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โš™๏ธย ap physics c: mech

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๐Ÿชย Unit 7

ย ย โ€ขย ย โฑ๏ธ2 min read

7.2 Orbits of Planets and Satellites

Daniella Garcia-Loos

daniella garcia-loos

โฑ๏ธ November 7, 2020

๐Ÿ“…

7.2 Orbits of Planets and Satellites

One of the most common applications of gravitation that AP enjoys is the orbits of planets and satellites!

Fast Facts of Planets and Satellites in Orbit:

  • Angular momentum is conserved

  • Mechanical energy is conserved

  • GPE is at 0 at an infinite distance away

  • โ€œEscape speedโ€ is the minimum speed required to escape the gravitational field

Here is the derivation for Gravitational Potential Energy:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-nfDDtoomhUS0.png?alt=media&token=e6de12f3-4bdc-470b-bfcc-8172b413fe0e

This also means that the kinetic energy of a satellite in orbit can be simplified down to only be dependent on the satellite and its distance.

We can connect centripetal/circular motion to gravitational force too!

The centripetal force acting on a satellite/planet system is caused by the gravitational force.

This means that we can derive some formulas to describe the orbits of these systems that move in a circular path.

Let's try to find tangential velocity:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-vuNfuzb8WRuI.png?alt=media&token=27333bab-196c-4bbe-8e1f-0c959216562c

Additionally, I'll leave the proof of this one as an exercise for you, but we can use a similar method to determine the period of the orbit.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-vb7SO1FCyhGs.png?alt=media&token=e1947e32-0adf-4ef7-b744-541fd9f6784f

There is another version of this formula which is known as Kepler's Third Law, which is only applicable for objects in circular orbit:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-AjQVEEpLTcKs.png?alt=media&token=59583a38-8af4-4b05-9f93-6debdd053d92

This is commonly rewritten as below in order to simplify the relationship:

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-yCQpU90WAnEk.png?alt=media&token=513d8ecb-7d07-4332-af4d-ca01cc4d429c

Lastly, in non-orbiting situations in relation to satellites, you can use the conservation of energy in order to help you describe its motion.

Unit 7.2 Practice Problems

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-uvzfoFOI0Bym.png?alt=media&token=f241832e-b27a-490c-abb9-36b14cb2c34a

Taken from College Board

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-vAYaaMdDTm0h.png?alt=media&token=9e092df5-2443-4240-9590-5223af5405c0

Answer

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-yzPyyW3H32zo.png?alt=media&token=20a667d0-58de-43e7-bf16-67134f353559

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-XW2S8ZY8w5UO.png?alt=media&token=b6f065fa-3590-4282-9871-107300100399

Answer

Most Physics C: Mechanics classes tend to skip over Kepler's Laws, so note that it is perfectly acceptable to justify your answer with other physics you know.

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-jNMkJcQWJxXu.png?alt=media&token=95cc2cb2-6348-4aba-a83e-888ac01a0ace

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-ic2kPoOUtfyy.png?alt=media&token=6b6a2000-9236-478f-84bb-45353b4ee307

Answer

https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-b0oD0z5lvxK7.png?alt=media&token=3b28b920-42af-41b1-8500-b0c41a2ae032

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