Possible Orbits in V = k r^4 Potential

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In summary, the V = k r^4 potential is a potential energy function used in classical mechanics to describe the force between two objects with a distance r between them. It is derived from the inverse square law of gravitational force and can be used to describe real-life systems, such as planetary orbits. Possible orbits in this potential include elliptical, hyperbolic, and parabolic orbits. While it has advantages in its simplicity and versatility, it is limited in its accuracy due to not accounting for other factors.
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yxgao
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Assume energy greater than 0.
Consider cases in which k is either + or - and l is either 0 or +.


Using your tuition, what are the possible orbits?


k<0:
l=0: falls to the center
l=+: hyperbola or it falls to the center

k>0:
l=0: hyperbola
l=+: hyperbola or it falls to the center

what about circular orbits?
 
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Is this question more difficult than it sounds? There are no responses from anyone! :)
 
  • #3


In this potential, circular orbits are not possible since the potential is not symmetric. The only possible stable orbits are hyperbolas or falling to the center.
 

FAQ: Possible Orbits in V = k r^4 Potential

What is the V = k r^4 potential?

The V = k r^4 potential is a potential energy function used in classical mechanics to describe the force between two objects with a distance r between them. It is also known as the quartic potential due to the exponent of r being 4.

How is the V = k r^4 potential derived?

The V = k r^4 potential is derived from the inverse square law of the gravitational force between two objects. By assuming a spherical symmetry and using the shell theorem, the potential energy function can be simplified to V = k r^4, where k is a constant.

What are possible orbits in the V = k r^4 potential?

Possible orbits in the V = k r^4 potential include elliptical, hyperbolic, and parabolic orbits. These orbits depend on the initial conditions and the total energy of the system.

Can the V = k r^4 potential be used to describe real-life systems?

Yes, the V = k r^4 potential can be used to describe real-life systems, such as the orbits of planets around the sun. It is an approximation of the actual gravitational potential, but it is useful for understanding the behavior of objects in space.

What are the advantages and limitations of using the V = k r^4 potential?

The advantages of using the V = k r^4 potential include its simplicity and its ability to describe a wide range of orbits. However, it is only an approximation and does not take into account other factors such as the effects of relativity and the presence of other objects in the system.

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