Deriving an expression for the radius of a satellite's orbit around Earth(?)

In summary, to derive an expression for the radius of a satellite's orbit around Earth, you need to set the centrifugal force equal to the force of gravity and rearrange the equations using the formulas for gravitational attraction and centrifugal force. This will give you the final equation: r = \sqrt[3]{\frac{T^{2}Gm_{E}}{4\prod^{2}}}, where T is the period of revolution, G is the universal gravitation constant, and m_E is Earth's mass. Remember to use the centripetal force formula with period in it when solving.
  • #1
AudioGeek
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Homework Statement


Derive an expression for the radius of a satellite's orbit around Earth in terms of the period of revolution, the universal gravitation constant, and Earth's mass.


Homework Equations


The final equation should be: r = [tex]\sqrt[3]{\frac{T^{2}Gm_{E}}{4\prod^{2}}}[/tex]


The Attempt at a Solution


I have no idea how to do this. To be honest, I'm not even really sure what it means to derive an equation. Someone please help me with this?
 
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  • #2
A satelite (or anything else) in orbit has a gravitational force inward toward Earth balanced by a centrifugal force outward.
You need the formulae for gravitational attraction (depends on the radius, mass of Earth and satellite) and the centrifugal force (depends on radius and mass of satellite)
Set the two equations equal to each other and rearrange them to get radius
 
Last edited:
  • #3
All circular satellite calculations begin with "centripetal force = force of gravity".
Fill in the formulas for the two forces and solve for the quantity you want.
Be sure to use the centripetal force formula with period in it since that is specified in the question.
 
  • #4
Delphi51 said:
All circular satellite calculations begin with "centripetal force = force of gravity".
Fill in the formulas for the two forces and solve for the quantity you want.
Be sure to use the centripetal force formula with period in it since that is specified in the question.

Once it was explained like this, it was dead easy!

Thanks a bunch =).
 

1. How is the radius of a satellite's orbit around Earth calculated?

The radius of a satellite's orbit around Earth can be calculated using the formula: r = (G * M * T^2 / 4π^2)^(1/3), where G is the gravitational constant, M is the mass of the Earth, and T is the orbital period of the satellite.

2. What factors affect the radius of a satellite's orbit around Earth?

The radius of a satellite's orbit around Earth is affected by the mass of the Earth, the mass of the satellite, and the orbital period of the satellite. Other factors that can affect the radius include atmospheric drag, solar radiation pressure, and other gravitational forces from celestial bodies.

3. How does the radius of a satellite's orbit around Earth change over time?

The radius of a satellite's orbit around Earth can change over time due to various factors such as atmospheric drag, solar radiation pressure, and gravitational forces from other celestial bodies. These changes can cause the orbit to either increase or decrease in size.

4. Can the radius of a satellite's orbit around Earth be adjusted?

Yes, the radius of a satellite's orbit around Earth can be adjusted by changing the satellite's velocity. This can be done by using thrusters or gravity assists from other celestial bodies. Scientists and engineers carefully plan and execute these maneuvers to achieve the desired orbit.

5. How does the radius of a satellite's orbit around Earth affect its speed?

The radius of a satellite's orbit around Earth is directly proportional to its speed. This means that the larger the radius, the slower the satellite's speed and vice versa. This relationship is governed by the law of conservation of angular momentum, which states that the product of an object's mass, velocity, and distance from the center of rotation remains constant.

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