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Chapter 13 Newton’s Universal Law of Gravity Today: • Force • Energy • Circular Orbits Weds: • Kepler’s Laws mm • Types of Orbits FrG 12ˆ • Escape Velocity 12r2 12 • Tidal Forces • General Relativity Sun at Center Orbits are Circular Tycho Brahe 1546-1601 Tycho was the greatest observational astronomer of his time. Tycho did not believe in the Copernican model because of the lack of observational parallax. He didn’t believe that the Earth Moved. The Rejection of the Copernican Heliocentric Model Tycho Brahe 1546-1601 Kepler worked for Tycho as his mathematician. Kepler derived his laws of planetary motion from Tycho’s observational data. Kepler’s Laws are thus empirical - based on observation and not Theory. Kepler’s 3 Laws of Planetary Motion 1: The orbit of each planet about the sun is an ellipse with the sun at one focus. 2. Each planet moves so that it sweeps out equal areas in equal times. 3. The square of the orbital period of any planet is proportional to the 2 cube of the semimajor axis of the T1 elliptical orbit. 3 constant r1 Planet Orbits are Elliptical "The next question was - what makes planets go around the sun? At the time of Kepler some people answered this problem by saying that there were angels behind them beating their wings and pushing the planets around an orbit. As you will see, the answer is not very far from the truth. The only difference is that the angels sit in a different direction and their wings push inward." -Richard Feynman Man of the Millennium Sir Issac Newton (1642 -1727) Milky Way Galaxy Orbital Speed of Solar System: 220 km/s Orbital Period: 225 Million Years Mercury: 48 km/s Venus: 35 km/s Earth: 30 km/s Mars: 24 km/s Jupiter: 13 km/s Neptune: 5 km/s Although the Moon is always lit from the Sun, we see different amounts of the lit portion from Earth depending on where the Moon is located in its month-long Orbital Speed of Moon: ~ 1 km/s orbit. Gravitational Force is Universal The same force that makes the apple fall to Earth, causes the moon to fall around the Earth. Universal Law of Gravity 1687 Every particle in the Universe attracts every other particle with a force along a line joining them. The force is directly proportional to the product to their masses and inversely proportional to the square of the distance between them. M d m mM F ~ d 2 Measuring G: Cavendish 1798 mM GmM F ~ 2 F d d 2 Nm2 Gx 6.67 1011 kg 2 G is the same everywhere in the Universe. G’s small size is a measure of relative strength of gravity. By comparison, the proportionality constant for the electric force is k~109!! Universal Law of Gravity 2 11 Nm Minus because of the direction Gx 6.67 10 2 of the unit vector. Attractive kg Central Forces are negative! Ignore when finding magnitude! The sign is critical for solvng mm FrG 12ˆ Energy Problems. 12r2 12 Gravity: Inverse Square Law GmM F d 2 Gravitational Force INSIDE the Earth How would the force of gravity and the acceleration due to gravity change as you fell through a hole in the Earth? What would your motion be? Assume you jump from rest. Ignore Air Resistance. Gravitational Force INSIDE the Earth Inside the Earth the Gravitational Force is Linear. Acceleration decreases as you fall to the center (where your speed is the greatest) and then the acceleration increases but in the opposite direction, slowing you down to a stop at the other end…but then you would fall back in again, bouncing back and forth forever! Earth-Moon Gravity Calculate the magnitude of the force of gravity between the Earth and the Moon. The distance between the Earth and Moon centers is 3.84x108m GmM F EM d 2 6.673x 1011 Nm 2 / kg 2 5.98 x 10 24 kg 7.35 x 10 22 kg FEM 2 3.84xm 108 20 FEM 2.01 x 10 N Earth-Moon Gravity Calculate the acceleration of the Earth due to the Earth-Moon gravitational interaction. FEM aE mE 2.01xN 1020 5.98x 1024 kg 52 aE 3.33 x 10 m / s Earth-Moon Gravity Calculate the acceleration of the Moon due to the Earth-Moon gravitational interaction. FEM aM mM 2.01xN 1020 7.35x 1022 kg 32 aM 2.73 x 10 m / s Earth-Moon Gravity The acceleration of gravity at the Moon due to the Earth is: 32 aM 2.73 x 10 m / s The acceleration of gravity at the Earth due to the moon is: 52 aE 3.33 x 10 m / s Why the difference? FORCE is the same. Acceleration is NOT!!! BECAUSE MASSES ARE DIFFERENT! Force is not Acceleration! FFEarth on Moon Moon on Earth The forces are equal but the accelerations are not! Superposition of Gravitational Force Vectors What is the net force on the mass at the origin? Problem Solving Strategy 1. Draw FBD 2. Calculate Force Magnitudes 3. Resolve into components 4. Add components mm FrG 12ˆ 5. Express as ijk vector 12r2 12 6. Find magnitude and direction of resultant force. Problem Finding little g Calculate the acceleration of gravity acting on you at the surface of the Earth. What is g? Gm M you E F m a F 2 you RE Source of the Force Response to the Force Gmyou M E This is your 2 mayou WEIGHT! RE Independent of your mass! GM E This is why a rock and a 2 feather fall with the RE same acceleration! Finding little g Calculate the acceleration of gravity acting on you at the surface of the Earth. What is g? Gmyou M E F myou a F 2 RE GM Source of the Force E Reaction to the Force a 2 RE 6.673x 1011 Nm 2 / kg 2 5.98 x 10 24 kg a 2 6.38xm 106 a 9.81 m/ s2 = g! In general, g for any Planet: GM g planet planet R2 planet g field is an acceleration field The gravitational field describes the “effect” that any object has on the empty space around itself in terms of the force that would be present if a second object were somewhere in that space F GM grg ˆ mr2 In Sum, Gravitational Force & Field Gm12 m FGG rˆ F m g r 2 2 Gravity field is the force that +1kg would feel if it were placed at this location. It is an acceleration field. The gravitational Field of the Earth at the surface of the Earth is simply, g: GM Fg GM ggrE 9.81 N / kg ˆ 9.81 m / s2 r2 mr2 Physics 42 Electric Force and Field qq Fr k 12ˆ 12 r2 F qE Two flavors of charge (+/-) kq Electric Field E due to q : E 1 1 r2 Curvature of Earth Curvature of the Earth: Every 8000 m, the Earth curves by 5 meters! If you threw the ball at 8000 m/s off the surface of the Earth (and there were no buildings or mountains in the way) how far would it travel in the vertical direction in 1 second? 1 y gt2~ 5 m / s 2 (1 s 2 ) 5 m 2 The ball will achieve orbit. Orbital Velocity If you can throw a ball at 8000m/s, the Earth curves away from it so that the ball continually falls in free fall around the Earth – it is in orbit around the Earth! Ignoring air resistance. Above the atmosphere Projectile Motion/Orbital Motion Projectile Motion is Orbital motion that hits the Earth! Orbital Motion| & Escape Velocity 8km/s: Circular orbit Between 8 & 11.2 km/s: Elliptical orbit 11.2 km/s: Escape Earth 42.5 km/s: Escape Solar System! Orbits Circular Orbit Elliptical Orbit Circular Orbits As the ball falls around the Earth in a circular orbit, does the acceleration due to gravity change its orbital speed? It only changes its direction! Ignoring air resistance. Above the atmosphere Circular Orbital Velocity The force of gravity is perpendicular to the velocity of the ball so it doesn’t speed it up – it changes only the direction of the ball. Gravity provides a centripetal acceleration the keeps it in a circle! The PE and KE are the same throughout the orbit. Since F is parallel to r, angular momentum is also conserved. v Elliptical Orbits As the satellite falls around the Earth in an elliptical orbit, does the acceleration due to gravity change its orbital speed? There is a component of force (and A acceleration) in the direction of motion! Gravity changes the satellite’s speed when in elliptical orbits. Mechanical Energy is conserved but KE and PE change throughout the orbit. Is angular momentum conserved? Why? Where is the speed greatest- A or B? B Satellite Orbits Circular Orbit Speed With Increasing Altitude g and v Above the Earth’s Surface GMm If an object is some F distance h above the r2 Earth’s surface, r GM g E becomes RE + h 2 RhE v2 The tangential speed of an g r object is its orbital speed v2 GM and is given by the E Rh 2 centripetal acceleration, g: E RhE GM Orbital speed decreases with v E increasing altitude! ()RhE Orbit Question Find the orbital speed of a satellite 200 km above the Earth. 24 6 Assume a circular orbit. MEE5.97 x 10 kg , R 6.38 x 10 m v2 msE M G F F ms a a r 2 r 2 msE M G v MGE Fm2 s v rr RhE What is this? (5.97x 1024 kg )(6.67 x 10 11 Nm 2 / kg 2 ) v 6.58xm 106 Notice that this 3 v 7.78 x 10 m / s is less 8km/s! Orbit Question What is the period of a satellite orbiting 200 km above the Earth? M5.97 x 1024 kg , R 6.38 x 10 6 m Assume a circular orbit.
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