Short Version : 8. Gravity

1. Short Version : 8. Gravity

2. Retrograde Motion Retrograde motion of Mars. As explained by Ptolemy.

3. Ptolemaic (Geo-Centric) System equant  epicycle deferent  swf

4. Cassini Apparent Sun Venus

5. 8.1. Toward a Law of Gravity 1543: Copernicus – Helio-centric theory. 1593: Tycho Brahe – Planetary obs. 1592-1610: Galileo – Jupiter’s moons, sunspots, phases of Venus. 1609-19: Kepler’s Laws 1687: Newton – Universal gravitation. Phases of Venus: Size would be constant in a geocentric system.

6. Kepler’s Laws Explains retrograde motion Mathematica

7. 8.2. Universal Gravitation Newton’s law of universal gravitation: m1 & m2 are 2 point masses. r12 = position vector from 1 to 2. F12 = force of 1 on 2. G = Constant of universal gravitation = 6.67  1011 N m2 / kg2 . F12 m2 r12 m1 Law also applies to spherical masses.

8. Example 8.1. Acceleration of Gravity • Use the law of gravitation to find the acceleration of gravity • at Earth’s surface. • at the 380-km altitude of the International Space Station. • on the surface of Mars.  (a) (b) (c) see App.E

9. Cavendish Experiment: Weighing the Earth ME can be calculated if g, G, & REare known. Cavendish: G determined using two 5 cm & two 30 cm diameter lead spheres.

10. 8.3. Orbital Motion Orbital motion: Motion of object due to gravity from another larger body. E.g. Sun orbits the center of our galaxy with a period of ~200 million yrs. Newton’s “thought experiment” g = 0 Condition for circular orbit Speed for circular orbit projectiles orbit Orbital period Kepler’s 3rd law

11. Example 8.3. Geosynchronous Orbit What altitude is required for geosynchronous orbits? Altitude = r  RE Earth circumference = Earth not perfect sphere  orbital correction required every few weeks.

12. Elliptical Orbits Projectile trajectory is parabolic only if curvature of Earth is neglected. ellipse Orbits of most known comets, are highly elliptical. Perihelion: closest point to sun. Aphelion: furthest point from sun.

13. Open Orbits Open (hyperbola) Closed (circle) Borderline (parabola) Closed (ellipse) Mathematica

14. 8.4. Gravitational Energy How much energy is required to boost a satellite to geosynchronous orbit? U = 0 on this path U12 depends only on radial positions. … so U12 is the same as if we start here.

15. Zero of Potential Energy  Gravitational potential energy E > 0, open orbit Open Closed E < 0, closed orbit Bounded motion Turning point

16. Energy in Circular Orbits  Circular orbits: r > r 0 K K  E E K K U U Higher K or v  Lower E & orbit (r) . To catch the satellite, the shuttle needs to lose energy. It does so by turning to fire its engine opposite its direction of motion. It drops lower, turns again , and fires its engine to achieve a circular orbit, now faster and lower than before. Mathematica

17. energy Altitude 0  K K > K h E = K+U = U / 2 h < h E = K+U = U / 2 E = K+U < E ( K < K ) U U < U UG 0

18. 8.5. The Gravitational Field • Two descriptions of gravity: • body attracts another body (action-at-a-distance) • Body creates gravitational field. • Field acts on another body. near earth Near Earth: Large scale: Action-at-a-distance  instantaneous messages Field theory  finite propagation of information Only field theory agrees with relativity. Great advantage of the field approach: No need to know how the field is produced. in space

19. Application: Tide Moon’s tidal (differential) force field at Earth’s surface Moon’s tidal (differential) force field near Earth Two tidal bulges Mathematica Sun + Moon  tides with varying strength. Tidal forces cause internal heating of Jupiter’s moons. They also contribute to formation of planetary rings.