Consider an LC circuit in which L $=465$ mH and C $=0\mathrm{.}123$F$.$

$($a$)$ What is the resonance frequency $0?$

$($b$)$ If a resistance of $1\mathrm{.}27$ k$$ is introduced into this circuit, what is the frequency of the damped oscillations?

$($c$)$ By what percentage does the frequency of the damped oscillations differ from the resonance frequency?

Step $1$

The LC circuit will oscillate without a power supply if it is connected with the capacitor initially charged. The values of the inductance and capacitance determine the angular frequency, which we estimate to be on the order of thousands of rad$/$s$.$ The introduction of a large resistor into the circuit would strongly damp the oscillations or prevent them altogether. Introducing a resistance of $1\mathrm{.}27$ k$$ may be enough to produce a discernable decrease in frequency.

Step $2$

For frequency, we will calculate angular frequency $$ in radians per second. We will use the equations for oscillating LC circuits with and without resistance.

Step $3$

$($a$)$ The angular frequency of undamped oscillations is given by

$0=$

$1$

LC

$$

$$

$=$

$1$

$465$e$-3$

$0\mathrm{.}465$

H

$0\mathrm{.}123$e$-6$

$1\mathrm{.}23$e$-07$

F

$$

$=4\mathrm{.}181$

$4\mathrm{.}18$

$103$ rad$/$s$.$

Step $4$

$($b$)$ With the resistance in the circuit, the damped angular frequency

$$d

is given by

$$d $=$

$1$LC

$$

R$2$L

$2$

$$

$1/2$

$$

$=$

$1$

H

F

$$

$103$

$2$

H

$2$

$$

$1/2$

$$

$=103$ rad$/$s$.$