To solve this problem, we need to analyze the given circuit and determine the output voltage $v_o$.

Given Information:

• The voltage source is $v_s(t) = 10 \cos(10t + 75^\circ) \, \text{V}$.
• The components in the circuit:
  • Resistor: $R = 10 \, \Omega$
  • Inductor: $L = 0.5 \, \text{H}$
  • Capacitor: $C = \frac{1}{20} \, \text{F}$

Step 1: Determine the Impedances

The circuit consists of a resistor, inductor, and capacitor connected in series. The impedance of each component is given by:

1. Resistor $R$: $Z_R = R = 10 \, \Omega$

2. Inductor $L$: $Z_L = j\omega L = j(10)(0.5) = j5 \, \Omega$

3. Capacitor $C$: $Z_C = \frac{1}{j\omega C} = \frac{1}{j(10)\left(\frac{1}{20}\right)} = \frac{20}{j10} = -j2 \, \Omega$

Step 2: Total Impedance $Z_{total}$

The total impedance $Z_{total}$ is the sum of the impedances in series:

$Z_{total} = Z_R + Z_L + Z_C = 10 + j5 - j2 = 10 + j3 \, \Omega$

Step 3: Calculate the Current $I(t)$

Using Ohm's Law:

$I(t) = \frac{v_s(t)}{Z_{total}} = \frac{10 \cos(10t + 75^\circ)}{10 + j3}$

To simplify this, we convert $Z_{total}$ into polar form:

$|Z_{total}| = \sqrt{10^2 + 3^2} = \sqrt{100 + 9} = \sqrt{109} \approx 10.44 \, \Omega$ $\theta_Z = \tan^{-1}\left(\frac{3}{10}\right) \approx 16.7^\circ$

So,

$Z_{total} \approx 10.44 \angle 16.7^\circ \, \Omega$

Thus, the current is:

$I(t) = \frac{10 \angle 75^\circ}{10.44 \angle 16.7^\circ} = \frac{10}{10.44} \angle (75^\circ - 16.7^\circ) \approx 0.958 \angle 58.3^\circ \, \text{A}$

So,

$I(t) = 0.958 \cos(10t + 58.3^\circ) \, \text{A}$

Step 4: Determine $v_o(t)$ Across the Resistor $R$

Since $v_o(t)$ is the voltage across the resistor $R$:

$v_o(t) = I(t) \cdot R = 0.958 \cos(10t + 58.3^\circ) \times 10 = 9.58 \cos(10t + 58.3^\circ) \, \text{V}$

Final Answer:

The output voltage $v_o(t)$ is:

$v_o(t) = 9.58 \cos(10t + 58.3^\circ) \, \text{V}$

Would you like further details or have any questions?

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Here are some related questions to consider:

1. How would the output voltage change if the inductor value were doubled?
2. What effect would decreasing the capacitance have on the total impedance?
3. Can you determine the phase difference between the source voltage and the output voltage?
4. How would the current change if the source frequency were increased?
5. What happens to the circuit if we replace the resistor with a capacitor?

Tip: When analyzing AC circuits, always remember to convert all quantities into their phasor (complex) form to simplify the calculations.