Question

A simple harmonic oscillator oscillates with frequency f when its amplitude is
a. if the amplitude is now doubled to 2a, what is the new frequency

Answer 1

If the amplitude is doubled, the frequency is still f.

In fact, the frequency of a simple harmonic oscillator does not depend on the amplitude. The frequency of the oscillator in fact is:
$f= \frac{1}{2 \pi} \sqrt{ \frac{k}{m} }$
where
k is the spring constant
m is the mass attached to the spring

We can see that the frequency does not depend on the amplitude A, but only on the properties of the oscillator, so if the amplitude is doubled, the frequency does not change.