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Obtain an expression for angular frequen...

Obtain an expression for angular frequency of LC oscillations ?

Text Solution

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By differentiating equation twice, we get
`q(t)=Q_(m)cos(omegat+phi)" "...(1)`
`(d^(2)q)/(dt)=-Q_(m)omega^(2)cos(omegat+phi)" "...(2)`
Substituting equations (1) and (2) in equation
`(dU)/(dt)=(1)/(2)L(2i(di)/(dt))+(1)/(2C)(2q(dq)/(dt))=0orL(d^(2)q)/(dt^(2))+(1)/(C)q=0`
we obtain
`L[-Q_(m)omega^(2)cos(omegat+phi)]+(1)/(C)Q_(m)cos(omegat+phi)=0`
Rearranging the terms, the angular frequency of LC oscillationsdf is given by
`omega=(1)/(sqrt(LC))`
This equation is the same as that obtained from qualitative analogy.
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