Physics 400 Practice Orbits Mr. Youker | Planet Venus Earth Mars (Sun) Mass (kg) 4.87 x1024 5.97 x1024 6.42 x1023 1.99 x1030 Radius (m) 6.05 x106 6.38 x106 3.40 x106 6.96 x108 Orbital Radius (m) 1.08 x1011 1.50 x1011 2.28 x1011 --- 1. What is the size of the gravitational force of attraction between the Earth and the Sun? (Hint: r = the orbital radius of the Earth.) Fg = GMm/r2 Fg = (6.67 x10-11)(1.99 x1030)(5.97 x1024) (1.50 x1011)2 Fg = 7.924 x1044 2.25 x1022 Fg = 3.52 x1022 N Why do we ignore the actual radius of the Sun and Earth in this calculation? The additional separation produced by adding the radii of the Earth and Sun only contributes 0.5% to the distance. 2. What is the size of the gravitational force of attraction between the Earth and a 1000 kg satellite with an altitude of 2000 km (2.00 x106 m above the Earth’s surface)? Fg = GMm/r2 Fg = (6.67 x10-11)(5.97 x1024)(1000) (6.38 x106 + 2.00 x106)2 Fg = 3.98 x1017 7.02 x1013 Fg = 5668 N Why do we not ignore the actual radius of the Earth in this calculation? The altitude of the satellite is 24% of the total separation. 3. What would be the weight of a 600 kg rock on the surface of Mars? Fg = GMm/r2 Fg = (6.67 x10-11)(6.42 x1023)(600) (3.40 x106)2 Fg = 2.57 x1016 1.156 x1013 Fg = 2,223 N 4. If you wished to fly your spaceship to Mars and place it in orbit at an altitude of 2000 km above the Martian surface, then what tangential velocity must you give to your spaceship? vorbit2 = GM/r vorbit2 = (6.67 x10-11)(6.42 x1023) (2.0 x106 + 3.40 x106) 2 vorbit = 7.93 x106 vorbit = 2816 m/s 5. How frequently (in Earth years) do the planets Venus and Mars align with one another during their orbits around the Sun? Venus Torbit2 = 42r3/GM Torbit2 = (42)(1.08 x1011)3 (6.67 x10-11)(1.99 x1030) 2 Torbit = 3.75 x1014 Torbit = 1.94 x107 s Mars Torbit2 = 42r3/GM Torbit2 = (42)(2.28 x1011)3 (6.67 x10-11)(1.99 x1030) 2 Torbit = 3.53 x1015 Torbit = 5.94 x107 s ∆t = TmarsT venus/(T mars – Tvenus) ∆t = (5.94 x107)( 1.94 x107) (5.94 x107 – 1.94 x107) ∆t = 2.99 x107 s ∆t = 8002.5 hr ∆t = 333 days 6. A geosynchronous satellite is one that possesses an orbital period equivalent to the rotational period of the planet around which it orbits. At what altitude must an Earth satellite be placed in order to be geosynchronous? Torbit2 = 42r3/GM 86,4002 = (42)(r)3 (6.67 x10-11)(5.97 x1024) 3 r = 7.53 x1022 rorbit = 4.22x107 m -6.38 x106 m alt.orbit = 3.58 x107 m alt.orbit = 35,800 km 7. In an attempt to determine the mass of Venus, NASA officials send a probe into orbit around the planet. This probe has a mass of 5000 kg and orbits in a 14,000 km diameter circular orbit. The satellite completes one revolution in two hours. What is the mass of Venus that can be determined from this data? Torbit2 = 42r3/GM 72002 = (42)(7 x106)3 (6.67 x10-11)(M) M = 3.92 x1024 kg