Semester 1 2008 gfl/Lecture Physics 1901 (Advanced) A/Prof Geraint F. Lewis Rm 560, A29 [email protected] gfl/Lecture

Semester 1 2008 http://www.physics.usyd.edu.au/~gfl/Lecture

Physics 1901 (Advanced)

A/Prof Geraint F. LewisRm 560, [email protected]/~gfl/Lecture


Gyroscope


Gyroscope: Not rotating


Gyroscope: Rotating


Precision SpeedOver 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.


Gravitation


Using GravitationNewton 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?


Using GravitationNewton 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?


Weight

At the surface of the Earth

We know RE, g and G, so can calculate ME


Falling through the EarthAssume 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!


Gravitational Potential EnergyImagine 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).


Escape VelocityImagine 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


Motion of SatellitesGravity 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.


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.


Non-circular OrbitsWhat 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 )


Kepler’s 1st LawBefore 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.


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


Kepler’s 2nd Law

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


Kepler’s 2nd LawRemember 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.


Kepler’s 3rd LawKepler’s 3rd Law is relates the period of an elliptical orbit with semi-major axis a

http://hyperphysics.phy-astr.gsu.edu/hbase/kepler.html

Note that this is not dependent upon the mass of the orbiting object.


Orbits

The Solar SystemThe Galactic Centre


Orbits: In generalIn 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!


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.


THE END

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Semester 1 2008 gfl/Lecture Physics 1901 (Advanced) A/Prof Geraint F. Lewis Rm 560, A29 [email protected] gfl/Lecture