A weakly damped harmonic oscillator of frequency `n_1` is driven by an external periodic force of frequency `n_2`. When the steady state is reached, the frequency of the oscillator will be

Options

A. `n_(1)`
B. `n_(2)`
C. `(n_(1)+n_(2))/(2)`
D. `(n_(1)+n_(2))`

Explanation

To solve the problem of a weakly damped harmonic oscillator driven by an external periodic force, we need to analyze the relationship between the natural frequency of the oscillator and the frequency of the driving force. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the System
A weakly damped harmonic oscillator has a natural frequency denoted as \( n_1 \). When this oscillator is subjected to an external periodic force with frequency \( n_2 \), we need to determine the frequency of the oscillator when it reaches a steady state.

**Hint:** Identify the key components of the system: the natural frequency of the oscillator and the frequency of the external force.

### Step 2: Recognize Forced Oscillation
In forced oscillation, the oscillator is influenced by an external periodic force. The equation of motion for such a system includes the restoring force from the spring and a damping force, along with the external driving force.

**Hint:** Recall that the external force can be represented as a function of time, typically in the form of a cosine function with its own frequency.

### Step 3: Analyze the Resonance Condition
For a weakly damped oscillator, when the frequency of the external driving force \( n_2 \) is close to the natural frequency \( n_1 \), the system can resonate. At resonance, the amplitude of oscillation reaches its maximum.

**Hint:** Remember that resonance occurs when the driving frequency matches the natural frequency.

### Step 4: Determine the Frequency of the Oscillator
When the steady state is reached, the frequency of the oscillator will match the frequency of the driving force. Thus, the frequency of the oscillator in steady state will be equal to \( n_2 \).

**Hint:** In steady state, the system oscillates at the frequency of the external force, not the natural frequency.

### Step 5: Conclusion
The frequency of the oscillator when the steady state is reached is \( n_2 \).

**Final Answer:** The frequency of the oscillator will be \( n_2 \).