Troubleshooting Coordinates System from Cone

Thread starter Reg_S
Start date Sep 27, 2022
Tags

Cone Coordinates System Troubleshooting

Click For Summary

The discussion centers on issues related to using a coordinate system derived from a cone, with the original poster expressing frustration over not achieving the desired results. Participants emphasize the importance of detailing the specific problem and presenting any attempted solutions to facilitate better assistance. Suggestions include clarifying the intended application of the coordinate system and sharing any relevant calculations or visual aids. The thread highlights the collaborative nature of troubleshooting in technical contexts. Engaging with the community by providing more information is essential for effective problem-solving.

Sep 27, 2022

Reg_S

Homework Statement: A point particle of mass m slides without friction on inside the surface of a truncated cone. The cone is fixed , with half angle α, a bottom radius a, and a top radius b. You are to assume that the particle always remains in contact with the cone. Use ρ as a coordinate describing a location along the cone surface and θ to describe the azimuthal angle locating the particle.

Relevant Equations: a) Consider first circular orbits ( Constant ρ) contained within the truncated cone. What range of total energy is available?
b) Find the frequency of small oscillations about a particular contained circular orbit assumed to be ρ=ρ0 ? Express in terms of a, b, α and ω0.
c) Now consider a non-circular orbit (ρ is not constant) . Assume a particle is released the top rim with an initial velocity with components ρ(0)=0 and θ(0)= ω' >0. For what range of ω' is the motion entirely contained within the truncated cone.

I tried using coordinates system from cone, but not got what actually want to get. Any idea from you will greatly appreciated. Thanks

Physics news on Phys.org

Sep 27, 2022

erobz

Gold Member

Don't know how to solve it, but I know what those who can are going to say... Present your attempt at a solution.

Thread 'Help needed with Elliptic Integrals'

I want to find the solution to the integral ##\theta = \int_0^{\theta}\frac{du}{\sqrt{(c-u^2 +2u^3)}}## I can see that ##\frac{d^2u}{d\theta^2} = A +Bu+Cu^2## is a Weierstrass elliptic function, which can be generated from ##\Large(\normalsize\frac{du}{d\theta}\Large)\normalsize^2 = c-u^2 +2u^3## (A = 0, B=-1, C=3) So does this make my integral an elliptic integral? I haven't been able to find a table of integrals anywhere which contains an integral of this form so I'm a bit stuck. TerryW

View full post »