This demonstration illustrates parametric resonance, which is a resonance phenomenon that arises because some parameter of the system is varying periodically in time. This happens, for example, when you periodically extend and retract your legs at an appropriate frequency while sitting on a playground swing. If a pendulum consists of a mass connected to a frictionless hinge by an idealized rod of length L, then the angle θ between the rod and vertical obeys the differential equation
θ" + (g/L) sinθ=0
If the length of the pendulum varies with time, the differential
equation
becomes
θ" + 2(L'(t)/L(t))θ'+(g/L(t)) sinθ=0
In this Applet, we choose the length of the pendulum to be
L(t)=1+A cos(wt) and we choose g=4π2. With this choice
of g, solutions of the A=0, linearized equation θ"+gθ=0 have period
1.
At time t, the length of the blue line in the Applet below is L(t). The angle between the blue line and vertical is θ(t). For comparison purposes, the gray pendulum provides the same data for the A=0, g=4π2, linearized equation
θ" + g θ =0
So, at time t, the length of the gray line is 1 and
the angle between the gray line and vertical is a constant (determined
by the initial speed that I have chosen) times sin(2πt).
The demonstration below is an applet. Google Chrome, Firefox, Safari and Microsoft Edge no longer execute applets natively because of security issues with NPAPI plugins. However applets can be played in the Google Chrome browser by using the CheerpJ Applet Runner extension.