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!

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!