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.