Comparing the L-C oscillations with the oscillations of a spring-block system

Options

A. L is equivalent to m
B. C is equivalent to K
C. current is equivalent to speed
D. rate of change of current is equivalent to accelerate

Answer

To compare L-C oscillations with the oscillations of a spring-block system, we can analyze the equations governing both systems and identify their similarities and differences.

### Step-by-Step Solution:

1. **Understanding L-C Oscillation:**
- An L-C circuit consists of an inductor (L) and a capacitor (C) connected together. When the circuit is closed, the capacitor discharges through the inductor, creating oscillations.
- The charge (Q) on the capacitor and the current (I) in the circuit are related. The governing equation for the L-C circuit can be derived using Kirchhoff's voltage law (KVL):
\[ L \frac{di}{dt} + \frac{Q}{C} = 0 \]
- The current (I) can be expressed as the rate of change of charge (Q), so we can rewrite the equation as:
\[ L \frac{d^2Q}{dt^2} + \frac{Q}{C} = 0 \]
- This is a second-order differential equation representing L-C oscillations.

2. **Understanding Spring-Block Oscillation:**
- A spring-block system consists of a mass (m) attached to a spring with spring constant (k). When the mass is displaced from its equilibrium position, it experiences a restoring force.
- The governing equation for the spring-block system is given by:
\[ m \frac{d^2x}{dt^2} + kx = 0 \]
- Here, \(x\) is the displacement from the equilibrium position, and this is also a second-order differential equation representing simple harmonic motion (SHM).

3. **Comparing the Two Systems:**
- Both equations are second-order differential equations that describe oscillatory motion.
- By comparing the equations:
- For the L-C circuit: \(L \frac{d^2Q}{dt^2} + \frac{Q}{C} = 0\)
- For the spring-block system: \(m \frac{d^2x}{dt^2} + kx = 0\)

4. **Identifying Analogies:**
- From the comparison, we can establish the following analogies:
- Inductance (L) is analogous to mass (m).
- The spring constant (k) is analogous to \(\frac{1}{C}\) (inverse of capacitance).
- Charge (Q) is analogous to displacement (x).
- Current (I) is analogous to velocity (\(v = \frac{dx}{dt}\)).
- The rate of change of current (\(\frac{di}{dt}\)) is analogous to acceleration (\(\frac{d^2x}{dt^2}\)).

5. **Conclusion:**
- The L-C oscillation and the spring-block oscillation exhibit similar mathematical forms, indicating that they are analogous systems. The relationships established allow us to draw parallels between electrical and mechanical oscillations.