There is a LCR circuit , If it is compared with a damped oscillation of mass m oscillating with force constant k and damping coefficient 'b'. Compare the terms of damped oscillation with the devices in LCR circuit.

Options

A. `L to m, C to (1)/(k),R to b`
B. `L to m , c to R to b`
C. `L to K, C to b, R to m`
D. `L to (1)/(m) ,C to (1)/(k),R to (1)/(b)`

Answer

To compare the terms of damped oscillation with the devices in an LCR circuit, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Damped Oscillation**:
The equation for damped oscillation can be expressed as:
\[ m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = 0 \]
Here, \( m \) is the mass, \( b \) is the damping coefficient, and \( k \) is the force constant.

2. **Understanding the LCR Circuit**:
In an LCR circuit, we can apply Kirchhoff's voltage law, which gives us:
\[ -iR - L \frac{di}{dt} - \frac{Q}{C} = 0 \]
where \( i \) is the current, \( R \) is the resistance, \( L \) is the inductance, and \( C \) is the capacitance.

3. **Relating Charge and Current**:
The current \( i \) is related to charge \( Q \) by:
\[ i = \frac{dQ}{dt} \]
Thus, substituting \( i \) into the equation gives:
\[ -R \frac{dQ}{dt} - L \frac{d^2Q}{dt^2} - \frac{Q}{C} = 0 \]

4. **Rearranging the LCR Equation**:
Rearranging the equation leads to:
\[ L \frac{d^2Q}{dt^2} + R \frac{dQ}{dt} + \frac{Q}{C} = 0 \]

5. **Identifying Corresponding Terms**:
Now, we can compare the coefficients of the two equations:
- From the damped oscillation equation, we have:
- \( m \) corresponds to \( L \)
- \( b \) corresponds to \( R \)
- \( k \) corresponds to \( \frac{1}{C} \)

6. **Final Relationships**:
Therefore, we can summarize the relationships as:
- \( m \equiv L \)
- \( b \equiv R \)
- \( k \equiv \frac{1}{C} \)

### Conclusion:
The relationships between the parameters of damped oscillation and the components of the LCR circuit can be summarized as:
- Mass \( m \) is equivalent to inductance \( L \)
- Damping coefficient \( b \) is equivalent to resistance \( R \)
- Force constant \( k \) is equivalent to \( \frac{1}{C} \)