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Let the pseudoforce be $f$. Then the equation of motion in the non inertial frame is
$$m \ddot{x} = -kx + f = -k\left(x - \frac{f}{k}\right)$$
Change variables to $X =x - \frac{f}{k}$, then the equation of motion is
$$m\ddot{X} = -kX$$
which is the equation for shm with solution
$$X = A \cos (\omega t + \phi)$$
where $\omega = \sqrt{k/m}$ and $A$ is the ...
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The first question is: Does your pendulum have a string or a rod? If you were to, say, suddenly introduce 20 meters of string without moving the mass at the end, it will fall straight down, and then the behaviour can be all sorts of things depending on how elastic the string is.
If it is a rod (and so the mass moves with the added length), then it depends on ...
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