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Effectus
3 months ago
9

A mass of 20 g stretches a spring 5 cm. Suppose that the mass is also attached to a viscous damper with a damping constant of 40

0 dyn·s/cm. If the mass is pulled down an additional 2 cm and then released, find its position u at any time t. Plot u versust. Determine the quasi frequency and the quasi period. Determine the ratio of the quasi period to the period of the corresponding undamped motion. Also find the time τ such that |u(t)| < 0.05 cm for all t > τ.

Physics
1 answer:
kicyunya [3.2K]3 months ago
8 0
Quasi frequency = 4√6Quasi period = π√6/12t ≈ 0.4045Explanation: The given data is: Mass, m = 20gτ = 400 dyn.s/cmk = 3920u(0) = 2u'(0) = 0General differential equation:mu" + τu' + ku = 0Substituting the known values yields:20u" + 400u' + 3920u = 0Now, dividing each side by 20 gives:u" + 20u' + 196u = 0To establish the characteristic equation, replace y" with r², y' with r, and y with 1 in the differential equation:r² + 20r + 196 = 0Now, solving for the roots results in:r = -10 ± 4√6iThe general solution for two complex roots is: y = c₁ eᵃt cosbt + c₂ eᵃt sinbtwith a being the real part of the roots and b representing the imaginary part. Given that a = -10 and b = 4√6,u(t) = c₁e⁻¹⁰^t cos 4√6t + c₂e⁻¹⁰^t sin 4√6twhere u(0) = 2 and u'(0) = 0(b)Quasi frequency: μ = (c)(d)Quasi period: T = 2π / μ(d)|u(t)| < 0.05 cmu(t) = |2e⁻¹⁰^t cos 4√6t + 5√6/6 e⁻¹⁰^t sin 4√6t| < 0.05solving for t:τ = t ≈ 0.4045
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