A 0.25 kg harmonic oscillator has a total mechanical energy of 4.1 j. if the oscillation amplitude is 20.0 cm, what is the oscillation frequency?

When the oscillator is at maximum extension, we know all of its energy is in Potential Energy, so if the total oscillation energy is 4.1 J, we know that at maximum displacement of 0.2 m, that

energy = 1/2 kA^2 where A = 0.2 m

k = 2E / A^2 = 2*4.1 J / 0.2^2=200 N/m

the frequency of oscillation is (1/2pi) sqrt[k/m]

knowing k and m, we can substitute values and find frequency