Motion of a Pendulum in the Wind

This Demonstration models the behavior of a simple pendulum subject to an additional force from air flow.
The pendulum has a spherical bob attached to a rigid rod and is suspended from a frictionless pivot. The direction of the air flow is assumed to be parallel to the plane in which the pendulum is swinging. The air flow causes a "drag force" on the pendulum bob. This force will alternatively be in the same or in the opposite direction of the moving pendulum and this way will either accelerate or decelerate the bob.
The pendulum starts from a horizontal position opposite the origin of the flow. The angular displacement of the pendulum is shifted away from the vertical depending on the air flow speed and the diameter and density of the bob. The pendulum eventually finds an equilibrium at an angle away from the vertical.
An animation of the pendulum is shown together with its phase-space diagram.


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


It is assumed that for air speeds less than 30 m/sec, the flow is nonturbulent and the drag force on the bob can be expressed as follows: , where
is the dimensionless drag coefficient; it depends on the geometry of the bob (0.47 is the accepted value for a sphere),
is the density of air (1.293 ),
is the cross-sectional area exposed to the air flow , and
is the air flow speed (m/sec).
Then the equilibrium of the forces acting on the bob becomes:
, with and .
Reducing this to the one degree of freedom of the system for , this gives the following equation of motion: .


    • 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+