mass m moving with a speed
  v in the vicinity of a massive object M
• M gtgt m
• We can assume M is at rest
• The total energy of the two object system is E
  K Ug

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Energy, cont.

• Since Ug goes to zero as r goes to infinity, the
  total energy becomes

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Energy, Circular Orbits

• For a bound system, E lt 0
• Total energy becomes
• This shows the total energy must be negative for
  circular orbits
• This also shows the kinetic energy of an object
  in a circular orbit is one-half the magnitude of
  the potential energy of the system

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Energy, Elliptical Orbits

• The total mechanical energy is also negative in
  the case of elliptical orbits
• The total energy is
• r is replaced with a, the semimajor axis

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Fig 11.10
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Escape Speed from Earth

• An object of mass m is projected upward from the
  Earths surface with an initial speed, vi
• Use energy considerations to find the minimum
  value of the initial speed needed to allow the
  object to move infinitely far away from the Earth

Fig 11.11
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Escape Speed From Earth, cont

• This minimum speed is called the escape speed
• Note, vesc is independent of the mass of the
  object
• The result is independent of the direction of the
  velocity and ignores air resistance

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Escape Speed, General

• The Earths result can be extended to any planet
• The table at right gives some escape speeds from
  various objects

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Escape Speed, Implications

• This explains why some planets have atmospheres
  and others do not
• Lighter molecules have higher average speeds and
  are more likely to reach escape speeds
• This also explains the composition of the
  atmosphere

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Black Holes

• A black hole is the remains of a star that has
  collapsed under its own gravitational force
• The escape speed for a black hole is very large
  due to the concentration of a large mass into a
  sphere of very small radius
• If the escape speed exceeds the speed of light,
  radiation cannot escape and it appears black

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Exercises of Chapter 11

• 6, 9, 10, 14, 17, 18, 28, 31, 48, 59