The beautiful rings of Saturn consist of countless centimeter-sized ice crystals, all orbiting the planet under the influence of gravity. Chapter Goal: To use Newton s theory of gravity to understand the motion of satellites and planets. Kepler s laws, as we call them today, state that 1. Planets move in elliptical orbits, with the sun at one focus of the ellipse. 2. A line drawn between the sun and a planet sweeps out equal areas during equal intervals of time. 3. The square of a planet s orbital period is proportional to the cube of the semimajor-axis length. 1
Newton proposed that every object in the universe attracts every other object. Suppose an object of mass m is on the surface of a planet of mass M and radius R. The local gravitational force may be written as where we have used a local constant acceleration: F 1on2 = F 2on1 = Gm 1m 2 r 2 G = 6.67 10 11 Nm 2 /kg 2 (Gravitational constant) On earth near sea level it can be shown that g surface = 9.80 m/s 2. 5. 5. 2
Force of Gravity Your weight is the gravitational force the Earth pulls on you Force between two people sitting 1 m (3.3 feet) apart : 10-7 lb = weight of a flea If you travel to the Moon: Your mass does not change, your weight will be 6 times less! If the Earth pulls you with 150 lb of force, you also pull on the Earth with 150 lbs! (Actionreaction) Gravity Surprises At an altitude of 100 miles, 99.999% of the atmosphere is below you. The gravitational force is less than at the Earth s surface, 95% as great What happens if the Sun suddenly turns into a Black hole? Nothing changes! Black holes do not suck everything? Satellite Orbits On which orbit would you place a spy satellite? A) LEO B)MEO C) GEO D) not shown here Spy Satellites They are on LEO Fast, one full orbit every 90 mins Looks at a desired location for about a min Geosynchronous Satellites Satellites complete on full orbit in one day Geostationary: above the equator, looks at the same spot all the time Communication satellites 3
Global Positioning System (GPS) and Medium- Earth Orbit (MEO) Orbit Simulation GPS satellites are in MEO One full orbit in 12 hrs 24 of them http://phet.colorado.edu When two isolated masses m 1 and m 2 interact over large distances, they have a gravitational potential energy of QUESTION: where we have chosen the zero point of potential energy at r =, where the masses will have no tendency, or potential, to move together. Note that this equation gives the potential energy of masses m 1 and m 2 when their centers are separated by a distance r. 4
We calculated the escape speed for a space ship to be 11.2 km/s. If we increase the payload of this ship significantly so that the mass is now 100 times more, what will be the new escape speed? A) More than 11.2 km/s B) Less than 11.2 km/s C) No change We calculated the energy needed to send a space ship to infinity. If we increase the payload of this ship significantly so that the mass is now 100 times more, what will be the new energy needed? QUESTION: A) No change B) 10 times more C) 70 times more D) 100 times more E) 700 times more The mathematics of ellipses is rather difficult, so we will restrict most of our analysis to the limiting case in which an ellipse becomes a circle. Most planetary orbits differ only very slightly from being circular. If a satellite has a circular orbit, its speed is 5
We know that for a satellite in a circular orbit, its speed is related to the size of its orbit by v 2 = GM/r. The satellite s kinetic energy is thus QUESTION: But GMm/r is the potential energy, U g, so If K and U do not have this relationship, then the trajectory will be elliptical rather than circular. So, the mechanical energy of a satellite in a circular orbit is always: 6
A satellite orbits the earth with constant speed at a height above the surface equal to the earth s radius. The magnitude of the satellite s acceleration is A. g on earth. B. g on earth. C. g on earth. D. 4g on earth. E. 2g on earth. 7
A satellite orbits the earth with constant speed at a height above the surface equal to the earth s radius. The magnitude of the satellite s acceleration is A. g on earth. B. g on earth. C. g on earth. D. 4g on earth. E. 2g on earth. The figure shows a binary star system. The mass of star 2 is twice the mass of star 1. Compared to, the magnitude of the force is A. one quarter as big. B. half as big. C. the same size. D. twice as big. E. four times as big. The figure shows a binary star system. The mass of star 2 is twice the mass of star 1. Compared to, the magnitude of the force is A. one quarter as big. B. half as big. C. the same size. D. twice as big. E. four times as big. A planet has 4 times the mass of the earth, but the acceleration due to gravity on the planet s surface is the same as on the earth s surface. The planet s radius is A. R e. B. R e. C. 4R e. D. R e. E. 2R e. A planet has 4 times the mass of the earth, but the acceleration due to gravity on the planet s surface is the same as on the earth s surface. The planet s radius is A. R e. B. R e. C. 4R e. D. R e. E. 2R e. Rank in order, from largest to smallest, the absolute values U g of the gravitational potential energies of these pairs of masses. The numbers give the relative masses and distances. In absolute value: A. U e > U d > U a > U b = U c B. U b > U c > U d > U a > U e C. U e > U a = U b = U d > U c D. U e > U a = U b >U c > U d E. U b > U c > U a = U d > U e 8
Rank in order, from largest to smallest, the absolute values U g of the gravitational potential energies of these pairs of masses. The numbers give the relative masses and distances. In absolute value: A. U e > U d > U a > U b = U c B. U b > U c > U d > U a > U e C. U e > U a = U b = U d > U c D. U e > U a = U b >U c > U d E. U b > U c > U a = U d > U e Two planets orbit a star. Planet 1 has orbital radius r 1 and planet 2 has r 2 = 4r 1. Planet 1 orbits with period T 1. Planet 2 orbits with period A. T 2 = T 1. B. T 2 = T 1 /2. C. T 2 = 8T 1. D. T 2 = 4T 1. E. T 2 = 2T 1. Two planets orbit a star. Planet 1 has orbital radius r 1 and planet 2 has r 2 = 4r 1. Planet 1 orbits with period T 1. Planet 2 orbits with period A. T 2 = T 1. B. T 2 = T 1 /2. C. T 2 = 8T 1. D. T 2 = 4T 1. E. T 2 = 2T 1. 9