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


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


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.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+