Physics 1901 (Advanced)

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Physics 1901 (Advanced)

• Prof Geraint F. Lewis
• Rm 560, A29
• gfl_at_physics.usyd.edu.au
• www.physics.usyd.edu.au/gfl/Lecture

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Gyroscope
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Gyroscope Not rotating
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Gyroscope Rotating
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Precision Speed

• Over a small time interval, there is a change in
  angular momentum given by

As dL is perpendicular to L, only the direction
of L changes, not magnitude.
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Gravitation
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Using Gravitation

• Newton realised that using his gravitational
  formula was actually pretty tricky. If we have a
  randomly shaped object, what is the force it
  produces on a test (small) mass locate near by?

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Using Gravitation

• Newton realised that a spherical shell of matter
  has special properties. If you are outside the
  shell, he was able to show that the gravitational
  force was the same as if all the mass were
  concentrated at the centre of the shell (Rev.
  12.6)

Hence any spherical symmetric body (ie built of a
series of shells) behaves as if all the mass were
concentrated at the centre of the shells. Newton
realised he could treat planets as basically
being points! What about inside a spherical shell?
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Weight

• At the surface of the Earth

We know RE, g and G, so can calculate ME
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Falling through the Earth

• Assume a mass is dropped down a tunnel in a
  uniform density Earth. What is its equation of
  motion? How long does it take to return?

We have a wave equation again!
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Gravitational Potential Energy

• Imagine bringing two mass from far apart to close
  together. There is a change in the gravitational
  potential given by (remember work done and
  potential energy in a uniform gravitational
  field).

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Escape Velocity

• Imagine a projective is fired straight up. How
  fast must it be traveling not to fall back?
• Using conservation of energy

• Earth 11km/s
• Sun 618km/s
• Neutron star 200000km/s

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Motion of Satellites

• Gravity provides a centripetal acceleration. If a
  satellite has the correct velocity, v, it will
  move in a circular orbit, continually falling
  towards the Earth, but not getting any closer.

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Motion of Satellites

• The period of the orbit is

With this, you can calculate the period for a
geostationary orbit. The total energy is
is negative! Orbit is bound.
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Non-circular Orbits

• What if the velocity is too small for circular
  motion?
• There is to much centripetal force and the
  objects radial position changes. By conservation
  of energy, it speeds up, then being too fast for
  circular motion.

Newton showed that the resultant motion is
elliptical, or if the velocity is much greater
than circular, the orbit is unbound and
hyperbolic. (The derivation is straight forward
and can be found at ??????????????????????????????
????)
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Keplers 1st Law

• Before Newton derived the mathematical form of
  orbits, Kepler determined three empirical laws .
• His 1st Law says that orbits in the solar system
  are elliptical, with the Sun at one focus.

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Is this always true?

• A pair of equal mass stars will orbit their
  centre of mass (the barycentre), apparently
  orbiting nothing at all!
• What motion do you expect the barycentre to have?

Borrowed from Wikipedia
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Keplers 2nd Law

• The 2nd Law says that the area an orbit sweeps
  out in a fixed time is a constant.

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Keplers 2nd Law

• Remember that the angular momentum is

For an elliptical orbit, r and ? are continually
changing, but L remains a constant. Given this,
we see that
Keplers 2nd Law is simply an expression of the
conservation of angular momentum.
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Keplers 3rd Law

• Keplers 3rd Law is relates the period of an
  elliptical orbit with semi-major axis a

Note that this is not dependent upon the mass of
the orbiting object.
http//hyperphysics.phy-astr.gsu.edu/hbase/kepler.
html
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Orbits
The Solar System
The Galactic Centre
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Orbits In general

• In general, mass distributions are not point-like
  or spherical, so the overall potential does not
  have a 1/R form.
• It turns out that closed elliptical orbits only
  occur in 1/R potentials, and generally orbits are
  more complicated, often having rosette-like
  patterns and are often not closed!

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Round Up

• This is the end of mechanics, and you should now
  be familiar with the concepts of force, momentum,
  energy and the action of gravity.
• Remember, that it is important to understand the
  underlying concepts and build on these to
  understand the evolution of physical systems.
• The laws of mechanics are universal and can be
  applied throughout science and the Universe.

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• THE END