10. A harmonic oscillator has a spring constant k and a mass m. It has a natural

Question

10. A harmonic oscillator has a spring constant k and a mass m. It has a natural oscillation frequency w. The oscillator is hooked up to an external force that is periodic with frequency wf. The external force is adjusted so that wf = and it is in resonance with the oscillator. The frequency of the external force is then changed to wf 3w. It is attached to a set of harmonic oscillators with different values of k and m. With which of the following oscillators will the external force be in resonance? (a) (3k, m) (b) (32k, 4m) (c) (16k, 4m) (d) (9k, 3m) (e) (27k, 3m)

Explanation / Answer

angular frequency W = 2pi f = sqrt(k/m)

sqraing on both sides

W^2 = 4 pi^2 f^2 = k/m

i.e w^2 = f^2 = cosntant * K/m

when Wf = W,   f^2 = K/m

but

when Wf = 3 W

(3f)^2 = 3 K/m

9 f^2 = 3 K/m

3 f^2 = k/m

now if   k = 9K and m = 3 m

then

3 f^2 = 9k/3m = 3 k/m

f^2 = k/m

so   this is Optipn D (9k, 3m)

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