Planetary Motion with satellite

In summary, the problem involves finding the radius of a planet with unknown mass, given the mass and orbit parameters of a satellite in circular orbit around it. The solution involves using Kepler's second law and Newton's law of universal gravitation to approximate the mass and radius of the planet. The acceleration of the satellite at different points in its orbit is also used to find the radius of the planet.
  • #1
teme92
185
2

Homework Statement



A 20 kg satellite has a circular orbit with a period 2.4 h and radius 8.0×106m around a planet of unknown mass. If the magnitude of the gravitational
acceleration on the surface of the planet is 8.0 m/s2, what is the radius of the
planet?

Homework Equations



F = Gm1m2/r2
Fc = m1v2/r
v=D/T, D=[itex]\pi[/itex]r

The Attempt at a Solution



So I got the velocity of the the satellite by using speed=distance/time where the distance is the circumference of the orbit or [itex]\pi[/itex]r and T is the period in seconds. I'm having trouble because there is an unknown mass of the planet and its radius. How do I go about finding this values? Any help is much appreciated.
 
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  • #2
This is from the 2011 paper, it's actually a lot easier than it looks.

T = period ## T = (2.4)(3600) = 8640 s ##

Okay so I did this question using Kepler's second law (law of periods). It's a very useful law and is worth learning.

<content deleted -- complete worked solutions are not permitted> -- gneill, PF Mentor
 
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  • #3
I thought the radius value was RTotal = RSatellite + RPlanet.

Also using your method I got r = 5.3x106. Did you use g=9.8 or is there some other difference between or values? Also thanks for the help.
 
  • #4
It's not the acceleration of the Earth its the acceleration of the planet ## a_g = 8.0 m/s^2 ## So let's break this down, we use Kepler's second law, using a little bit of algebra this relation provides is with an approximation of the mass of the planet. Having the mass of the planet and the acceleration of the planet we have the radius of the planet.

## ∑F = ma_g = \frac{GMm}{r^2} ## which implies ## a_g = \frac{GM}{r^2} ##
 
  • #5
What is the radial acceleration of the satellite at 8x106 m? You know the radial acceleration at this location, you know radial acceleration at the surface, and you know that the radial acceleration is inversely proportional to the square of the radius.

Chet
 
  • #6
patrickmoloney said:
This is from the 2011 paper, it's actually a lot easier than it looks.

T = period ## T = (2.4)(3600) = 8640 s ##

Okay so I did this question using Kepler's second law (law of periods). It's a very useful law and is worth learning.

<worked solution content deleted -- gneill, PF Mentor>

You're not supposed to solve the problem. Just explain how the solution works.
 
Last edited by a moderator:
  • #7
Ok thanks for clearing that up and all the help guys
 

1. What is the purpose of using satellites for planetary motion?

Satellites are used to study and observe the motion of planets in our solar system. They provide valuable data and images that help scientists understand the dynamics of planetary motion, including gravitational forces and orbital paths.

2. How do satellites track the motion of planets?

Satellites use advanced technology, such as cameras and sensors, to continuously monitor the position and movement of planets. They also use mathematical calculations to predict future positions based on their current trajectory.

3. How do satellites help us understand the laws of planetary motion?

By tracking the motion of planets, satellites provide evidence for the laws of planetary motion proposed by scientists such as Johannes Kepler and Isaac Newton. These laws help us understand the relationship between gravitational forces and the movements of planets in our solar system.

4. Can satellites be used to study the motion of other objects in space?

Yes, satellites can also track the motion of other celestial objects such as comets and asteroids. They can also observe the effects of gravitational forces on these objects, providing valuable insights into the dynamics of our solar system.

5. How do satellites contribute to our understanding of the universe?

Satellites play a crucial role in studying the motion of planets and other celestial objects, providing valuable data and images that help expand our knowledge of the universe. They also help us make predictions and discoveries about the nature of our solar system and beyond.

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