Balancing act 2

We have a pendulum made of a rigid, massless rod of length \(l\) and a small ball of mass \(m\) attached to the end of the rod. The period of small amplitude oscillations is \(T_0= 2 \pi \sqrt{\frac lg}\).

The pivot point (axis of rotation) of the pendulum is not fixed, but it is attached to a mechanism that shakes the pivot point in the vertical directions with amplitude \(a\ll l\) and period \(T\ll T_0\). (For \(a=0\) we get back the traditional pendulum.)

We set up the rod such that it is in the vertical, unstable position with the ball right above the pivot point. When \(a=0,\) this position is unstable; for any small error in positioning the ball, the pendulum will turn over.

Is it possible that if we switch on the shaking \(a\ne 0,\) the position becomes stable and the ball will remain over the pivot point indefinitely?

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