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To address this problem, let's first recall the analogy between a mechanical harmonic oscillator (mass-spring system with damping) and an RLC circuit.

The differential equation of an RLC series circuit for charge \( q(t) \) is:

\[
L \frac{d^2q(t)}{dt^2} + R \frac{dq(t)}{dt} + \frac{1}{C}q(t) = V(t)
\]

This is analogous to the differential equation of a damped harmonic oscillator (mass-spring-damper system):

\[
m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = F(t)
\]

Where:
- \( m \) is the mass.
- \( b \) is the damping constant (related to drag).
- \( k \) is the spring constant.
- \( F(t) \) is the external force.

### Now, let's match terms between the two equations:

1. **Mass \( m \)** corresponds to **Inductance \( L \)**.
2. **Damping constant \( b \)** corresponds to **Resistance \( R \)**.
3. **Spring constant \( k \)** corresponds to \( \frac{1}{C} \), where \( C \) is the capacitance.

### Now, let's compute the values for each:

- **Inductance \( L = 0.05 \, \text{H} \)** (Henrys) represents the mass \( m \).
- **Resistance \( R = 69 \, \Omega \)** (Ohms) represents the damping constant \( b \).
- **Capacitance \( C = 0.00012 \, \text{F} \)** (Farads) relates to the spring constant \( k \) as \( k = \frac{1}{C} \).

#### a. Numerical value of the quantity analogous to mass in the mechanical differential equation:
\[
m = L = 0.05 \, \text{kg}
\]

#### b. Numerical value of the quantity analogous to the spring constant in the mechanical differential equation:
\[
k = \frac{1}{C} = \frac{1}{0.00012} \, \text{F}^{-1} = 8333.33 \, \text{N/m}
\]

#### c. Numerical value of the quantity analogous to the damping constant in the mechanical differential equation:
\[
b = R = 69 \, \text{Ns/m}
\]

### Final answers:
- a. \( 0.05 \, \text{kg} \)
- b. \( 8333.33 \, \text{N/m} \)
- c. \( 69 \, \text{Ns/m} \)

Let me know if you would like further clarifications or details!

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Here are 5 additional questions to deepen your understanding:
1. How does the frequency response of an RLC circuit relate to the natural frequency of a mass-spring system?
2. What happens to the system when the resistance (damping) is significantly increased?
3. How does the quality factor of the RLC circuit relate to mechanical oscillators?
4. Can you derive the resonant frequency for the given RLC circuit and compare it with the mechanical analogy?
5. What is the effect of changing the capacitance on the dynamics of the system?

**Tip:** In both mechanical and electrical oscillators, increasing damping (or resistance) reduces oscillations but increases stability!

An RLC series circuit containing a 69 Ω resistor, a 0.05 H inductor, and a 0.00012 F capacitor is driven by an AC voltage source. The differential equation that describes the distribution and dynamics of electric charge in this circuit bears a striking resemblance to the mechanical differential equation that describes the position of a mass on a spring that is subject to a drag force whose magnitude is proportional to the velocity of the mass.

What is the numerical value of the quantity in the RLC differential equation that is most analogous to the mass in the mechanical differential equation?
What is the numerical value of the quantity in the RLC differential equation that is most analogous to the spring constant in the mechanical differential equation?
What is the numerical value of the quantity in the RLC differential equation that is most analogous to the proportionality constant that relates the drag force magnitude to the velocity in the mechanical differential equation?

Explore the analogy between an RLC series circuit and a mechanical oscillator. Learn how to calculate the mass, spring constant, and damping constant from the differential equations of the system, involving a 69 Ω resistor, a 0.05 H inductor, and a 0.00012 F capacitor. This topic is ideal for undergraduate students in Physics or Engineering.

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