1. Physics for Scientists and Engineers, 3rd edition
Lecture PowerPoint
Physics for Scientists and Engineers, 3rd edition
Fishbane Gasiorowicz Thornton
© 2005 Pearson Prentice Hall
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2. Chapter 12
Gravitation

3. Main Points of Chapter 12 Early observations of planetary motion
Kepler’s laws
Universal gravitation
Potential energy
Planets and satellites
Escape speed
Orbits
Gravitation and extended objects

4. Main Points of Chapter 12 Tides
Equality of inertial and gravitational mass
Einstein’s theory of gravitation
Equivalence principle

5. 12-1 Early Observations of Planetary Motion
Just looking at the sky, one would assume that the earth is the center of the universe, and that everything we see is in orbit around us.
Early models of the solar system were geocentric, but some of the planets were observed to exhibit retrograde motion:

6. 12-1 Early Observations of Planetary Motion
The geocentric model was made more elaborate, with epicycles upon epicycles, but ultimately was unable to explain retrograde motion.

7. 12-1 Early Observations of Planetary Motion
Kepler’s laws:
Planets move in planar elliptical paths with the Sun at one focus of the ellipse.

8. 12-1 Early Observations of Planetary Motion
Kepler’s laws:
2. During equal time intervals the radius vector from the Sun to a planet sweeps out equal areas.

9. 12-1 Early Observations of Planetary Motion
Kepler’s laws:
3. If T is the time that it takes for a planet to make one full revolution around the Sun, and if R is half the major axis of the ellipse (R reduces to the radius of the planet’s orbit if that orbit is circular), then:
(12-1)
where C is a constant whose value is the same for all planets.

10. 12-2 Newton’s Inverse-Square Law
Newton wanted to explain Kepler’s laws; found that:
Force must be central
Inverse-square law
Possible paths must be conic sections:

11. 12-2 Newton’s Inverse-Square Law
Law of universal gravitation includes all requirements:
(12-4)
G is a constant that can be measured using known masses; find:
(12-7)

12. 12-2 Newton’s Inverse-Square Law
Potential energy can be derived from force:
(12-8)

13. 12-3 Planets and Satellites
Newton realized that falling with a sufficiently large initial horizontal velocity is orbiting – that is, the same force that causes the apple to fall from the tree also keeps the Moon in its orbit.

14. 12-3 Planets and Satellites
Escape speed is outward speed needed to escape from Earth’s gravitational potential well (that is, to make total energy nonnegative):
(12-10)

15. 12-3 Planets and Satellites
Types of Orbits
Total energy:
(12-9)
If E is negative, orbits are closed – elliptical or circular
If E is positive, orbits are open – hyperbola

16. 12-3 Planets and Satellites
Some facts and definitions:
Orbits are all conical sections
Ellipse has semimajor and semiminor axes
Ellipse equation:
(12-11)
Closest approach to sun is called perihelion; farthest is aphelion

17. 12-3 Planets and Satellites
Semimajor axis:
Eccentricity:
(12-12)
(12-13)

18. 12-3 Planets and Satellites
Types of Orbits: circle, ellipses, parabola, hyperbola

19. 12-4 Gravitation and Extended Objects
The Gravitational Force Due to a Spherically Symmetric Object
Within a hollow shell, the gravitational force is zero – forces from opposite sides cancel

20. 12-4 Gravitation and Extended Objects
The Gravitational Force Due to a Spherically Symmetric Object
Within the object, force at r is due to mass inside r:

21. 12-4 Gravitation and Extended Objects
Dark Matter:
Discovered within and around galaxies and galaxy clusters
Within galaxies: needed to explain rotational speed of outer areas
Within galaxy clusters: needed to make clusters gravitationally bound
No matter of any kind is detectable where it is needed – scientists still figuring it out

22. 12-4 Gravitation and Extended Objects
Acceleration of gravity, g, varies with altitude above Earth’s surface, due to changing distance from Earth’s center:
(12-19)
This is a very small difference, even at the top of Mt. Everest!

23. 12-4 Gravitation and Extended Objects
Tidal Forces
Tidal forces occur because extended objects are not points. Near side of Earth is closer to Moon than far side, so the magnitude of the gravitational force is not the same on both sides.
This difference in gravitational forces on different parts of an extended object is the tidal force.

24. 12-4 Gravitation and Extended Objects
Tidal Forces
Different accelerations at A and B:

25. 12-5 A Closer Look at Gravitation
Two massive bodies will exert equal and opposite forces on each other – does that mean they orbit around each other?
Yes – actually, they both orbit around their common center of mass.
For a small planet and a large star, the difference between this and orbiting the star’s center is tiny.
For a large planet or a double star, it can be a large effect.

26. 12-5 A Closer Look at Gravitation
If several objects are exerting gravitational forces on each other, net force is vector sum of forces, and net potential energy is sum of potential energies.

27. 12-5 A Closer Look at Gravitation
Equality of inertial and gravitational masses:
The m in F = ma is the inertial mass, a measure of how resistant an object is to an external force.
The m in the gravitational force equation is the gravitational mass, a measure of the gravitational force the object exerts on another mass.
Apparently the inertial and gravitational masses of an object are equal, but why?

28. 12-6 Einstein’s Theory of Gravitation
Equivalence of gravitational and inertial mass led Einstein to formulate the equivalence principle:
No experiment can distinguish between the following two situations:
(1) a physical system at rest that is subject to a uniform gravitational force; and
(2) a physical system that is uniformly accelerating in the absence of gravity.

29. 12-6 Einstein’s Theory of Gravitation
Predictions of equivalence principle:
Light falls under the influence of gravity
Gravitational lenses
Black holes
Precession of planetary orbits

30. Summary of Chapter 12 Kepler’s three laws:
1. Planets move in planar elliptical paths with the Sun at one focus of the ellipse.
2. During equal time intervals, the radius vector from the Sun to a planet sweeps out equal areas.
3. If T is the time that it takes for a planet to make one full revolution around the Sun, and if R is half the major axis of the ellipse (R reduces to the radius of the orbit of the planet if that orbit is circular), then
(12-1)
where C is a constant whose value is the same for all planets.

31. Summary of Chapter 12, cont.
Newton showed that Kepler’s laws result from universal law of gravitation:
(12-4)
Not all masses are pointlike; if spherically symmetric, can treat all mass within radius of measurement to be concentrated at center
Differences in gravitational force across an extended object are tidal forces

32. Summary of Chapter 12, cont.
Einstein’s general theory of relativity gives a much more complete description of gravity; starts with equivalence principle for inertial and gravitational mass