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Chapter 4 & 5 Oscillator with external forcing-I & II

Problem 5: A mildly damped oscillator driven by an external force is known to have a resonance at an angular frequency somewhere near ω = 1 MHz with a quality factor of 1100. Further, for the force (in Newtons) F (t) = 10 cos(ωt) the amplitude of oscillations is 8.26mm at ω = 1.0 KHz and 1.0 µm at 100MHz. a. What is the spring constant of the oscillator? b. What is the natural frequency 0 of the oscillator? c. What is the FWHM? d. What is the phase difference between the force and the oscillations at ω = ω0 +FWHM/2? Answer: Form equation (5.4) we have the amplitude |x(t)| =

F m

q

(ω02 − ω 2 )2 + 4β 2 ω 2

.

(1)

Here F = 10, ω0 ∼ 1 MHz and β = ω0 /2Q ∼ 106 /2 · 1100 ∼ 103 . a. ω02 >> ω 2 at ω = 1.0 KHz. We can ignore terms containing ω 2 in the denominator of equation 1. Then we have |x(t)| = 8.26 × 10−3 =

10 . mω02

(2)

So k=1210 N/m . b. ω02

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Problem 5: A mildly damped oscillator driven by an external force is known to have a resonance at an angular frequency somewhere near ω = 1 MHz with a quality factor of 1100. Further, for the force (in Newtons) F (t) = 10 cos(ωt) the amplitude of oscillations is 8.26mm at ω = 1.0 KHz and 1.0 µm at 100MHz. a. What is the spring constant of the oscillator? b. What is the natural frequency 0 of the oscillator? c. What is the FWHM? d. What is the phase difference between the force and the oscillations at ω = ω0 +FWHM/2? Answer: Form equation (5.4) we have the amplitude |x(t)| =

F m

q

(ω02 − ω 2 )2 + 4β 2 ω 2

.

(1)

Here F = 10, ω0 ∼ 1 MHz and β = ω0 /2Q ∼ 106 /2 · 1100 ∼ 103 . a. ω02 >> ω 2 at ω = 1.0 KHz. We can ignore terms containing ω 2 in the denominator of equation 1. Then we have |x(t)| = 8.26 × 10−3 =

10 . mω02

(2)

So k=1210 N/m . b. ω02

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