Gravitation and Circular Orbits

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SUMMARY

The energy required to launch a satellite of mass m into a circular orbit at altitude h is defined by the equation: \( \frac{GMm}{R} \cdot \frac{R+2h}{2(R+h)} \), where R represents the Earth's radius and M denotes the Earth's mass. The discussion emphasizes the importance of gravitational potential energy (PE) and kinetic energy (KE) in this context. Participants suggest using the work-energy principle, specifically \( W = -\Delta U_g \), to derive the necessary energy calculations. The conversation highlights the need to equate gravitational forces to understand orbital mechanics.

PREREQUISITES
  • Understanding of gravitational potential energy (PE) and kinetic energy (KE)
  • Familiarity with the concepts of circular motion and centripetal force
  • Knowledge of the gravitational constant (G) and its application in orbital mechanics
  • Basic algebraic manipulation skills for equation derivation
NEXT STEPS
  • Study the derivation of gravitational potential energy in the context of satellite motion
  • Learn about centripetal force and its role in maintaining circular orbits
  • Explore the implications of Earth's mass and radius on satellite launch energy requirements
  • Investigate the work-energy theorem and its applications in physics problems
USEFUL FOR

Students and professionals in physics, aerospace engineering, and anyone interested in understanding the principles of satellite dynamics and orbital mechanics.

kevi555
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Just a question about gravitation equations:

Neglecting Earth's rotation, show that the energy needed to launch a satellite of mass m into circular orbit at altitude h is equal to:

(\frac {GMm}{R})(\frac{R+2h}{2(R+h)})

Where R = the radius of the Earth and M = the mass of the Earth.

I've tried subbing in r=h+R and it hasn't given me much help.

Any help would be truly appreciated! Thanks all.
 
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Try expanding the equation and see what you get. (HINT: think W=-\Delta U_g)
 
"W" as in work? or what variable?
 
kevi555 said:
"W" as in work? or what variable?

Work.
 
kevi555 said:
Just a question about gravitation equations:

Neglecting Earth's rotation, show that the energy needed to launch a satellite of mass m into circular orbit at altitude h is equal to:

(\frac {GMm}{R})(\frac{R+2h}{2(R+h)})

Where R = the radius of the Earth and M = the mass of the Earth.

I've tried subbing in r=h+R and it hasn't given me much help.

Any help would be truly appreciated! Thanks all.

What is the force that makes a body go round in a circle? What is supplying that force here? You have to equate the two.
 
Some addition which I didn't bother to mention:

Energy reqd = (PE + KE) in orbit - PE on surface of earth. You have to use the previous concept anyway.
 

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