Semester Physics 1901 (Advanced) A/Prof Geraint F. Lewis Rm 560, A29

Semester Gyroscope

Semester Gyroscope: Not rotating

Semester Gyroscope: Rotating

Semester 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.

Semester Gravitation

Semester 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?

Semester 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?

Semester Weight At the surface of the Earth We know R E, g and G, so can calculate M E

Semester 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!

Semester 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).

Semester 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

Semester 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.

Semester 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.

Semester 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 rbit )

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

Semester 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

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

Semester Kepler’s 2 nd 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 2 nd Law is simply an expression of the conservation of angular momentum.

Semester Kepler’s 3 rd Law Kepler’s 3 rd 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.

Semester Orbits The Solar System The Galactic Centre

Semester 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!

Semester 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.

Semester THE END