InterviewSolution
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InterviewSolution
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Explain mechanicalanalogy of LC oscillations by quantitative treament. |
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Answer» Solution :i. The energy E remains constant for varying vaues of x and v. Differentiating E with respect to time, we get `(dE)/(dt)=(1)/(2)(2v(dv)/(dt))+(1)/(2)k(2x(dx)/(dt))=0` or `m(d^(2)x)/(dt^(2))+kx=0" since "(dx)/(dt)=vand(dv)/(dt)=(d^(2)x)/(dt^(2))` ii. This is the differential equation of the OSCILLATIONS of the spring-mass system. The general solution of eqution is of the from `x(t)=X_(m)cos(omegat+phi)` where `X_(m)` is the maximum value of `x(t),omega` the angular frequency and `phi` the phase constant. iii. Similarly, the electromagnetic energy of the LC system is given by `U=(1)/(2)LI^(2)+(1)/(2)((1)/(C))q^(2)="constant"` Differentiating U with respect to time, we get `(dU)/(dt)=(1)/(2)L(2I(di)/(dt))+(1)/(2C)(2q(dq)/(dt))=0` `orL(d^(2)q)/(dt^(2))+(1)/(C)q=0""...(1)` since `i=(dq)/(dt)and(di)/(dt)=(d^(2)q)/(dt2)` iv. The general solution of equation (1) is of the form `q(t)=Qmcos(omegat+phi)` v. where `Q_(m)` is the maximum value of q (t), `omega` the angular frequency and `phi` the phase consatnt. |
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