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Free Oscillations and Rotations of a Rigid Pendulum

(A Virtual Lab for Undergraduate Students)

The most familiar example of a nonlinear mechanical oscillatory system is an ordinary pendulum in the field of gravity, e.g., any rigid body that can swing and rotate about some fixed horizontal axis (a physical, or compound pendulum), or a massive small bob on a rigid rod or spoke of negligible mass (an ideal simple or mathematical pendulum). We exploit a rigid rod rather than a flexible string or wire in order to examine full revolutions of the pendulum as well as its swinging back and forth.

A physical pendulum is characterized by its mass m, the moment of inertia I about the axis of rotation, and the distance a between the axis and the center of mass. The differential equation of motion for the physical pendulum is the same as the differential equation for a simple pendulum, whose length L equals I/ma. Consequently, these two systems are dynamically equivalent – they have common variety of possible motions, and it is sufficient to study the simple pendulum. If we use the period of indefinitely small oscillations of the pendulum as a time unit in our mathematical model, then the single parameter remains that completely defines all properties of the system, namely, the dimensionless quality factor Q, that characterises the intensity of viscous friction.

A computer model of the pendulum is presented on this page. The Java-applet requires some time to load, so please be patient while it is starting. After the applet loaded, you can switch to off-line mode. The model allows us to observe the motion of the pendulum. You can vary the time scale by moving the slider (named "Delay" – down on the panel to the left side of the pendulum) for convenient observation. The model allows you also to display the graphs of time dependence of the angle and angular velocity (by checking the corresponding check-boxes on the same panel), as well as the graphs of energy transformations and phase diagram. You can resize and drag with the mouse the panels with graphs and the phase trajectory to any convenient place on the screen.

The panel on the right-hand side of the pendulum allows to vary parameters of the pendulum (absence or presence of viscous friction, and quality factor Q ), and also to change initial conditions of excitation (initial angle of deflection and initial angular velocity). You can change the values either by dragging the sliders, or by typing the desired values from the keyboard (editing the corresponding number fields). In the last case you should press "Enter" key after editing. The model will accept the new values after you press the button "Accept new values".

You can also start the simulation or make a pause at any moment by clicking here.

Principal goals of the lab:

  • To study free oscillations in a nonlinear physical system on the example of the pendulum – the most familiar nonlinear mechanical system.
  • To study experimentally the dependence of the period of natural oscillations on the amplitude and to compare the experimental results with an approximate theoretical formula for the period.
  • To get acquainted with the phase portrait of a nonlinear oscillator, with the phase plane, separatrix, and with special points of different types in the phase plane.
  • To study the energy transformations occurring during the oscillations with large amplitudes and revolutions of the pendulum.
  • To study the limiting motion of the pendulum in the absence of friction.
  • To measure the period of oscillations with amplitudes approaching 180 degrees, and the period of revolutions, and to compare experimental results with relevant theoretical calculations.
  • To study the influence of viscous friction on the phase portrait of the pendulum.
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