Pendulum on a Rotating Door — Forces, Equilibria, and Small-Angle Oscillations
A point mass hangs from a string on a door that rotates steadily. The motion is described in a spherical frame attached to the mass. String tension is purely radial and balances inertia and weight components
Coriolis force appears tangent to the path in the rotating frame. Centrifugal force has radial and tangential parts along the sphere directions. The angular equation of motion couples gravity and door rotation. Equilibria exist at the downward angle and at a tilted angle when rotation is high. There is a critical angular speed that switches which equilibrium is stable. For small oscillations around the downward position the motion is harmonic. The phase of the cosine solution follows from initial angle and angular speed.
