Angular acceleration of a pendulum on an accelerating train

[Harmony]

If I hang a pendulum on the ceiling of an accelerating train (with an initial angular displacement), how would the angular acceleration be affected?

My intuition told me that the angular acceleration of the pendulum should be affected by the acceleration of the train. But...whether the train accelerate or not, the force acting on the tangential direction of the pendulum is still the mass of the pendulum right? So it seems tat the angular acceleration of the pendulum will be affected by length and gravitational acceleration alone, and not the acceleration of the train?

 

[Ibix]

Acceleration due to gravity is no different from acceleration for any other reason. Thus you can simply add the two acceleration vectors to get a total acceleration that replaces $g$ in the equations of motion.

If the acceleration of the train is constant then the result follows trivially. The pendulum oscillates about the direction of the total acceleration vector with a period dependent on the magnitude of the sum.

If the acceleration isn't constant you can write down the differential equations governing the motion as in my first paragraph, but they won't be analytically soluble in general. Although if you require that the change in acceleration be slow compared to the period of the pendulum then I expect it wouldn't be far off the obvious "oscillates around slowly changing 'vertical' with slowly changing period".

Note that I've assumed throughout that the pendulum is free to oscillate in the plane defined by the gravitational and train acceleration vectors. If it isn't, life gets more complicated.