`{:("List - I","List - II"),("(A) Frequency of seconds pendulum","(E)"A omega),("(B) Angular frequency of oscillating spring","(F)" (1)/(2)),("(C) Maximum velocity of simple harmonic oscillator","(G)"A omega^(2)),("(D) Maximum acceleration of simple harmonic oscillator","(H)"sqrt((k)/(m))):}`

Options

A. a - e, b - f, c - g, d - h
B. a - f, b - h, c - g, d - e
C. a - f, b - h, c - e, d - g
D. a - f, b - e, c - h, d - g

Answer

To solve the problem, we need to match the items from List - I with the corresponding items from List - II based on the properties of oscillations. Let's analyze each item step by step.

### Step 1: Frequency of Seconds Pendulum
- A seconds pendulum is defined as one that takes 2 seconds to complete one full oscillation (i.e., one complete swing back and forth).
- The frequency \( f \) is given by the formula:
\[ f = \frac{1}{T} \]
where \( T \) is the time period. For a seconds pendulum, \( T = 2 \) seconds.
- Therefore, the frequency is:
\[ f = \frac{1}{2} \text{ Hz} \]
- This corresponds to option (E) in List - II.

### Step 2: Angular Frequency of Oscillating Spring
- The angular frequency \( \omega \) for a mass-spring system is given by:
\[ \omega = \sqrt{\frac{k}{m}} \]
where \( k \) is the spring constant and \( m \) is the mass attached to the spring.
- This corresponds to option (H) in List - II.

### Step 3: Maximum Velocity of Simple Harmonic Oscillator
- The maximum velocity \( V_{\text{max}} \) of a simple harmonic oscillator occurs at the mean position (where displacement \( x = 0 \)).
- It is given by:
\[ V_{\text{max}} = A \omega \]
where \( A \) is the amplitude of the motion.
- This corresponds to option (E) in List - II.

### Step 4: Maximum Acceleration of Simple Harmonic Oscillator
- The maximum acceleration \( a_{\text{max}} \) occurs at the maximum displacement (where \( x = A \)).
- It is given by:
\[ a_{\text{max}} = \omega^2 A \]
- This corresponds to option (G) in List - II.

### Summary of Matches
Now, we can summarize the matches:
- (A) Frequency of seconds pendulum → (E) \( \frac{1}{2} \)
- (B) Angular frequency of oscillating spring → (H) \( \sqrt{\frac{k}{m}} \)
- (C) Maximum velocity of simple harmonic oscillator → (E) \( A \omega \)
- (D) Maximum acceleration of simple harmonic oscillator → (G) \( \omega^2 A \)

### Final Matches
- (A) → (E)
- (B) → (H)
- (C) → (E)
- (D) → (G)