Motion of Pendulum with Varying Length

We consider a pendulum consisting of a bob hanging on a string attached to the top of a stationary cylindrical drum. When the pendulum is released from a position away from the vertical, it swings with a varying length, tracing a spiral-shaped path.
If the bob can swing to its maximum angle, it will reach its initial height (assuming there is no friction). This is a consequence of the conservation of energy.
  • Contributed by: Erik Mahieu
  • With additional contributions by: Franz Brandhuber

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Lagrangian mechanics can be used to solve this problem.
Let be the radius of the drum, the length of the string, and the angle between the free part of the string and the vertical.
The potential energy is ,
the kinetic energy is ,
the Lagrangian is ,
and the resulting equation of motion is .
Since the string cannot support compression forces, the maximum initial angle has to satisfy the equation , keeping the bob under the lowest point of the drum.
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