Central force as a conic: How to show it?

[kthouz]

If F denotes the central force and is given as the inverse of the square distance
F=-k/r^2. where k>0. How can i show that F the path of the partilce under that force is a conic?
What i can see is that it is the second law of Kepler. And when i try to compare F with the former equation of conics r= (pe)/(1+ecosB). I can find p!
Just i need your help

 

[Shooting Star]

https://www.physicsforums.com/showthread.php?t=205469" is a thread which does exactly that. You have to follow the steps patiently.

 

[charng]

I would suggest approaching this problem by first understanding the properties of a conic section. A conic section is a curve that is formed by intersecting a plane with a double-napped cone. It can take the shape of a circle, ellipse, parabola, or hyperbola depending on the angle of intersection and the distance between the plane and the vertex of the cone.

In order to show that the path of a particle under the central force F=-k/r^2 is a conic, we need to show that the equation of the path satisfies the properties of a conic. One way to do this is by using the polar coordinates system, where r represents the distance from the origin and θ represents the angle from a fixed reference line. In this system, the central force can be written as F=-k/r^2.

Using the second law of Kepler, which states that the line joining a planet and the sun sweeps out equal areas in equal times, we can establish that the particle under the influence of the central force will move in a path that is symmetric about the focus (the center of the force). This means that the particle will always be at a fixed distance from the focus, which is a characteristic of a conic section.

Next, we can compare the central force equation with the general equation of a conic section in polar coordinates, r= (pe)/(1+ecosθ), where p is the distance from the focus to the directrix and e is the eccentricity. By equating the two equations, we can see that p=k and e=1. This means that the path of the particle is a conic section with a focus at the center of the force and a directrix at a distance of k from the focus.

In conclusion, the central force F=-k/r^2 can be shown to produce a conic section as the path of the particle under its influence. This is supported by the second law of Kepler and the comparison of the central force equation with the general equation of a conic section. Further analysis and calculations could also be done to show the specific shape of the conic (circle, ellipse, parabola, or hyperbola) depending on the value of k.