Lecture27-2A_000

# Lecture27-2A_000 - Chapter 9 Gravitation II Kepler's Laws...

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Chapter 9 Chapter 9 Gravitation II Gravitation II

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Kepler Kepler s s Laws of Planetary Motion Laws of Planetary Motion Kepler’s three laws make no effort to explain the planetary motion. Instead, they are mathematical descriptions of the planet’s motion. 1. The planets orbit the Sun in ellipses with the Sun at one focus. Discuss Keper’s breakthrough with Brahe’s data. 2. A line joining the Sun and a planet sweeps out equal areas in equal times. This is a straightforward result of the conservation of angular momentum.
Kepler Kepler s s Laws of Planetary Motion Laws of Planetary Motion 3. The square of the planet’s orbital period is proportional to the cube of the semimajor axis of its orbit. In units of Earth years and Astronomical Units, the average distance from the Earth to the Sun, this law is expressed as T 2 = a 3 . This final observation occurred several years after the first two. It was Newton’s prediction of these observations using his law of gravity that resulted in a basic understanding of orbital motion and (weak) gravity in general. In fact Kepler’s third law (in SI units) is a straightforward extension of our knowledge of the angular velocity of an orbiting object. 2 4 2 T 2 GM r o 3 T 2 4 2 GM r o 3

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Newton Newton s Law of Gravity s Law of Gravity Newton realized that the motion of the falling apple and the motion of the moon around the Earth were due to the same force. They were both falling toward the Earth due to the force of gravity. F g GMm r 2 Universal Gravitation This force obeys the inverse square law. Also the minus sign indicates that this force is attractive. G is the universal constant of gravitational attraction and is given by G = 6.673 x 10 -11 Nm 2 /kg 2 Strictly speaking it only applies to point objects. However, for spherically symmetrical objects r is the distance between their centers. As long as the size of the object is small compared to r , then it is simply the distance between them.
Orbital Motion Orbital Motion An object orbiting the Earth (or any other object orbiting a large massive object) is accelerating toward the center of the Earth. The blue lines indicate the path of an object in the absence of gravity. From our study of circular motion we know that gravity must provide the force for radial acceleration. This leads to the period for a circular orbit: GMm r 2 mv 2 r m 2 r 2 GM r 3 4 2 T 2 GM r 3 T 2 4 2 GM r 3 We have proved Kepler’s 3 rd law for circular orbits. Note that this expression is independent of the object’s mass . This law is the primary way astronomers measure the product GM of objects throughout our galaxy.

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Cavendish Experiment Cavendish Experiment Astronomers measure the product GM for orbiting objects, but what about G
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Lecture27-2A_000 - Chapter 9 Gravitation II Kepler's Laws...

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