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C. Spirit
I would like you to derive it yourself. It is very simple; just use the fact that this system conserves the total energy and that, by definition, ω2 = k / m. As for the second question, you are confusing the angular frequency of the period of the motion with the angular velocity of the mass. It is unfortunate that both have the same symbol for the particular case you are working with but use $\dot{\theta }$ instead for the angular velocity.