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PHY183-Lecture41 - Review Kepler's Laws 1 All planets move...

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April 10, 2006 Physics for Scientists&Engineers 1 1 Physics for Scientists & Physics for Scientists & Engineers 1 Engineers 1 Spring Semester 2006 Lecture 41 April 10, 2006 Physics for Scientists&Engineers 1 2 Review: Kepler Kepler’ s s Laws Laws 1. All planets move in elliptical orbits with the Sun at one focal point 2. A straight line connecting the center of the Sun and the center of any planet sweeps out an equal area in any given time interval, dA/dt = const. 3. The square of the period of a planet is proportional to the cube of the semi-major axis of the orbit T 1 2 r 1 3 = T 2 2 r 2 3 (due to conservation of L) April 10, 2006 Physics for Scientists&Engineers 1 3 Typical orbits April 10, 2006 Physics for Scientists&Engineers 1 4 ! Kepler’s third Law (just shown): ! Solve for M : ! Insert numbers for Earth’s orbit ( T =1 y, r =150 million km ) ! In the same way we can determine the mass of Earth from the Moon’s orbit, or the mass of the galaxy from stars orbits Mass of the Sun T 2 r 3 = 4 ! 2 GM M = 4 ! 2 r 3 GT 2 M = 4 ! 2 (1.496 " 10 11 m) 3 (6.67 " 10 # 11 m 3 kg -1 s # 2 )(3.15 " 10 7 s) 2 = 1.99 " 10 30 kg
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April 10, 2006 Physics for Scientists&Engineers 1 5 Planets Planets’ Data Planet Radius (km) Mass (10 24 kg) g (m/s 2 ) Escape v (km/s) r -orbit (10 6 km) Eccent. T -orbit (years) Mercury 2,440 0.330 3.7 4.3 57.9 0.205 0.241 Venus 6,050 4.87 8.9 10.4 108.2 0.007 0.615 Earth 6,380 5.97 9.8 11.2 149.6 0.017 1 Mars 3,400 0.642 3.7 5.0 227.9 0.094 1.88 Jupiter 71,500 1,890 23.1 59.5 778.6 0.049 11.9 Saturn 60,300 568 9.0 35.5 1433 0.057 29.4 Uranus 25,600 86.8 8.7 21.3 2872 0.046 83.8 Neptune 24,800 102 11.0 23.5 4495 0.009 164 Pluto 1,200 0.012 0.6 1.1 5870 0.249 248 Sun 696,000 1,990,000 274 618 - - - April 10, 2006 Physics for Scientists&Engineers 1 6 Satellite Orbits View from above North Pole, June 23 2004 Low orbit satellites: Hubble, space station, Iridium system, spy sat.s Geostationary communication sat.s GPS, research sat.s April 10, 2006 Physics for Scientists&Engineers 1 7 Geostationary Satellite ! Where do you have to put a satellite so that it is always at the same location in the sky? ! Must have circular orbit with same period as rotation period of Earth and same rotation axis as Earth ! Rotation axis aligned with that of Earth: equatorial orbit ! What is the orbital radius? ! Solve Kepler’s 3rd Law for radius: ! Radius of orbit is 6.6 times radius of Earth ! Geostationary satellites are 35,790 km above sea level T 2 r 3 = 4 ! 2 GM " r = GMT 2 4 ! 2 # $ % & ( 1/3 r = (6.674 ! 10 " 11 m 3 kg -1 s " 2 )(5.9742 ! 10 24 kg)(86,164 s) 2 4 # 2 $ % & ( ) 1/3 = 42,168 km April 10, 2006 Physics for Scientists&Engineers 1 8 Quiz 41.1 ! A mass m is inside a uniform spherical shell at the position shown in the figure below. Which direction will the mass accelerate?
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