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Chapter 13 Gravitation

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Chapter 13

Gravitation

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

- 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

Chapter 13

Problem 9

- 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

- 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

Gravity force near the surface of Earth

In reality gis not a constant because:

Earth is rotating,

Earth is approximately an ellipsoid

with a non-uniform density

Gravity force near the surface of Earth

Weight of a crate measured at the equator:

- 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

- 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

Chapter 13

Problem 20

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

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

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

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

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

Chapter 13

Problem 33

Johannes Kepler

(1571-1630)

Tycho Brahe/

Tyge Ottesen

Brahe de Knudstrup

(1546-1601)

- 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

- First Kepler’s law
- Elliptical orbits of planets are described by a semimajor axisa and an eccentricitye
- For most planets, the eccentricities are very small (Earth's e is 0.00167)

- 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

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

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

- Satellites
- For an elliptic orbit it can be shown
- Orbits with different ebut the same a have the same total mechanical energy

Chapter 13

Problem 50

Answers to the even-numbered problems

Chapter 13:

Problem 2

2.16

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

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

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

Answers to the even-numbered problems

Chapter 13:

Problem 54

(a) 8.0 × 108 J;

(b) 36 N