Hi, I am a physics teacher and am working on a problem for a general solution to the problem of the swing carousel ride. Most physics students learn the to solve the problem of finding the angle of the swings given the tangential speed , or vice versa. This is easy.

A harder task is to find the angle as a function of angular velocity, and radius of the platform from which hang the swings. Looking only for the steady state solution, I decided I would have to work at it from the non-inertial rotation frame of reference, and use the pseudoforce centrifugal force. It make it easier to get an equation for the angle that only depends on the constant parameters of angular speed , platform radius , chain length , and gravity g.

the problem is I don't know how to solve for the angle. I could get a numerical solution but that is not what I want.

The equation boils down to

Tan x = A + B sin x

where A and B are constants that depend on the parameters. What should I do to solve for x?

Thanks for any help!

Eric