The time periods - hence the frequencies (f) and the amplitudes (A) of the driven pendulums in steady state are measured. The variation of A with f is correctly shown by the graph.

Options

A.
B.
C.
D.

Explanation

To solve the problem regarding the variation of amplitude (A) with frequency (f) for driven pendulums in steady state, we need to analyze the behavior of the system as frequency changes. Here’s a step-by-step breakdown of the solution:

### Step 1: Understanding the System
In a driven pendulum, the amplitude of oscillation depends on the frequency of the driving force. As the frequency approaches the natural frequency of the pendulum, resonance occurs, leading to a significant increase in amplitude.

**Hint:** Consider how resonance affects the amplitude of oscillation in a driven system.

### Step 2: Analyzing the Options
We have four options that describe the relationship between amplitude and frequency. We need to evaluate each option based on the expected behavior of a driven pendulum.

**Hint:** Look for patterns in the graphs that indicate how amplitude changes as frequency is varied.

### Step 3: Evaluating Option 1
Option 1 suggests a linear increase in amplitude with frequency. This implies that as frequency increases, amplitude continuously increases without any limit.

**Hint:** Think about whether amplitude can increase indefinitely with frequency in a real system.

### Step 4: Evaluating Option 2
Option 2 shows a curve that increases with frequency until it reaches a maximum (resonance) and then decreases. This is characteristic of a driven oscillator where amplitude peaks at resonance.

**Hint:** Recall the concept of resonance and how it affects amplitude in oscillatory systems.

### Step 5: Evaluating Option 3
Option 3 indicates that amplitude remains constant regardless of frequency. This scenario does not reflect the behavior of a driven pendulum, especially near resonance.

**Hint:** Consider if it's physically plausible for amplitude to remain unchanged with varying frequency.

### Step 6: Evaluating Option 4
Option 4 suggests that amplitude decreases linearly as frequency increases. This is contrary to the expected behavior of a driven pendulum.

**Hint:** Think about the relationship between frequency and amplitude in the context of resonance.

### Step 7: Conclusion
After evaluating all options, Option 2 is the correct representation of the relationship between amplitude and frequency for a driven pendulum in steady state. The graph shows that amplitude increases with frequency until it reaches a maximum at the resonant frequency, after which it decreases.

**Final Answer:** The correct option is Option 2.