Pendulum on moving pivot
d
∗ ∗ A pendulum is at rest with its bob pointing toward the center of the Earth. The support of the pendulum is moved horizontally with uniform acceleration a, and the pendulum starts to swing. Neglect rotation of the Earth.
Consider the motion of the pendulum as the pivot moves over a small distance d subtending an angle θ0≈ d/R_e at the center of the Earth. Show that if the period of the pendulum is 2πRe/g, the pendulum will continue to point toward the center of the Earth, if effects of order θ02 and higher are neglected. (One way to make such a long-period pendulum would be to place the pivot point of a long rod close to its center of mass.)