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Suppose we have a "conical pendulum" consisting of a bob $P$ of mass $m$ attached to a rigid chord of length $l$ which is attached at its other end to a fixed point $O$. We displace the bob so that it forms an angle $\theta_0$ with the vertical axis and give it some initial velocity $\vec{v_0}$.

What are the possible motions of the pendulum?

We can first consider the easy case where $\vec{v_0}$ is horizontal.

Note: "conical pendulum" just means a pendulum which is allows to move in 3 dimmensions as opposed to the classic planar pendulum motion

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    $\begingroup$ ? conical $\hspace{10mm}$ $\endgroup$
    – daniel
    May 7, 2017 at 15:58

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A superposition of circular and oscillatory. Take the components of the initial velocity in the radial as well as circumnavigating directions. The radial velocity will give rise to oscillatory and the circumnavigating to circular.

Conserve the angular momentum and the total energy.

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  • $\begingroup$ Angular momentum is conserved. Is there an easy way to see this, because I just proved it by expressing the acceleration in cylindrical coordinates and noting that there is no component of force in the $\vec{e_\theta}$ direction. Also could you please elaborate on what you said, at least flesch out the main ideas a bit because I don't understand why we would get oscillator equations. I found two coupled equations (for $z$ and for $r$), they both depend on the tension in the chord. $\endgroup$
    – math_lover
    May 7, 2017 at 16:34
  • $\begingroup$ Only the z-component of angular momentum is conserved IMO because the force along x and y are zero hence torque's z component is zero. $\endgroup$
    – Abhijeet Melkani
    May 7, 2017 at 16:56
  • $\begingroup$ Sorry I don't have a pen and paper at this moment or I could have worked it out more clearly. $\endgroup$
    – Abhijeet Melkani
    May 7, 2017 at 16:56
  • $\begingroup$ Hint try spherical coordinates. I am confident you won't get a coupled system then. $\endgroup$
    – Abhijeet Melkani
    May 7, 2017 at 16:58

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