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=0If we turn the pendulum upside down and shake its pivot point vertically, the differential equation becomes
θ" - (1/L)(g+h''(t)) sinθ=0If we shake the pivot point of the pendulum horizontally instead of vertically the differential equation becomes
θ" - (g/L)sinθ+(w''(t)/L) cosθ=0In 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 and Microsoft Edge no longer execute applets because of security issues with NPAPI plugins. Some browsers that still play applets are Internet Explorer, Safari and the Pale Moon browser [ windows, linux, Mac ].