Gravitation

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Chapter 12

• Gravitation

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Theories of Gravity

• Newtons
• Einsteins

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Newtons Law of Gravitation

• any two particles m1, m2 attract each other
• Fg magnitude of their mutual gravitational force

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Newtons Law of Gravitation

• Fg magnitude of the force that
• m1 exerts on m2
• m2 exerts on m1
• direction along the line joining m1, m2

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Newtons Law of Gravitation

• Also holds if m1, m2 are two bodies with
  spherically symmetric mass distributions
• r distance between their centers

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Newtons Law of Gravitation

• universal law
• G fundamental constant of nature
• careful measurements G6.6710-11Nm2/kg2

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How Cavendish Measured G(in 1798, without a
laser)
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1798 small masses (blue) on a rod gravitate
towards larger masses (red), so the fiber twists
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2002 we can measure the twist by reflecting
laser light off a mirror attached to the fiber
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If a scale calibrates twist with known forces, we
can measure gravitational forces, hence G
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What about g 9.8 m/s2 ?

• How is G related to g?
• Answer

Show that g GmE/RE2
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Other planets, moons, etc?

• gpacceleration due to gravity at planets
  surface
• density assumed spherically symmetric, but not
  necessarily uniform

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ExampleEarths density is not uniform
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Yet to an observer (m) outside the Earth, its
mass (mE) acts as if concentrated at the center
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Newtons Law of Gravitation

• This is the magnitude of the force that
• m1 exerts on m2
• m2 exerts on m1
• What if other particles are present?

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Superposition Principle

• the gravitational force is a vector
• so the gravitational force on a body m due to
  other bodies m1 , m2 , ... is the vector sum

Do Exercise 12-8 Do Exercise 12-6
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Superposition Principle

• Example 12-3
• The total gravitationalforce on the mass at Ois
  the vector sum

Do some of Example 12-3 and introduce Extra
Credit Problem 12-42
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Gravitational Potential Energy, U

• This follows from

Derive U - G m1m2/r
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Gravitational Potential Energy, U

• Alternatively a radial conservative force has a
  potential energy U given by F dU/dr

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Gravitational Potential Energy, U

• U is shared between both m1 and m2
• We cant divide up U between them

Example Find U for the Earth-moon system
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Superposition Principle for U

• For many particles,U total sharedpotential
  energy of the system
• U sum of potentialenergies of all pairs

Write out U for this example
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Total Energy, E

• If gravity is force is the only force acting, the
  total energy E is conserved
• For two particles,

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Application Escape Speed

• projectile m
• Earth mE
• Find the speed that m needs to escape from the
  Earths surface

Derive the escape speed Example 12-5
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Orbits of Satellites
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Orbits of Satellites

• We treat the Earth as a point mass mE
• Launch satellite m at A with speed v toward B
• Different initial speeds v give different orbits,
  for example (1) (7)

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Orbits of Satellites

• Two of Newtons Laws predict the shapes of
  orbits
• 2nd Law
• Law of Gravitation

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Orbits of Satellites

• Actually
• Both the satellite and the point C orbit about
  their common CM
• We neglect the motion of point C since it very
  nearly is their CM

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Orbits of Satellites

• If you solve the differential equations, you find
  the possible orbit shapes are
• (1) (5) ellipses
• (4) circle
• (6) parabola
• (7) hyperbola

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Orbits of Satellites

• (1) (5) closed orbits
• (6) , (7) open orbits
• What determines whether an orbit is open or
  closed?
• Answer escape speed

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Escape Speed

• Last time we launched m from Earths surface (r
  RE)
• We set E 0 to find

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Escape Speed

• We could also launch m from point A (any r gt RE)
• so use r instead of RE

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Orbits of Satellites

• (1) (5) ellipses
• launch speed v lt vesc
• (6) parabola
• launch speed v vesc
• (7) hyperbola
• launch speed v gt vesc

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Orbits of Satellites

• (1) (5) ellipses
• energy E lt 0
• (6) parabola
• energy E 0
• (7) hyperbola
• energy E gt 0

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Circular Orbits
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Circular Orbit Speed v
Derive speed v

• uniform motion
• independent of m
• determined by radius r
• large r means slow v

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Compare to Escape Speed

• If you increase your speed by factor of 21/2
  you can escape!

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Circular Orbit Period T
Derive period T

• independent of m
• determined by radius r
• large r means long T

Do Problem 12-45
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Circular Orbit Energy E
Derive energy E

• depends on m
• depends on radius r
• large r means large E

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Orbits of Planets
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Same Math as for Satellites

• Same possible orbits, we just replace the Earth
  mE with sun ms
• (1) (5) ellipses
• (4) circle
• (6) parabola
• (7) hyperbola

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Orbits of Planets

• Two of Newtons Laws predict the shapes of
  orbits
• 2nd Law
• Law of Gravitation
• This derives Keplers Three Empirical Laws

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Keplers Three Laws

• planet orbit ellipse(with sun at one focus)
• Each planet-sun line sweeps out equal areas in
  equal times
• For all planet orbits, a3/T2 constant

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Keplers First Law

• planet orbit ellipse
• P planet
• S focus (sun)
• S focus (math)
• a semi-major axis
• e eccentricity0 lt e lt 1e 0 for a circle

Do Problem 12-64
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Keplers Second Law

• Each planet-sun line sweeps out equal areas in
  equal times

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Keplers Second Law
Present some notes on Keplers Second Law
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Keplers Third Law

• We proved this for a circular orbit (e 0)
• T depends on a, not e

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Keplers Third Law

• Actually
• Since both the sun and the planet orbit about
  their common CM

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Theories of Gravity

• Newtons
• Einsteins

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Einsteins Special Relativity

• all inertial observers measure the same value c
  3.0108 m/s2 for the speed of light
• nothing can travel faster than light
• special means not general
• spacetime ( space time) is flat

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Einsteins General Relativity

• nothing can travel faster than light
• but spacetime is curved, not flat
• matter curves spacetime
• if the matter is dense enough, then a black
  hole forms

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Black Holes
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Black Holes

• If mass M is compressed enough, it falls inside
  its Schwarzschild radius, Rs
• This curves spacetime so much that a black hole
  forms

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Black Holes

• Outside a black hole, v and r for circular orbits
  still obey the Newtonian relationship
• Also from far away, a black hole obeys Newtonian
  gravity for a mass M

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Black Holes

• Spacetime is so curved, anything that falls into
  the hole cannot escape, not even light
• Light emitted from outside the hole loses energy
  (redshifts) since it must do work against the
  extremely strong gravity
• So how could we detect a black hole?

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Black Holes
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AnswerAn Accretion Disk (emits X-rays)
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Matter falling towards the black hole enters
orbit, forming a hot disk and emitting X-rays