Question

In the figure below, let R = 7.50 Ω, L = 2.10 mH, and C = 1.60 µF.

(a) Calculate the frequency of the damped oscillation of the circuit when the switch is thrown to position b. ? kHz

(b) What is the critical resistance for damped oscillations? ? Ω

Answer 1

Part A

damped oscillation frequency is given by

\(w=\left[\frac{1}{L \cdot C}-\left(\frac{R}{2 \cdot L}\right)^{2}\right]^{\frac{1}{2}}\)

where:

resistance

\(R=7.50 \Omega\)

inductance

\(L=2.10 \mathrm{mH} \cdot \frac{1 H}{10^{3} \mathrm{H}}=2.10 \cdot 10^{-3} \mathrm{H}\)

capacitance

\(C=1.60 \mu F \cdot \frac{10^{-6} F}{1 \mu F}=1.60 \cdot 10^{-6} F\)

evaluated numerically

\(w=\left[\frac{1}{2.10 \cdot 10^{-3} H \cdot 1.60 \cdot 10^{-6} F}-\left(\frac{7.50 \Omega}{2 \cdot 2.10 \cdot 10^{-3} H}\right)^{2}\right]^{\frac{1}{2}}=\left[2.98 \cdot 10^{8} \frac{r a d^{2}}{s^{2}}-3.19 \cdot 10^{6} \frac{r a d^{2}}{s^{2}}\right] \frac{1}{2}=\)

\(17170 \frac{r a d}{s}\)

\(w=17170 \frac{\mathrm{rad}}{\mathrm{s}}\)

but the frequency is equal to

\(w=2 \cdot \pi \cdot f \Rightarrow f=\frac{w}{2 \cdot \pi}=f=\frac{17170 \frac{\mathrm{rad}}{\mathrm{s}}}{2 \cdot \pi \mathrm{rad}}=2733 \mathrm{~Hz}\)

\(f=2733 H z \cdot \frac{1 k H z}{1000 H z}=2.73 K H z\)

part C

The critical value of resistance to damped oscillations is

\(R_{c}=\sqrt{\frac{4 \cdot L}{C}}=\sqrt{\frac{4 \cdot 2 \cdot 10 \cdot 10^{-3} H}{1.60 \cdot 10^{-6} F}}=72.5 \Omega\)