1. Chapter 13
Gravitation

2. Newton’s law of gravitation
Any two (or more) massive bodies attract each other
Gravitational force (Newton's law of gravitation)
Gravitational constant G = 6.67*10 –11 N*m2/kg2 = 6.67*10 –11 m3/(kg*s2) – universal constant

3. Gravitation and the superposition principle
For a group of interacting particles, the net gravitational force on one of the particles is
For a particle interacting with a continuous arrangement of masses (a massive finite object) the sum is replaced with an integral

4. Chapter 13
Problem 9

5. Shell theorem
For a particle interacting with a uniform spherical shell of matter
Result of integration: a uniform spherical shell of matter attracts a particle that is outside the shell as if all the shell's mass were concentrated at its center

6. Gravity force near the surface of Earth
Earth can be though of as a nest of shells, one within another and each attracting a particle outside the Earth’s surface
Thus Earth behaves like a particle located at the center of Earth with a mass equal to that of Earth
g = 9.8 m/s2
This formula is derived for stationary Earth of ideal spherical shape and uniform density

7. Gravity force near the surface of Earth
In reality g is not a constant because:
Earth is rotating,
Earth is approximately an ellipsoid
with a non-uniform density

8. Gravity force near the surface of Earth
Weight of a crate measured at the equator:

9. Gravitation inside Earth
For a particle inside a uniform spherical shell of matter
Result of integration: a uniform spherical shell of matter exerts no net gravitational force on a particle located inside it

10. Gravitation inside Earth
Earth can be though of as a nest of shells, one within another and each attracting a particle only outside its surface
The density of Earth is non-uniform and increasing towards the center
Result of integration: the force reaches a maximum at a certain depth and then decreases to zero as the particle reaches the center

11. Chapter 13
Problem 20

12. Gravitational potential energy
Gravitation is a conservative force (work done by it is path-independent)
For conservative forces (Ch. 8):

13. Gravitational potential energy
To remove a particle from initial position to infinity
Assuming U∞ = 0

14. Escape speed
Accounting for the shape of Earth, projectile motion (Ch. 4) has to be modified:

15. Escape speed
Escape speed: speed required for a particle to escape from the planet into infinity (and stop there)

16. Escape speed If for some astronomical object
Nothing (even light) can escape from the surface of this object – a black hole

17. Chapter 13
Problem 33

18. Kepler’s laws Three Kepler’s laws
Tycho Brahe/
Tyge Ottesen
Brahe de Knudstrup
( )
Johannes Kepler
( )
Kepler’s laws
Three Kepler’s laws
1. The law of orbits: All planets move in elliptical orbits, with the Sun at one focus
2. The law of areas: A line that connects the planet to the Sun sweeps out equal areas in the plane of the planet’s orbit in equal time intervals
3. The law of periods: The square of the period of any planet is proportional to the cube of the semimajor axis of its orbit

19. First Kepler’s law
Elliptical orbits of planets are described by a semimajor axis a and an eccentricity e
For most planets, the eccentricities are very small (Earth's e is )

20. Second Kepler’s law
For a star-planet system, the total angular momentum is constant (no external torques)
For the elementary area swept by vector

21. Third Kepler’s law For a circular orbit and the Newton’s Second law
From the definition of a period
For elliptic orbits

22. Satellites For a circular orbit and the Newton’s Second law
Kinetic energy of a satellite
Total mechanical energy of a satellite

23. Satellites For an elliptic orbit it can be shown
Orbits with different e but the same a have the same total mechanical energy

24. Chapter 13
Problem 50

25. Answers to the even-numbered problems
Chapter 13:
Problem 2
2.16

26. Answers to the even-numbered problems
Chapter 13:
Problem 4
2.13 × 10−8 N;
(b) 60.6º

27. Answers to the even-numbered problems
Chapter 13:
Problem 20
G(M1 +M2)m/a2;
(b) GM1m/b2;
(c) 0

28. Answers to the even-numbered problems
Chapter 13:
Problem 32
2.2 × 107 J;
(b) 6.9 × 107 J

29. Answers to the even-numbered problems
Chapter 13:
Problem 54
(a) 8.0 × 108 J;
(b) 36 N