9458

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.

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DETAILS

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: .

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