Let the angle between the pendulum and the vertical line is and the angular velocity

And the angle of the pendulum (projected to x-y plane) with the x-axis is , and it's angular velocity

The [url=http://en.wikipedia.org/wiki/Lagrangian_mechanics]lagrange equation[/url] for the system is

The equation of the motion is

...... from

and

Angular momentum is conserved. ...... from

And the following is the simulation of such a system:

When the checkbox ([b]circular loop[/b]) is checked,

The vertical component tangential of the string balanced with the mass m, and the horizontal component tangential provide the centripetal force for circular motion.

You can uncheck it and change the period ,

and you will find out the z-coordinate of the pendulum will change with time when

or

You can also drag the blue dot to change the length of the pendulum.

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