Comparing the L-C oscillations with the oscillations of a spring-block system (force constant of spring=k and mass of block=m), the physical quantity mk is similar to

Options

A. CL
B. V`(1)/(CL)`
C. `(C)/(L)`
D. `(L)/(C)`

Answer

To solve the problem of comparing L-C oscillations with the oscillations of a spring-block system, we can follow these steps:

### Step 1: Understand the systems
We have two systems to compare:
1. An L-C circuit (inductor-capacitor circuit) that undergoes oscillations.
2. A spring-block system that oscillates under the influence of a restoring force.

### Step 2: Write the equations of motion
For the L-C circuit, we can apply Kirchhoff's law:
\[ E + V = 0 \]
Where \(E\) is the electromotive force and \(V\) is the potential across the capacitor. The equation can be expressed as:
\[ L \frac{dI}{dt} + \frac{Q}{C} = 0 \]
Where \(Q\) is the charge on the capacitor and \(I\) is the current.

### Step 3: Substitute for charge
We know that charge \(Q\) can be expressed in terms of current:
\[ Q = I \cdot t \]
Substituting this into the equation gives:
\[ L \frac{dI}{dt} + \frac{I \cdot t}{C} = 0 \]

### Step 4: Differentiate with respect to time
Differentiating the equation with respect to time, we have:
\[ L \frac{d^2I}{dt^2} + \frac{I}{C} = 0 \]

### Step 5: Compare with the spring-block system
The equation for a spring-block system is given by:
\[ M \frac{d^2x}{dt^2} + Kx = 0 \]
Where \(M\) is the mass and \(K\) is the spring constant.

### Step 6: Identify the similarities
From the equations:
- We can see that \(L\) (inductance) corresponds to \(M\) (mass).
- The term \(\frac{1}{C}\) corresponds to \(K\) (spring constant).

Thus, we have:
\[ L \sim M \quad \text{and} \quad K \sim \frac{1}{C} \]

### Step 7: Calculate the quantity \(mk\)
We are asked to find the quantity \(mk\):
\[ mk = M \cdot K \]
Substituting the relationships we found:
\[ mk = L \cdot \frac{1}{C} = \frac{L}{C} \]

### Conclusion
Thus, the physical quantity \(mk\) is similar to:
\[ \frac{L}{C} \]

### Final Answer
The correct answer is \( \frac{L}{C} \).

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