1. Copyright © 2010 Pearson Education, Inc. Chapter 12: Gravitation

2. Copyright © 2010 Pearson Education, Inc. History of Gravitational Theory Ancient Greece & Ptolemy Nicolas Copernicus Tycho Brahe Johannes Kepler Isaac Newton Albert Einstein

3. Copyright © 2010 Pearson Education, Inc. Ancient Greece & Ptolemy Early theory had the Earth at the center of the universe. –“Geocentric” All stars were fixed at the same distance from the Earth. –Think of them on a sphere surrounding Earth. Saw objects that seemed to “wander” through the night sky.

4. Copyright © 2010 Pearson Education, Inc. Ancient Greece & Ptolemy Aristotle & Plato said the planets circle perfectly around the Earth. 2 nd century AD Ptolemy created a complicated system of circles and epicycles which could explain all types of movements.

5. Copyright © 2010 Pearson Education, Inc. Nicolas Copernicus (1473-1543) Born in Poland, 1473. Became a priest but maintained an interest in sky observations. Age 41, gave friends an anonymous manuscript arguing Ptolemy’s model would be greatly simplified if the sun were at the center of the universe.

6. Copyright © 2010 Pearson Education, Inc. Tycho Brahe (1546-1601) Danish noblemen born in 1546. Discovered a new star when he was 26, securing funding for the best astronomical observatory at the time. Rejected Copernican view of the universe and adopted his own. –Planets orbit the sun, which orbits Earth the center. Meticulous data collection.

7. Copyright © 2010 Pearson Education, Inc. Johannes Kepler (1571-1630) Born in Germany, spent early years as a math teacher. Brahe needed Kepler’s mathematical prowess, Kepler needed Brahe’s data. After Brahe’s death, realized planets orbited in ellipse’s, not circles. Kepler’s Laws of Planetary Motion.

8. Copyright © 2010 Pearson Education, Inc. Sir Isaac Newton (1642-1727) Invented calculus. Apple observation? Law of Universal Gravitation Held the Lucasian Chair of Mathematics at Cambridge University

9. Copyright © 2010 Pearson Education, Inc. Albert Einstein (1879-1955) Challenged Newton’s Theory of Gravity Special Theory of Relativity (1905) General Theory of Relativity (1915) –Gravity bends the space around objects.

10. Copyright © 2010 Pearson Education, Inc. 12-1 Newton’s Law of Universal Gravitation Newton’s insight: The force accelerating an apple downward is the same force that keeps the Moon in its orbit. Hence, Universal Gravitation.

11. Copyright © 2010 Pearson Education, Inc. 12-1 Newton’s Law of Universal Gravitation The gravitational force is always attractive, and points along the line connecting the two masses: The two forces shown are an action-reaction pair.

12. Copyright © 2010 Pearson Education, Inc. 12-1 Newton’s Law of Universal Gravitation The radius term in Newton’s Law of Universal Gravitation is the center-to-center distance of the two objects.

13. Copyright © 2010 Pearson Education, Inc. Exercise 12-1 A man takes his dog for a walk on a deserted beach. Treating people and dogs as point objects for the moment, find the force of gravity between the 105-kg man and his 11.2-kg dog when they are separated by a distance of (a) 1.00 m and (b) 10.0 m.

14. Copyright © 2010 Pearson Education, Inc. Question 12.1b Earth and Moon II a) one quarter b) one half c) the same d) two times e) four times If the distance to the Moon were doubled, then the force of attraction between Earth and the Moon would be:

15. Copyright © 2010 Pearson Education, Inc. 12-1 Newton’s Law of Universal Gravitation G is a very small number; this means that the force of gravity is negligible unless there is a very large mass involved (such as the Earth). If an object is being acted upon by several different gravitational forces, the net force on it is the vector sum of the individual forces. This is called the principle of superposition.

16. Copyright © 2010 Pearson Education, Inc. Force of Gravity During a Lunar Eclipse

17. Copyright © 2010 Pearson Education, Inc. 12-2 Gravitational Attraction of Spherical Bodies Gravitational force between a point mass and a sphere: the force is the same as if all the mass of the sphere were concentrated at its center.

18. Copyright © 2010 Pearson Education, Inc. 12-2 Gravitational Attraction of Spherical Bodies What about the gravitational force on objects at the surface of the Earth? The center of the Earth is one Earth radius away, so this is the distance we use: Therefore,

19. Copyright © 2010 Pearson Education, Inc. 12-2 Gravitational Attraction of Spherical Bodies The acceleration of gravity decreases slowly with altitude:

20. Copyright © 2010 Pearson Education, Inc. Example 12-2 If you climb to the top of Mt. Everest, you will be about 8.86 km above sea level. What is the acceleration due to gravity at this altitude?

21. Copyright © 2010 Pearson Education, Inc. 12-2 Gravitational Attraction of Spherical Bodies Once the altitude becomes comparable to the radius of the Earth, the decrease in the acceleration of gravity is much larger:

22. Copyright © 2010 Pearson Education, Inc. 12-2 Gravitational Attraction of Spherical Bodies The Cavendish experiment allows us to measure the universal gravitation constant:

23. Copyright © 2010 Pearson Education, Inc. 12-2 Gravitational Attraction of Spherical Bodies Even though the gravitational force is very small, the mirror allows measurement of tiny deflections. Measuring G also allowed the mass of the Earth to be calculated, as the local acceleration of gravity and the radius of the Earth were known.

24. Copyright © 2010 Pearson Education, Inc. 12-3 Kepler’s Laws of Orbital Motion Johannes Kepler made detailed studies of the apparent motions of the planets over many years, and was able to formulate three empirical laws: 1. Planets follow elliptical orbits, with the Sun at one focus of the ellipse.

25. Copyright © 2010 Pearson Education, Inc. Anatomy of an Ellipse Perihelion Aphelion A B Focus C

26. Copyright © 2010 Pearson Education, Inc. Anatomy of an Ellipse Perihelion (or perigee): point in the orbit of a planet, asteroid or comet where it is nearest to the sun. Apehelion (or apogee): point in the orbit of a planet, asteroid or comet where it is farthest from the sun. Semi-Major Axis (A): farthest distance from ellipsoid center to the edge of the ellipsoid. Semi-Minor Axis (B): shortest distance from ellipsoid center to the edge of the ellipsoid.

27. Copyright © 2010 Pearson Education, Inc. 12-3 Kepler’s Laws of Orbital Motion 2. As a planet moves in its orbit, it sweeps out an equal amount of area in an equal amount of time.

28. Copyright © 2010 Pearson Education, Inc. The Earth’s orbit is slightly elliptical. In fact, the Earth is closer to the Sun during the (northern hemisphere’s) winter than it is during the summer. Is the speed of the Earth during winter (a) greater than, (b) less than, or (c) the same as its speed during the summer?

29. Copyright © 2010 Pearson Education, Inc. Conservation of Angular Momentum

30. Copyright © 2010 Pearson Education, Inc. 12-3 Kepler’s Laws of Orbital Motion 3. The period, T, of a planet increases as its mean distance from the Sun, r, raised to the 3/2 power. This can be shown to be a consequence of the inverse square form of the gravitational force.

31. Copyright © 2010 Pearson Education, Inc. Derivation of Kepler’s Third Law

32. Copyright © 2010 Pearson Education, Inc. 12-3 Kepler’s Laws of Orbital Motion A geosynchronous satellite is one whose orbital period is equal to one day. If such a satellite is orbiting above the equator, it will be in a fixed position with respect to the ground. These satellites are used for communications and and weather forecasting.

33. Copyright © 2010 Pearson Education, Inc. The National Weather Service wants to put a new geosyncrhonous weather satellite into low Earth orbit. Find the altitude above the Earth’s surface where a satellite orbits with a period of one day.

34. Copyright © 2010 Pearson Education, Inc. 12-3 Kepler’s Laws of Orbital Motion GPS satellites are not in geosynchronous orbits; their orbit period is 12 hours. Triangulation of signals from several satellites allows precise location of objects on Earth.

35. Copyright © 2010 Pearson Education, Inc. 12-3 Kepler’s Laws of Orbital Motion Kepler’s laws also give us an insight into possible orbital maneuvers.