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Physics

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A 3-kg mass attached to a spring with k = 145 N/m is oscillating in a vat of oil, which damps the oscillations.

(a) If the damping constant of the oil is b = 14 kg/s, how long will it take the amplitude of the oscillations to decrease to 1% of its original value?
(b) What should the damping constant be to reduce the amplitude of the oscillations by 91% in 4 s?

The answers are 1.97s and 3.61 kg/s. How can I find these answers? Thank you in advance.

  • Physics - ,

    (a)
    The amplitude of damped oscillations depends on time as
    A=Aₒ•e^-(β•t),
    The damping coefficient β =b/2m, where b=14 kg/s is the damping constant,
    β =b/2m =14/2•3=2.33 s^-1
    Given:
    Aₒ/A=100%/1%= 100
    Aₒ/A= Aₒ/ Aₒ•e^-(β•t) = e^(β•t) =100,
    β•t =ln100,
    t=ln100/ β = 4.6/2.33=1.97 s.
    (b)
    Given:
    Aₒ/A=100%/9%= 100/9
    Aₒ/A= Aₒ/ Aₒ•e^-(β•t) = e^(β•t) =100/9,
    β•t =ln(100/9)=2.408,
    β =2.408/4=0.602 s^-1,

    b=2m β =2•3•0.602=3.61 kg/s.

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