Lecture 27b

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Example 5.2

• Astronomical
• measurements
• show that the
• orbital speed of
• masses in spiral
• galaxies rotating
• about their centers
• is approximately
• constant as a
• function of distance R from the galaxy center,
  as in the figure (for the Andromeda Galaxy). Show
  that this is inconsistent with the galaxy having
  its mass concentrated at its center can be
  explained if the galaxy mass increases with
  distance R from the center.

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• Solution Assume that the galaxy mass M has a
  spherically symmetric distribution ?. We can
  solve the problem, despite the fact that the
  radius R might be hundreds of light years! The
  orbital speed v of a small mass m in orbit at R
  about the galaxy center is found (Phys. I!) by
  setting the gravitational force (mass) ?
  (centripetal acceleration) or (GMm)/)(R2)
  m(v2)/R
• ? v (GM/R)½ (1)
• or v ?R-½ (2)
• If (2) is true, v vs. R would
• be like the dashed curve! For the
• solid (expt.) curve to be consistent
• with (1) requires M M(R) ? R ? CONCLUSION
  There must be more matter in a galaxy than is
  observed (dark matter 90 of mass in the
  universe). At the forefront of research.

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Example 5.3

• Consider a thin,
• uniform circular
• ring of radius a
• mass M. A small
• mass m is placed
• in the plane of the
• ring. Find a
• position of
• equilibrium
• determine if it is
• stable.

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• Use the result for a Line Distribution
• (M ???(r?)ds?, F - G???(r?)ds?/r
• Here, ?? const. ? ? M 2pa?
• The potential at the position of a
• point mass m a distance r? from
• the ring center a distance b from
• the differential mass dM
• F(r?) -?aG?(df/b)
• f angle shown in the figure, limits (0 ? f
  ? 2p)
• From the figure, b2 a2 (r?)2 2ar?cosf
• ? F - ?aG?dfa2 (r?)2 2ar?cosf-½
  
• - ?G?df1 (r?/a)2 2(r?/a)cosf-½

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• F - ?G?df1 (r?/a)2 2(r?/a)cosf-½ (1)
• Consider positions close to the center
• point (near r? 0) Expand (1) for
• r? ltlt a integrate term by term
• F(r?) ? -?G?df1 (r?/a)cosf
• ?(r?/a)2(3cos2f -1).
• (For those who know care, the
• integrand in this form is a series of Legendre
  Polynomials!)
• F(r?) ? -(MG/a)1 ?(r?/a)2
• The potential energy of mass m at r? is
• U(r?) mF(r?) ? -m(MG/a)1 ?(r?/a)2

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• U(r?) mF(r?) ? -m(MG/a)1 ?(r?/a)2
• The equilibrium position is given by
• (dU/dr?) 0 -m(MGr?)/(2a3)
• ? r? 0 is an equilibrium position!
• This should be obvious by symmetry!
• Stability (or not) is obtained from the
• 2nd derivative (d2U/dr?2)0 -m(MG)/(2a3) lt 0
• ? r? 0 is a position of unstable
  equilibrium!
• (Not obvious!)