## Satellite Orbits and Energy

12.4 Satellite Orbits and Energy

### Learning Objectives

By the end of this section, you will be able to:

- Describe the mechanism for circular orbits
- Find the orbital periods and speeds of satellites
- Determine whether objects are gravitationally bound

### Circular Orbits

**Practice!**

Practice 12.4.1 |
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A satellite moves in a circular orbit at a constant speed around the Earth. Which of the following statements is true? |

(a) No force acts on the satellite. |

(b) The satellite moves at constant speed and hence doesn’t accelerate. |

(c) The satellite has an acceleration directed away from the Earth. |

(d) The satellite has an acceleration directed toward the Earth. |

(e) Work is done on the satellite by the gravitational force. |

Practice 12.4.2 |
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A satellite is placed in a geosynchronous orbit. In this equatorial orbit with a period of 24 hours, the satellite hovers over one point on the equator. Which statement is true for a satellite in such an orbit? |

(a) There is no gravitational force on the satellite. |

(b) There is no acceleration toward the center of the Earth. |

(c) The satellite is in a state of free fall toward the Earth. |

(d) There is a tangential force that helps the satellite keep up with the rotation of the Earth. |

(e) The force toward the center of the Earth is balanced by a force away from the center of the Earth. |

Practice 12.4.3 |
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A satellite originally moves in a circular orbit of radius R around the Earth. Suppose it is moved into a circular orbit of radius 4R. What happens to the satellite’s speed? |

(a) eight times larger |

(b) four times larger |

(c) one-half as large |

(d) one-eighth as large |

(e) one-sixteenth as large |

Practice 12.4.4 |
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Two communications satellites are in orbit at the same height, but one weighs twice as much as the other. The speed of the heavier satellite is _____ the speed of the lighter one. |

(a) Less than |

(b) Equal to |

(c) Greater than |

(d) (need more information) |

Practice 12.4.5 |
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Two satellites A and B of the same mass are orbiting Earth. The distance of satellite B from Earth’s center is twice that of satellite A. What is the ratio of the speed of A to that of B? |

(a) 1/2 |

(b) 1 |

(c) 2 |

(d) |

(e) |

F_{net} = ma so , thus . The farther away you are, the slower you go. So if A is 1/2 the radius, v is times bigger.

**Discuss!**

How far away from the earth does an orbiting spacecraft have to be in order for the astronauts inside to be weightless?

### Orbital Energy

**Practice!**

Practice 12.4.6 |
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A planet in elliptical orbit around a star moves from the point in its orbit farthest from the star (A) to the closest point (P). The work done by the force of gravity during this movement is: |

(a) + |

(b) – |

(c) 0 |

Practice 12.4.7 |
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The planet executes one complete orbit starting from point A and returning to A. The work done by the force of gravity during this orbit is: |

(a) + |

(b) – |

(c) 0 |

Practice 12.4.8 |
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Which satellite has a larger kinetic energy, K? Assume orbits are circular and both satellites have the same mass. |

(a) K_{A} < K_{B} |

(b) K_{A} > K_{B} |

(c) K_{A} = K_{B} |

(d) Cannot be determined |

Practice 12.4.9 |
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Which satellite-earth system has a larger potential energy? Assume orbits are circular and both satellites have the same mass |

(a) Satellite-earth System A |

(b) Satellite-earth System B |

(c) Both have the same potential energy |

(d) Cannot be calculated |

Practice 12.4.10 |
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Which satellite-earth system has a larger total energy? Assume orbits are circular and both satellites have the same mass. |

(a) Satellite-earth System A |

(b) Satellite-earth System B |

(c) Both have the same total energy |

(d) Cannot be calculated |

Practice 12.4.11 |
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You put your spacecraft into a circular orbit around planet Z, which has half the radius of earth and has one-quarter earth’s mass. Compared to being in an orbit of the same size around earth, how are the kinetic energy K and gravitational potential energy U for your spacecraft different when in orbit around planet Z? |

(a) K is greater and U is less (more negative). |

(b) K is less and U is greater (less negative). |

(c) K is less and U is less (more negative). |

(d) K is greater and U is greater (less negative). |

(e) none of the above |

**Discuss!**

On July 15, 2004, NASA launched the Aura spacecraft to study the earth’s climate and atmosphere. This satellite was injected into an orbit 705 km above the earth’s surface.

(a) How many hours does it take this satellite to make one orbit?

(b) How fast (in km/s) is the Aura spacecraft moving?