Which of the following equation represents damped oscillation?

Options

A. `(dx^(2))/(dt^(2)) = - kx`
B. `(dx^(2))/(dt^(2)) = -kx + f_(0) sin omega_(0) t`
C. `(d^(2)x)/(dt^(2)) -kx + (dx)/(dt) = 0`
D. `(d^(2)x)/(dt^(2)) + (dx)/(dt) + (k)/(m) x =0`

Answer

To determine which of the given equations represents damped oscillation, we need to analyze the characteristics of damped oscillation equations. The standard form of the equation for damped oscillation is:

\[ m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = 0 \]

Where:
- \( m \) is the mass,
- \( b \) is the damping constant,
- \( k \) is the spring constant,
- \( x \) is the displacement.

Now, let's evaluate the given options step by step:

### Step 1: Analyze the first option
The first option is:

\[ \frac{d^2x}{dt^2} = 0 \]

This equation does not include a damping term or a restoring force term. It simply states that the second derivative of \( x \) with respect to time is zero, which indicates no oscillation.

**Conclusion**: This option does not represent damped oscillation.

### Step 2: Analyze the second option
The second option is:

\[ \frac{d^2x}{dt^2} + \frac{dx}{dt} = 0 \]

This equation lacks a restoring force term (kx). It only has a damping term and does not represent oscillatory motion.

**Conclusion**: This option does not represent damped oscillation.

### Step 3: Analyze the third option
The third option is:

\[ \frac{d^2x}{dt^2} + \frac{dx}{dt} - kx = 0 \]

This equation has the correct form with a second derivative, a first derivative, and a displacement term. However, the presence of a negative \( kx \) term suggests that the system is not oscillating but rather moving towards a stable point without oscillation.

**Conclusion**: This option does not represent damped oscillation.

### Step 4: Analyze the fourth option
The fourth option is:

\[ \frac{d^2x}{dt^2} + \frac{b}{m} \frac{dx}{dt} + \frac{k}{m} x = 0 \]

This equation can be rearranged to match the standard form of the damped oscillation equation. If we let \( b/m \) represent the damping ratio and \( k/m \) represent the spring constant per unit mass, we see that this equation has all the necessary components for damped oscillation.

**Conclusion**: This option represents damped oscillation.

### Final Answer
The correct option that represents damped oscillation is the **fourth option**.

---