Transcript ClassicalMechanics_6..

Physics 1901 (Advanced)
A/Prof Geraint F. Lewis
Rm 557, A29
[email protected]
www.physics.usyd.edu.au/~gfl/Lecture
http://www.physics.usyd.edu.au/~gfl/Lecture
Gyroscope
http://www.physics.usyd.edu.au/~gfl/Lecture
Gyroscope: Not rotating
http://www.physics.usyd.edu.au/~gfl/Lecture
Gyroscope: Rotating
http://www.physics.usyd.edu.au/~gfl/Lecture
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.
http://www.physics.usyd.edu.au/~gfl/Lecture
Gravitation
http://www.physics.usyd.edu.au/~gfl/Lecture
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?
http://www.physics.usyd.edu.au/~gfl/Lecture
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!
http://www.physics.usyd.edu.au/~gfl/Lecture
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).
http://www.physics.usyd.edu.au/~gfl/Lecture
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
http://www.physics.usyd.edu.au/~gfl/Lecture
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.
http://www.physics.usyd.edu.au/~gfl/Lecture
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.
http://www.physics.usyd.edu.au/~gfl/Lecture
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
http://en.wikipedia.rg/wiki/O
http://www.physics.usyd.edu.au/~gfl/Lecture
rbit)
Kepler’s 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.
http://www.physics.usyd.edu.au/~gfl/Lecture
Is this always true?
A pair of equal mass stars
will orbit their centre of
mass (the barycentre),
apparently orbiting nothing
at all!
Borrowed from Wikipedia
What motion do you expect
the barycentre to have?
http://www.physics.usyd.edu.au/~gfl/Lecture
Kepler’s 2nd Law
The 2nd Law says that
the area an orbit
sweeps out in a fixed
time is a constant.
http://www.physics.usyd.edu.au/~gfl/Lecture
Kepler’s 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
Kepler’s 2nd Law is simply an
expression of the conservation of
angular momentum.
http://www.physics.usyd.edu.au/~gfl/Lecture
Kepler’s 3rd Law
Kepler’s 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
http://www.physics.usyd.edu.au/~gfl/Lecture
Orbits
The Solar System
The Galactic Centre
http://www.physics.usyd.edu.au/~gfl/Lecture
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!
http://www.physics.usyd.edu.au/~gfl/Lecture
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.
http://www.physics.usyd.edu.au/~gfl/Lecture
THE END
http://www.physics.usyd.edu.au/~gfl/Lecture