Discussion
The equation of motion for a spring-mass system excited by a harmonic force is
\[ M\ddot{x} + Kx = F\cos(\omega t), \]
where \(M\) is the mass, \(K\) is the spring stiffness, \(F\) is the force amplitude and \(\omega\) is the angular frequency of excitation. Resonance occurs when \(\omega\) is equal to
Answer: D
Resonance occurs at \(\omega = \sqrt{K/M}\).
Resonance is excitation at the natural frequency of the undamped system. The free equation \(M\ddot{x} + Kx = 0\) gives \(\omega_{n} = \sqrt{K/M}\) in rad/s, and the forced amplitude \(F/(K - M\omega^{2})\) grows without limit as \(\omega\) approaches it.
\(\frac{1}{2\pi}\sqrt{K/M}\) is the same natural frequency in hertz, but \(\omega\) is an angular frequency, so the units do not match. \(2\pi\sqrt{K/M}\) multiplies where it should not, and \(\sqrt{M/K}\) is inverted (it has units of time).
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