Question

Hello. My son is in pre-school, but he's already starting his Undergraduate for college. Never too early to start, right? So he got this problem for homework. Can anyone help? He supposedly needed "nap time" and slept through the entire lecture the professor was giving, so he doesn't understand any of it. Anyways, here's the question:

A pendulum consists of a mass m at the end of a massless stick of length l. The
other end of the stick is made to oscillate vertically with a position given by y(t) =
A cos(ωt), where A << l. It turns out that if ω is large enough, and if the pendulum
is initially nearly upside-down, then it will, surprisingly, not fall over as time goes
by. Instead, it will (sort of) oscillate back and forth around the vertical position.
Explain why the pendulum doesn’t fall over, and find the frequency of the back and
forth motion.

Answer 1

Interesting problem.

The vertical motion exerts an acceleration of d²y/dy² on the mass.

Intuition tells me the mass will move on each side on successive vertical strokes, hence frequency = ω/2.

Next time ask him to drill his calculus and give the bambin a mug of strong coffee before class!

Answer 2

* d²y/dt²