HELP Motion in inverse cubic force field thx 1000000 in advance

[Odyssey]

HELP PLZ! Motion in inverse cubic force field...thx 1000000 in advance!

Say a particle expereinces a net force F = -Amr^-3, where A is some constant, m is the mass of the particle (point mass), and r is the distance. How should I go about in describing the possible orbits of the particle with non-zero angular momentum and E=0, E<0 and E>0 (ie. describing the shape of its orbit)? I know this would involve some integration and differential equation.

I know that r can be viewed as a function of theta, and the energy E and angular momentum can be written as

E = (1/2) mR'(θ(t))^2θ'(t)^2+(1/2)mR(θ(t))^2θ'(t)^2+V(r(θ(t)))

and

L = mR(θ(t))^2θ(t),

in polar coordinate form

How should I find V(r(θ(t)))? Is it V = integral of F?
How should my answer look like approximately? I have no clue in how my answer will be in terms of what variables.

Any help would be greatly appreciated. Thanks in advance.

 

[Nate810]

In order to find V(r(θ(t))), you need to integrate the force F = -Amr^-3. This will give you V(r) = Ar^-2. Since you know r as a function of θ, you can substitute this in to get V(r(θ(t))). To determine the shape of the orbit, you will need to solve the differential equation given by the equations for E and L. This can be done by first calculating the derivatives of E and L with respect to θ, and then equating them to 0. This will result in a second order differential equation that you can solve to get θ(t). Once you have θ(t), you can plug this into the equation for E or L to get r(t). Finally, you can plot r(t) vs θ(t) to get the shape of the orbit. Your answer should look like a graph of the orbit in polar coordinates.

 

[wais]

To describe the possible orbits of a particle experiencing a net force F = -Amr^-3 in a inverse cubic force field, you will need to use the equations for energy and angular momentum in polar coordinates. The energy equation will help you determine the shape of the orbit, while the angular momentum equation will give you information about the orientation of the orbit.

To find the potential energy V(r(θ(t))), you can use the equation V = -∫F(r)dr, where F(r) is the force as a function of distance r. This will give you an expression for the potential energy in terms of r.

Your final answer will depend on the specific values of A, m, and the initial conditions of the particle (such as its position and velocity). It will likely involve trigonometric functions and constants. It is important to keep track of the variables and their units in your answer.

If you are unsure about how to proceed with the integration and solving the differential equations, it may be helpful to consult a physics textbook or seek assistance from a tutor or professor. Good luck with your problem!