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

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

Orbital Motion Circular Orbits Keplers Laws Energy in Elliptical Orbits Homework


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SLIDE 1

Orbital Motion

  • Circular Orbits
  • Kepler’s Laws
  • Energy in Elliptical Orbits
  • Homework
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SLIDE 2

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

✆ ✝ ✞ ✟✡✠ ✁ ✂ ✆ ✝ ☛ ✆ ☞ ✁ ☎ ✁ ✂ ✄✍✌ ✎ ✄ ✠ ✆ ✁ ✂ ✏ ✌ ✄ ✎ ✄ ✏ ✌ ✠ ☞ ✁ ☎ ✄
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SLIDE 3

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
✗ ✠ ✘ ✙ ✚ ✠ ✛ ✒ ✁ ✂ ✏ ✌ ✆ ☞ ✁ ☎ ✁ ✂ ✄ ✠ ☞ ✁ ☎ ✁ ✂ ✒ ✄ ✆ ☞ ✁ ☎ ✁ ✂ ✄ ✠ ✆ ☞ ✁ ☎ ✁ ✂ ✒ ✄ ✗ ✠ ✆ ✘ ✠ ✛ ✒ ✚
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SLIDE 4

Example 1

A playful astronaut releases a bowling ball, of mass

= 7.20 kg, into circular orbit about Earth at an altitude

  • f 350 km. (a) What is the mechanical energy
  • f the ball in its orbit? (b) What is the

mechanical energy

✗ ✣
  • f 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?

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SLIDE 5

Kepler’s First Law

  • All planets move in elliptical orbits, with the Sun at one focus.
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SLIDE 6

Kepler’s Second Law

  • A line that connects a planet to the Sun sweeps out equal areas in the plane of the planet’s orbit

in equal times.

✆ ✝ ✤ ✠ ✆ ✝ ✄ ✥ ✆ ✝ ✞ ✟✦✠ ✆ ✝ ✄ ✥ ✞ ✟ ✧ ✄ ★ ✎ ✄ ✠ ✩ ✆ ✝ ✤ ✠ ✪ ✆ ✝ ✫ ✪✬ ✠ ✩ ✠ ✭ ✆ ✝ ✫ ✠ ✆ ✝ ✄ ✥ ✆ ✝ ✮ ✠ ✁ ✂ ✆ ✝ ✄ ✥ ✆ ✝ ✏ ✠ ✯✰✱ ✲ ✬ ☛ ✱ ✬ ✪ ✳ ✠ ✛ ✒✵✴ ✆ ✝ ✄ ✥ ✪ ✆ ✝ ✄ ✴ ✠ ✛ ✒ ✴ ✆ ✝ ✄ ✥ ✆ ✝ ✏ ✪✬ ✴ ✠ ✫ ✒ ✁ ✂ ✪ ✬ ✪ ✳ ✪ ✬ ✠ ✫ ✒ ✁ ✂ ✠ ✯✰ ✱ ✲ ✬ ☛ ✱ ✬
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SLIDE 7

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.

✑ ✌ ✠ ✶ ✷ ✸ ✔ ✓ ✌ ☞ ✁ ☎ ✹ ✺ ✻ ☛ ✕
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SLIDE 8

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

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SLIDE 9

Homework Set 11 - Due Mon. Feb. 9

  • Answer Questions 11.5 & 11.6
  • Do Problems 11.13, 11.18, 11.20 & 11.22