This demonstration illustrates that one can stabilize an upside down pendulum by shaking it vertically.
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 we turn the pendulum upside down and shake
its pivot point vertically, the differential
equation
becomes
θ" - (1/L)(g+h''(t)) sinθ=0
If we shake the pivot point of the pendulum horizontally instead
of vertically the differential equation becomes
θ" - (g/L)sinθ+(w''(t)/L) cosθ=0
In this Applet, we choose the h(t) and w(t) to be
1+A cos(wt) with the amplitude, A, and frequency, w, of the shaking
both adjustable.
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.