Pendulum on an accelerating train with changing length

Question

Based on my own research, I found a general solution that can model a pendulum found on an accelerating train. The following solutions are based on small angle approximations.
F=-mgsin θ
F≈-mgθ, applying small angle approximation
F=-(mg/L) s, by applying arc length formula
This is in the form, F = -kx(hooke's law)
Therefore k = mg/L
Applying newton's laws
F = ma = -kx
Solving this eqn, ma = -(mg/L)x
we get,

This general equation can help plot the sway angle of the pendulum. Another known fact is that the length of a pendulum affects its period.
I am wondering if the equation will hold if the length of the pendulum is changing. i.e Becoming shorter progressively.
I expect the period to get shorter as the length shortens. How will its amplitude be affected?

Appo · Answer

Your initial equation of motion is of the form:
$$m mathbf{a} = mathbf{F}(L, g, a). $$
However, when $L$ varies in time, $F$ also becomes time dependent. Thus, your initial differential equation differs from your new one in dynamics. Therefore, you should solve the problem again.
By the way, your solution seems not quiet working. Namely, using equivalence principle, a pendulum in accelerating train plus gravity is equivalent to a pendulum in a gravitational field that is the vector sum of $-mathbf{a}$ and $mathbf{g}$. Hence, the frequency of the pendulum should be dependent at least to $g$ and $a$. Moreover, the axis of oscillation should be change by something like $tan^{-1}(frac{-g}{a})$.

Pendulum on an accelerating train with changing length

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