Orbital Motion Circular Orbits Keplers Laws Energy in Elliptical - PowerPoint PPT Presentation

� � � � Orbital Motion Circular Orbits Kepler’s Laws Energy in Elliptical Orbits Homework

✄ ✁ ☛ ✆ ☞ ✁ ☎ ✁ ✂ ✎ ✄ ✠ ✆ ✂ � ✏ ✌ ✄ ✎ ✄ ✏ ✌ ✠ ☞ ✁ ☎ ✝ ✆ ✂ ✁ ✂ ✁ ✁ ☎ ☎ ✁ ✂ � ✄ ✂ ✆ ✝ ✞ ✁ ✁ Circular Orbits Consider a mass in a circular orbit of radius about a mass The relationship between the gravitational force on due to and the acceleration of is given by Newton’s second law of motion as ✟✡✠ ✄✍✌

✚ ✂ ✒ ✂ ✁ ☎ ✁ ☞ ✠ ✄ ✁ ✆ ☎ ✁ ☞ ✆ ✌ ✏ ✂ ✁ ✒ ✄ ☞ ✠ ✂ ✒ ✛ ✠ ✘ ✆ ✠ ✗ ✄ ✒ ✁ ✁ ☎ ✁ ☞ ✆ ✠ ✄ ✂ ✁ ☎ ✛ ✚ � � ✙ ✌ ✄ ✌ ✓ ✔ ✏ ✑ ✠ ✄ ✓ ✒ ✠ ✏ ✑ ✂ ✁ ✌ ✑ ☞ ✌ ✘ ✠ ✗ � ☎ ✁ ☞ ✁ ✓ ✔ ✠ ✌ ✑ ✄ ☎ Circular Orbits (cont’d) The velocity of the mass can also be written in terms of the orbital period (the time to complete one orbit) as Substituting this into the previous expression for yields ✄✖✕ The total mechanical energy of the orbiting satellite is

✜ ✢ ✗ ✗ ✣ Example 1 A playful astronaut releases a bowling ball, of mass = 7.20 kg, into circular orbit about Earth at an altitude of 350 km. (a) What is the mechanical energy of the ball in its orbit? (b) What is the mechanical energy of the ball on the launch pad at Cape Canaveral? (c) What is the change in the ball’s mechanical energy between the launch pad and the orbit?

� Kepler’s First Law All planets move in elliptical orbits, with the Sun at one focus.

✬ ✄ ✝ ✆ ✴ ✒ ✛ ✠ ✴ ✄ ✝ ✆ ✪ ✥ ✝ ✥ ✆ ✛ ✠ ✳ ✪ ✬ ✱ ☛ ✬ ✲ ✯✰✱ ✠ ✄ ✆ ✝ ✬ ✱ ☛ ✬ ✲ ✱ ✯✰ ✠ ✂ ✁ ✒ ✫ ✠ ✪ ✝ ✳ ✪ ✬ ✪ ✂ ✁ ✒ ✫ ✠ ✴ ✪✬ ✏ ✏ ✆ � ✄ ✆ ✩ ✠ ✄ ✎ ★ ✄ ✧ ✟ ✞ ✥ ✝ ✥ ✆ ✞ ✝ ✆ ✥ ✄ ✝ ✆ ✠ ✤ ✝ ✆ ✝ ✤ ✠ ✪ ✄ ✝ ✆ ✂ ✁ ✠ ✮ ✝ ✆ ✥ ✄ ✝ ✆ ✠ ✆ ✝ ✫ ✪✬ ✠ ✩ ✭ ✆ ✝ ✫ ✠ in equal times. A line that connects a planet to the Sun sweeps out equal areas in the plane of the planet’s orbit ✒✵✴ Kepler’s Second Law ✟✦✠

✕ ✓ ☛ ✻ ✺ ✹ ☎ ✁ ☞ ✌ ✔ � ✸ ✷ ✶ ✠ ✌ ✑ Kepler’s Third Law The square of the orbital period of any planet is proportional to the cube of the semimajor axis of the orbit.

✆ ✜ ☛ ✜ ☛ ✁ ✒ ✗ ✠ � ☞ ✁ Energy in Elliptical Orbits The total mechanical energy for a mass in an elliptical orbit about a mass is where is the semimajor axis of the orbit

� � Homework Set 11 - Due Mon. Feb. 9 Answer Questions 11.5 & 11.6 Do Problems 11.13, 11.18, 11.20 & 11.22