In case of damped oscillation frequency of oscillation is

Options

A. greater than natural frequency
B. less than natural frequency
C. equal to nature frequency
D. Both (a) and (b)

Explanation

To solve the question regarding the frequency of oscillation in the case of damped oscillation, we will follow these steps:

### Step-by-Step Solution:

1. **Understand the Forces Acting on the Oscillator**:
In a simple harmonic motion (SHM), the primary force acting on the mass (M) is the restoring force from the spring, which is given by Hooke's Law:
\[ F_{\text{spring}} = -kx \]
where \( k \) is the spring constant and \( x \) is the displacement from the equilibrium position.

2. **Introduce Damping Force**:
In the case of damped oscillation, there is an additional damping force acting on the mass, which opposes the motion. This force can be expressed as:
\[ F_{\text{damping}} = -bv \]
where \( b \) is the damping constant and \( v \) is the velocity of the mass.

3. **Calculate the Net Force**:
The net force acting on the mass can be expressed as the sum of the spring force and the damping force:
\[ F_{\text{net}} = F_{\text{spring}} + F_{\text{damping}} = -kx - bv \]

4. **Set Up the Equation of Motion**:
According to Newton's second law, the net force is also equal to the mass times acceleration:
\[ M \frac{d^2x}{dt^2} = -kx - b\frac{dx}{dt} \]
Rearranging this gives us:
\[ M \frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0 \]

5. **Identify the Damped Angular Frequency**:
The equation of motion for damped oscillations can be compared to the standard form of a second-order linear differential equation. The damped angular frequency \( \omega_d \) can be derived from the equation:
\[ \omega_d = \sqrt{\frac{k}{m} - \left(\frac{b}{2m}\right)^2} \]
Here, \( \frac{k}{m} \) is the square of the natural frequency \( \omega_0^2 \).

6. **Relate Damped Frequency to Natural Frequency**:
The damped frequency \( f_d \) can be related to the damped angular frequency as follows:
\[ f_d = \frac{\omega_d}{2\pi} \]
Since \( \omega_d < \omega_0 \), it follows that:
\[ f_d < f_0 \]
where \( f_0 \) is the natural frequency of the system.

### Conclusion:
In the case of damped oscillation, the frequency of oscillation is less than the natural frequency of the system. Therefore, the correct answer is that the damped frequency \( f_d \) is less than the natural frequency \( f_0 \).

---