(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A damped oscillator satisfies the equation

x'' + 2Kx' + [tex]\Omega[/tex]^2 *(x)

where K and [tex]\Omega[/tex] are positive constants with K < [tex]\Omega[/tex] (underdamping).

i)At time t =0 the particle is released from rest at the point x=a . Show that the subsequent motion is given by

x=a*exp(KT)(cos([tex]\Omega[/tex]_{D}*t) +K/([tex]\Omega[/tex]_{2})*sin([tex]\Omega[/tex]_{D}*t)

where [tex]\Omega[/tex]_{D}=([tex]\Omega[/tex]^2 - K^2)^^{1/2}.

ii)Find all the turning points of the function x(t) and show that the rati of successive maximum values of x is e^(-2*[tex]\pi[/tex]*K/([tex]\Omega[/tex]_{D})

iii)a certain damped oscillator has mass 10 kg , period 5 seconds and successive maximum values of its displacement are in the ratio 3:1. Find the values of the spring and damping constants [tex]\alpha[/tex] and [tex]\beta[/tex].

2. Relevant equations

3. The attempt at a solution

I had no trouble with part i) so I will skipped directly to part ii and iii.

ii) Not sure how to calculate the turning points at x(t) and why taking the ratio of those turning points is significant.

iii) How would knowing that finding the successive maximum values of its displacement are in the ratio 3 :1 aid me in finding the values of the spring and damping constants?

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# Homework Help: Damped oscillator

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