Newton’s Theory of Gravity and Planetary Motion AP Physics C: Mechanics

Geocentric View Points Aristotle (Greek) 384-322 BC Ptolemy (Alexandrian Greek) 85-65AD Heliocentric Viewpoints Aristarchus (Greek)310-230 BC Copernicus (Poland and Italy) 1473-1543 Galileo Galilei (Italian) 1564-1642

Tycho Brahe (Danish) 1546-1601

Tycho Brahe Compromise Theory: The Sun orbits the Earth, but the remaining planets orbit the Sun. Brahe passed along his observations to Johannes Kepler, his assistant.

Brahe’s Compromise Theory

What’s Wrong With This? As we all know, Brahe’s theory is not accurate for multiple reasons, but two should be immediately noticeable. The sun is the CENTER of our solar system. ALL planets, including Earth, revolve around the sun with a certain period that is related to its distance to the sun. The orbits are circular! We know that the planet’s orbits are not circular but elliptical.

Johannes Kepler (1571-1630) German Astronomer Kepler is credited with developing three laws that govern planetary motion:

Kepler’s First Law of Planetary Motion Kepler’s Laws of Planetary Motion 1. Planets move in elliptical orbits, with the sun at one focus of the ellipse. Slide 13-20

Kepler’s Second Law of Planetary Motion Kepler’s Laws of Planetary Motion 2. A line drawn between the sun and a planet sweeps out equal areas during equal intervals of time. WHY? Slide 13-21

Kepler’s 2nd Law (The Law of Areas) A line from the sun to a planet sweeps out equal areas in equal lengths of time.

Eccentricity of an Ellipse e=c/a For a circle e=0 Pluto has the highest e=0.25 in our system. Earth’s e=0.017

Kepler’s Third Law of Planetary Motion Kepler’s Laws of Planetary Motion 3. The square of a planet’s orbital period (or the amount of time it takes to complete one revolution) is proportional to the cube of the semimajor-axis length (distance from sun. T2 ~ r3 Slide 13-22

Kepler’s Third Law Kepler’s Third Law describe how long it will take a planet to revolve around the sun based on its semi-major axis. In order to calculate said period, we use the equation: *** G is the universal gravitational constant = 6.67 x 10-11 Nm2 / kg2

Kepler’s Third Law Since distances are normally very large, it is helpful to measure the distances in Astronomical Units (AU). 1 AU is the average radius of the orbit of the Earth about the sun. The period for 1 AU is approximately 1 year. However, this depends on the type of problem we use. We can use either one, but the AU is normally used to comparisons to the Earth. Remember, we use ratios to compare!

EXAMPLE: An asteroid revolves around the sun with a mean orbital radius of three times that of the earth. What is the period of the asteroid in earth years? Answer: 5.2y

Example: Astronomers have only recently seen evidence of planets orbiting nearby stars. These are called extrasolar planets. Suppose a planet is observed to have a 1200 day period as it orbits a star at the same distance as Jupiter is from the sun. What is the mass of the star? Assume r = 7.78x1011 meters. 2.59 x 1031 kg

Kepler’s Laws of Planetary Motion A circular orbit is a special case of an elliptical orbit. Slide 13-23

Questions on Kepler’s Laws

Newton’s Law of Universal Gravitation Isaac Newton Newton’s Law of Universal Gravitation Legend has it that Newton saw an apple fall from a tree, and it occurred to him that the apple was attracted to the center of the earth. If the apple was so attracted, why not the moon? Newton posited that gravity is a universal attractive force between all objects in the universe. Isaac Newton, 1642–1727. Slide 13-24

Newton’s Law of Gravity Newton’s Law of Universal Gravitation The moon is in free fall around the earth. Slide 13-25

Newton’s Law of Gravity Newton’s Law of Universal Gravitation The gravitational constant is a universal constant with the value G = 6.67  1011 N m2/kg2. Slide 13-26

Newton’s Law of Universal Gravitation QuickCheck 13.1 Newton’s Law of Universal Gravitation The force of Planet Y on Planet X is ___ the magnitude of . X on Y One quarter. One half. The same as. Twice. Four times. Slide 13-29 22

Newton’s Law of Universal Gravitation QuickCheck 13.1 Newton’s Law of Universal Gravitation The force of Planet Y on Planet X is ___ the magnitude of . X on Y One quarter. One half. The same as. Twice. Four times. Slide 13-30 23

Newton’s Law of Universal Gravitation Little g and Big G Newton’s Law of Universal Gravitation An object of mass m sits on the surface of Planet X. According to an observer on the planet, the gravitational force on m should be FG = mgsurface. According to Newton’s law of gravity, the gravitational force on m should be . These are the same if: Slide 13-36

Newton’s Law of Universal Gravitation QuickCheck 13.4 Newton’s Law of Universal Gravitation Planet X has free-fall acceleration 8 m/s2 at the surface. Planet Y has twice the mass and twice the radius of planet X. On Planet Y g = 2 m/s2. g = 4 m/s2. g = 8 m/s2. g = 16 m/s2. g = 32 m/s2. Slide 13-37 25

Newton’s Law of Universal Gravitation QuickCheck 13.4 Newton’s Law of Universal Gravitation Planet X has free-fall acceleration 8 m/s2 at the surface. Planet Y has twice the mass and twice the radius of planet X. On Planet Y g = 2 m/s2. g = 4 m/s2. g = 8 m/s2. g = 16 m/s2. g = 32 m/s2. Slide 13-38 26

gravity above the Earth’s surface r = RE + h Note: g decreases with increasing altitude As r ® , the weight of the object approaches zero

Decrease of g with Distance Gravity Depends on Height Slide 13-40

Questions on Universal Gravitation???

Speeds of Satellites In order to find the speed of a satellite, or any object revolving around the Earth, we can use the principles of centripetal force and Newton’s Law of Universal Gravitation to find it. Let’s take a look.

Gravitational Potential Energy As a particle moves from A to B, its gravitational potential energy changes by:

Gravitational Potential Energy of the Earth-particle system The reference point is chosen at infinity where the force on a particle would approach zero. Ui = 0 for ri = ∞This is valid only for r > = RE and not valid for r < RE U is negative because the object is getting closer to Earth.

Gravitational Potential Energy of any two particles The negative is used because the distance is decreasing, gravity is bringing the object closer! This is the same potential energy as mgh. However, this is used as a whole, while the other is used when we are closer to the surface of the earth.

Gravitational Potential Energy of any two particles QuickCheck 13.6 Gravitational Potential Energy of any two particles Which system has more (larger absolute value) gravitational potential energy? System A. System B. They have the same gravitational potential energy. Slide 13-46 34

Gravitational Potential Energy of any two particles QuickCheck 13.6 Gravitational Potential Energy of any two particles Which system has more (larger absolute value) gravitational potential energy? System A. System B. They have the same gravitational potential energy. Slide 13-47 35

Systems with Three or More Particles (Configuration of Masses) The total gravitational potential energy of the system is the sum over all pairs of particles Gravitational potential energy obeys the superposition principle

Systems with Three Particles The absolute value of Utotal represents the work needed to separate the particles by an infinite distance. Remember energy is a scalar quantity.

Ex #31 A system consists of three particles, each of mass 5.00g, located at the corners of an equilateral triangle with sides of 30.0cm. Calculate the potential energy of the system. If the particles are released simultaneously, where will they collide? Ans: a) -1.67x10-14 J

Energy in a Circular Orbit

Note: Energy in a Circular Orbit K>0 and is equal to half the absolute value of the potential energy. |E| = binding energy of the system. The total mechanical energy is negative.

Energy in a Circular Orbit 2r = diameter of orbit Elliptical Orbit 2a = semi-major axis

Energy in a Circular Orbit A satellite of mass m is in the elliptical orbit shown below around the Earth (radius rE, mass M). Determine the speed of the satellite at perigee. Write your answer in terms of r1, r2, M, and G. Determine the speed of the satellite when it’s at the point X in the figure.

Escape Speed In order to find the escape speed of a rocket or any object from a planet, we will use the principles of energy conservation. Before the rocket leaves the planet, it has both kinetic energy and gravitational potential energy due to gravity trying to bring it back down. Once the rocket leaves the planet is has no gravitational potential energy and no kinetic.

Example 13.2 Escape Speed Slide 13-48

Note: For a Two Particle Bound System Both the total energy and the total angular momentum are constant.

Compare the Kinetic Energy and Angular Momentum of a Satellite at orbit 1 and 2 Earth

How does the speed of a satellite at position 2 compare to the speed at position 1. The distance r2 =2r1. (Hint: Use conservation of angular momentum) Earth 2 1