Obtain an expression for angular frequency of LC oscillations ?

Explanation

i. The mechanical energy of the spring-mass system is given by
`E=(1)/(2)mv^(2)+(1)/(2)kx^(2)`
ii. The energy E remains constant for varying values of x and v. Differentiating E with respect to time, we get
`(dE)/(dt)=(1)/(2)m(2v(dv)/(dt))+(1)/(2)k(2x(dx)/(dt))=0`
`orm(d^(2)x)/(dt^(2))+kx=0`
since `(dx)/(dt)=vand(dv)/(dt)=(d^(2)x)/(dt^(2))" "...(1)`
iii. This is the differential equation of the oscillation of the spring-mass system. The general solution of equation (1) is of the form `x(t)=X_(m)cos(omegat+phi)
`
iv. where `X_(m)` is the maximum value of x (t), `omega` the angular frequency and `phi` the phase constant.
Similarly, the electronmagnetic 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
`U=(1)/(2)(2i(di)/(dt))+(1)/(2C)(2i(dq)/(dt))=0`
`orL(d^(2)q)/(dt^(2))+(1)/(C)q=0" "...(2)`
since `i=(dq)/(dt)and(di)/(dt)=(d^(2)q)/(dt^(2))`
v. The general solution of eqution (2) is of the form
`q(t)=Q_(m)cos(omegat+phi)" "(3)`
where `Q_(m) is the maximum value of q (t), `omega` the angular frequency and `phi` the phase constant.
Current in the LC circuit
The current flowing in the LC circuit is obtained by differentiating q (t) with respect to time.
`i(t)=(dq)/(dt)=(d)/(dt)[Q_(m)cos(omegat+phi)]`
`=-Q_(m)omegasin(omegat+phi)" "" since "I_(m)=Q_(m)omega` or `i(t)=-I_(m)sin(omegat+phi)" "...(4)`
The equation (4) clearly shows that current varies as a function of time t. In fact, it is a sinusoidally varying alternating current with angular frequency `omega.`
Angular frequency of LC oscillations
By differentiating equation (3) twice, we get
`(d^(2)q)/(dt)=-Q_(m)omega^(2)cos(omegat+phi)" "...(5)`
Substituting equations (3) and (5) in equation (2), 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 oscillations is given by
`omega=(1)/(sqrt(LC))`
This equation is the same as that obtained from qualitative analogy.