Motion of a Conical Pendulum

For the conical pendulum, gravity, mass, and the angular velocity determine the angle between the string and the vertical axis of the pendulum.
This angle is given by , where is the acceleration of gravity, is the angular velocity, and is the length of the string that suspends the mass . For simplicity, we set .
You can vary the time, angular velocity , gravity , and mass .

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The upward force produced by the tension balances the downward force from the weight of the pendulum:
. (1)
Also, because the pendulum system is conical, the mass moves in a circular orbit.
In a circular orbit, the pendulum experiences an inward centripetal force given . The horizontal component of tension balances this centripetal force:
. (2)
Dividing (2) by (1):
,
,
,
,
.
This means that the angle that the string makes with the vertical depends on the gravity , the length of the string , and the angular velocity, .
Snapshot 1: as the angular velocity increases, the angle between the string and the vertical increases
Snapshot 2: as gravity increases, the angle between the string and the vertical decreases
Snapshot 3: increasing the mass does not have any effect on the angle
Special thanks to the University of Illinois NetMath Program and the mathematics department at William Fremd High School.
This Demonstration was written with materials from Making Math [1].
References
[1] O'Reilly Media. "Making Math." (Jun 2, 2014) www.makingmath.com.
[2] R. Fitzpatrick. "The Conical Pendulum" in Classical Mechanics: An Introductory Course. (Jun 12, 2014) farside.ph.utexas.edu/teaching/301/lectures/node88.html.
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