Answer to Question #89130 in Physics for Shivam Nishad

Question #89130
For a weakly damped harmonic oscillator, the
instantaneous displacement is given by
x(t) = ao exp (-bt) cos (wd t + phy).
Calculate its average energy.
1
Expert's answer
2019-05-15T10:08:24-0400
E=K+U=0.5m(dxdt)2+0.5kx2E=K+U=0.5m(\frac{dx}{dt})^2+0.5kx^2

dxdt=aoexp(bt)[bcos(wdt+ϕ)+wdsin(wdt+ϕ)]\frac{dx}{dt}=-a_o \exp{(-bt)} [b \cos{(w_d t + \phi)}+w_d \sin{(w_d t + \phi)}]

Thus

K=0.5ma02e2bt[b2cos2(wdt+ϕ)+wd2sin2(wdt+ϕ)+bwdsin2(wdt+ϕ)]K=0.5ma_0^2e^{-2bt} [b^2 \cos^2{(w_d t + \phi)}+w_d^2 \sin^2{(w_d t + \phi)}+bw_d \sin{2(w_d t + \phi)}]

We have:


k=mwd2k=mw_d^2


U=0.5ma02wd2e2btcos2(wdt+ϕ)U=0.5ma_0^2w_d^2 e^{-2bt} \cos^2{(w_d t + \phi)}E=0.5ma02e2btA(t)\langle E \rangle=0.5ma_0^2 e^{-2bt} \langle A(t) \rangle

A(t)=(b2+wd2)cos2(wdt+ϕ)+wd2sin2(wdt+ϕ)+bwdsin2(wdt+ϕ)A(t)=(b^2+w_d^2) \cos^2{(w_d t + \phi)}+w_d^2 \sin^2{(w_d t + \phi)}+bw_d \sin{2(w_d t + \phi)}

A(t)=0.5((b2+wd2)+wd2)\langle A(t) \rangle=0.5((b^2+w_d^2)+w_d^2)

As for weakly damped harmonic oscillator


bwdb\ll w_d

A(t)=wd2\langle A(t) \rangle=w_d^2

E=0.5ma02wd2e2bt=E0e2bt\langle E \rangle=0.5ma_0^2w_d^2 e^{-2bt} =E_0e^{-2bt}


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment