1. Chapter 8
Gravity

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. Chapter 8
Problem 15
Two identical lead spheres with their centers 14 cm apart attract each other with a 0.25-µN force. Find their mass.

4. 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

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. Shell theorem
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

7. 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

8. 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

9. Gravitational field
A gravitational field exists at every point in space
When a particle is placed at a point where there is gravitational field, the particle experiences a force
The field exerts a force on the particle
The gravitational field is defined as:
The gravitational field is the gravitational force experienced by a test particle placed at that point divided by the mass of the test particle

10. Gravitational field
The presence of the test particle is not necessary for the field to exist
The source particle creates the field
The gravitational field vectors point in the direction of the acceleration a particle would experience if placed in that field
The magnitude is that of the freefall acceleration at that location

11. Gravitational potential energy
Gravitation is a conservative force (work done by it is path-independent)
For conservative forces (Ch. 7):

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

13. Gravitational potential energy

14. Orbits
Accounting for the shape of Earth, projectile motion (Ch. 3) has to be modified:

15. Orbits
The total mechanical energy E = K + U determines the type of orbit an object follows:
E < 0: The object is in a bound, elliptical orbit

16. Orbits
The total mechanical energy E = K + U determines the type of orbit an object follows:
Special cases include circular orbits and the straight-line paths of falling objects

17. Orbits
The total mechanical energy E = K + U determines the type of orbit an object follows:
E > 0: The orbit is unbound and hyperbolic

18. Orbits
The total mechanical energy E = K + U determines the type of orbit an object follows:
E = 0: The borderline case gives a parabolic orbit

19. Orbits
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. Orbits
The “parabolic” trajectories of projectiles near Earth’s surface are actually sections of elliptical orbits that intersect Earth

21. Orbits
The trajectories are parabolic only in the approximation that we can neglect Earth’s curvature and the variation in gravity with distance from Earth’s center

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

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

24. Chapter 8
Problem 40
A white dwarf is a collapsed star with roughly the Sun’s mass compressed into the size of Earth. What would be (a) the orbital speed and (b) the orbital period for a spaceship in orbit just above the surface of a white dwarf?

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

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

27. Chapter 8
Problem 54
A projectile is launched vertically upward from a planet of mass M and radius R; its initial speed is twice the escape speed. Derive an expression for its speed as a function of the distance r from the planet’s center.

28. 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

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

30. Chapter 8
Problem 23
The Mars Reconnaissance Orbiter circles the red planet with a 112-min period. What’s the spacecraft’s altitude?

31. Questions?

32. Answers to the even-numbered problems
Chapter 8
Problem 16
4.62 × 10−8 N

33. Answers to the even-numbered problems
Chapter 8
Problem 20
1.87 y

34. Answers to the even-numbered problems
Chapter 8
Problem 26
1.67 × 106 m