The motion of a conical pendulum in a rotating frame: The study of the paths, determination of oscillation periods, and the Bravais pendulum

In this paper, Newton's equations are solved to describe the motion of a conical pendulum in a rotating frame for the right- and left-hand conical oscillations. If the pendulum is started with the initial angular velocity (⁠ $± ω 0− Ω z)$⁠, $with ω 0$ being the frequency of pendulum oscillation and $Ω z$ the angular velocity of the rotating frame around the z-axis, the paths are shown to be circular, which would apparently indicate that the Coriolis force becomes non-effective in the rotating frame. This paper explains why the paths in the rotating frame are circular, and the difference between the periods for the right- and left-handed rotations is calculated analytically. It is also shown that in an inertial frame, the conical pendulum follows an elliptical path. As the Bravais pendulum is a conical pendulum oscillating at Earth's surface, its motion is also discussed.