1. Copyright © 2012 Pearson Education Inc. Orbital motion, final review Physics 7C lecture 18 Thursday December 5, 8:00 AM – 9:20 AM Engineering Hall 1200

2. Copyright © 2012 Pearson Education Inc. Newton’s law of gravitation The gravitational force can be expressed mathematically as F g = Gm 1 m 2 /r 2, where G is the gravitational constant. Note G is different from g. G = 6.67 E -11 N m 2 /kg 2

3. Copyright © 2012 Pearson Education Inc. Gravitation and spherically symmetric bodies The gravitational interaction of bodies having spherically symmetric mass distributions is the same as if all their mass were concentrated at their centers. This is exact, not approximation! Let’s prove it mathematically.

4. Copyright © 2012 Pearson Education Inc. Gravitational potential energy The gravitational potential energy of a system consisting of a particle of mass m and the earth is U = –Gm E m/r. This assumes zero energy at infinite distance.

5. Copyright © 2012 Pearson Education Inc. The motion of satellites The trajectory of a projectile fired from A toward B depends on its initial speed. If it is fired fast enough, it goes into a closed elliptical orbit (trajectories 3, 4, and 5 in Figure 13.14 below).

6. Copyright © 2012 Pearson Education Inc. Circular satellite orbits For a circular orbit, the speed of a satellite is just right to keep its distance from the center of the earth constant. (See Figure 13.15 below.) A satellite is constantly falling around the earth. Astronauts inside the satellite in orbit are in a state of apparent weightlessness because they are falling with the satellite. (See Figure 13.16 below.) Follow Example 13.6.

7. Copyright © 2012 Pearson Education Inc. Kepler’s laws and planetary motion Each planet moves in an elliptical orbit with the sun at one focus.

8. Copyright © 2012 Pearson Education Inc. Kepler’s laws and planetary motion A line from the sun to a given planet sweeps out equal areas in equal times (see Figure at the right).

9. Copyright © 2012 Pearson Education Inc. Kepler’s laws and planetary motion The periods of the planets are proportional to the 3 / 2 powers of the major axis lengths of their orbits.

10. Copyright © 2012 Pearson Education Inc. Some orbital examples The orbit of Comet Halley. See Figure below.

11. Copyright © 2012 Pearson Education Inc. A point mass inside a spherical shell If a point mass is inside a spherically symmetric shell, the potential energy of the system is constant. This means that the shell exerts no force on a point mass inside of it. what is the motion of the small ball?

12. Copyright © 2012 Pearson Education Inc. A point mass inside a spherical shell

13. Copyright © 2012 Pearson Education Inc. Black holes If a spherical nonrotating body has radius less than the Schwarzschild radius, nothing can escape from it. Such a body is a black hole. (See Figure 13.26 below.) The Schwarzschild radius is R S = 2GM/c 2. The event horizon is the surface of the sphere of radius R S surrounding a black hole. Follow Example 13.11.

14. Copyright © 2012 Pearson Education Inc. Detecting black holes We can detect black holes by looking for x rays emitted from their accretion disks. (See Figure below.)

15. Copyright © 2012 Pearson Education Inc. Review for final exam