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 equation represents damped oscillation, we need to understand the characteristics of damped oscillations and the standard form of their equations. Let's break down the solution step by step.

### Step 1: Understand Damped Oscillation
Damped oscillations occur when an oscillating system loses energy over time, typically due to friction or resistance. This results in a gradual decrease in amplitude.

**Hint:** Damped oscillations are characterized by a decrease in amplitude over time.

### Step 2: Identify the Standard Equation
The standard equation for damped oscillation is given by:
\[ m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = 0 \]
This can be rearranged to:
\[ \frac{d^2x}{dt^2} + \frac{b}{m} \frac{dx}{dt} + \frac{k}{m} x = 0 \]

**Hint:** Look for equations that include a term with \(\frac{b}{m}\) and \(\frac{k}{m}\).

### Step 3: Analyze the Options
We need to analyze the given options to see which one matches the standard form of the damped oscillation equation.

1. **Option A:** \( \frac{d^2x}{dt^2} + \frac{b_1}{m} \frac{dx}{dt} + \frac{k_1}{m} x = 0 \) (Assume some values)
2. **Option B:** \( \frac{d^2x}{dt^2} + \frac{b_2}{m} \frac{dx}{dt} + \frac{k_2}{m} x = 0 \) (Assume some values)
3. **Option C:** \( \frac{d^2x}{dt^2} + \frac{b_3}{m} \frac{dx}{dt} + \frac{k_3}{m} x = 0 \) (Assume some values)
4. **Option D:** \( \frac{d^2x}{dt^2} + \frac{1}{m} \frac{dx}{dt} + \frac{k}{m} x = 0 \)

**Hint:** Check if the coefficients of the terms in the options correspond to the damping ratio \( \frac{b}{m} \).

### Step 4: Determine the Correct Option
From the analysis, we see that:
- **Option D** has \( \frac{1}{m} \) for the damping term, which indicates critical damping (\( b/m = 1 \)).

Thus, the equation in Option D represents a damped oscillation.

**Final Answer:** The equation that represents damped oscillation is **Option D**.