What is g in the rod pendulum

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Harmonic approximation using the example of the rod pendulum

By a rod pendulum we want to understand a transducer that consists of a point-shaped mass piece that is attached to a rod of length that is assumed to be massless. In the following, we denote the deflection angle from the vertical with.

Rod pendulum for small deflections

Work order
  • Start the rod pendulum with different starting conditions!
  • Can one still speak of a harmonic oscillation with this movement?
  • Try to set up the equation of motion for the deflection angle for a rod pendulum with mass m!

solution

The differential equation for a rod pendulum of length is:

This cannot be solved easily. The approximation (Taylor expansion) applies to small angles. This simplifies the differential equation to:

Now this differential equation can be solved analogously to the differential equation for the spring pendulum:

Rod pendulum for large deflections

The aim now is to find out when this harmonic approximation is no longer suitable.

Work order

Investigate the following questions!

  • How does the period of oscillation depend on the deflection?
  • On the basis of the project, roughly determine the deflection (specified in) from which the period of oscillation deviates by more than the period of oscillation of the rod pendulum that can still be described as harmonic (at).

solution