Explaining Central Force of inverse r^3

[I Am Robot]

Homework Statement

Consider the central force F=-k/(r^3) u(hat in r direction). This is an attractive force. For various values of k, explain what the orbit of a particle about a force center looks like.

Homework Equations

Included within question.

The Attempt at a Solution

I am not quite sure where to begin. I know this is like modelling the tidal force separate of gravitation, or I think so. Looking for a starting point if anyone can help. (This is for a classical dynamics course.)

 

[mark726]

Hello, thank you for your post! The central force F=-k/(r^3) u(hat in r direction) is a type of inverse-square force, which means that the strength of the force decreases with the square of the distance from the force center. In this case, the force is also attractive, meaning that it pulls objects towards the force center.

The value of k determines the strength of the force. If k is small, then the force will be weak and the orbit of a particle will be close to a circular shape around the force center. As k increases, the force becomes stronger and the orbit becomes more elliptical.

When k reaches a critical value, the orbit becomes a parabola, meaning that the particle will approach the force center once and then escape. This is known as a parabolic orbit.

If k continues to increase, the orbit becomes a hyperbola, meaning that the particle will approach the force center twice and then escape to infinity. This is known as a hyperbolic orbit.

In summary, for smaller values of k, the orbit will be more circular, and as k increases, the orbit becomes more elliptical, parabolic, and finally hyperbolic. This behavior can be seen in many real-world systems, such as the orbits of planets around the sun.

I hope this helps to answer your question and provide a starting point for your solution. If you have any further questions, please don't hesitate to ask. Good luck with your classical dynamics course!