1.

For the damped oscillator shown in Fig. 14.19, the mass m of the block is 200 g, k = 90 N m^(-1) and the damping constant b is 40 g s^(-1). Calculate (a) the period of oscillation, (b) time taken for its amplitude of vibrations to drop to half of its initial value, and (c) the time taken for its mechanical energy to drop to half its initial value.

Answer» <html><body><p></p>Solution :(a) We see that <a href="https://interviewquestions.tuteehub.com/tag/km-1064498" style="font-weight:bold;" target="_blank" title="Click to know more about KM">KM</a> = `90x×0.2 = 18 kg Nm^(-1)=kg^(2)s^(-2),` therefore `sqrt(km)=4.243kgs^(-1)`, and `b = 0.04 kg s^(-1)`. Therefore, b is much less than `sqrt(km)` . Hence, the <a href="https://interviewquestions.tuteehub.com/tag/time-19467" style="font-weight:bold;" target="_blank" title="Click to know more about TIME">TIME</a> period T from Eq. (14.34) is given by <br/> `T=2pisqrt(m/k)` <br/> `=2pisqrt((0.2kg)/(90Nm^(-1)))` <br/> = 0.3 s <br/> (b) Now, from Eq. (14.33), the time, `T_(1//2)`, for the <a href="https://interviewquestions.tuteehub.com/tag/amplitude-859568" style="font-weight:bold;" target="_blank" title="Click to know more about AMPLITUDE">AMPLITUDE</a> to drop to half of its initial value is given by <br/> `T_(1//2)=("ln"(1//2))/(b//2m)` <br/> `=(0.693)/(40)xx2xx200s` <br/> `=6.93s` <br/> (c) For calculating the time, `t_(1//2)`, for its mechanical energy to drop to half its initial value we make use of Eq. (14.35). From this equation we have, <br/> `E(t_(1//2)//E(0)=exp(-bt_(1//2)//m)` <br/> Or `1//2=exp(-bt_(1//2)//m)` <br/> `"ln"(1//2)=-(bt_(1//2)//m)` <br/> Or `t_(1//2)=(0.693)/(40gs^(-1))xx200g` <br/> =3.46 s <br/> This is just half of the <a href="https://interviewquestions.tuteehub.com/tag/decay-945588" style="font-weight:bold;" target="_blank" title="Click to know more about DECAY">DECAY</a> period for amplitude. This is not surprising, because, according to Eqs. (14.33) and (14.35), energy depends on the square of the amplitude. <a href="https://interviewquestions.tuteehub.com/tag/notice-25787" style="font-weight:bold;" target="_blank" title="Click to know more about NOTICE">NOTICE</a> that there is a factor of 2 in the exponents of the two exponentials.</body></html>


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