A simple harmonic oscillator has an amplitude A and time period `6 pi` second. Assuming the oscillation starts from its mean position, Find the time required by it to travel from x = 0 to `x = frac{sqrt 3}{2}A`

Options

A. `pi/2` sec
B. `pi/6` sec
C. `pi/4` sec
D. `pi` sec

Answer

To solve the problem, we need to find the time taken by a simple harmonic oscillator to travel from its mean position (x = 0) to a position where x = (√3/2)A.

### Step-by-Step Solution:

1. **Identify the parameters**:
- Amplitude (A) = A
- Time period (T) = 6π seconds

2. **Determine the angular frequency (ω)**:
- The angular frequency (ω) is given by the formula:
\[ \omega = \frac{2\pi}{T} \]
- Substituting the value of T:
\[ \omega = \frac{2\pi}{6\pi} = \frac{1}{3} \text{ rad/s} \]

3. **Write the equation of motion**:
- The displacement in simple harmonic motion can be expressed as:
\[ x = A \sin(\omega t) \]
- Here, we need to find the time (t) when \( x = \frac{\sqrt{3}}{2}A \).

4. **Set up the equation**:
- Substitute \( x = \frac{\sqrt{3}}{2}A \) into the equation:
\[ \frac{\sqrt{3}}{2}A = A \sin(\omega t) \]
- Dividing both sides by A (assuming A ≠ 0):
\[ \frac{\sqrt{3}}{2} = \sin(\omega t) \]

5. **Find the angle corresponding to the sine value**:
- We know that \( \sin(\frac{\pi}{3}) = \frac{\sqrt{3}}{2} \).
- Therefore, we can equate:
\[ \omega t = \frac{\pi}{3} \]

6. **Solve for time (t)**:
- Substitute ω into the equation:
\[ \frac{1}{3} t = \frac{\pi}{3} \]
- Multiplying both sides by 3:
\[ t = \pi \text{ seconds} \]

### Final Answer:
The time required for the oscillator to travel from \( x = 0 \) to \( x = \frac{\sqrt{3}}{2}A \) is \( \pi \) seconds.

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