A simple harmonic oscillation has an amplitude `A` and time period `T`. The time required to travel from `x = A` to ` x= (A)/(2)` is

Options

A. `(T)/(6)`
B. `(T)/(4)`
C. `(T)/(3)`
D. `(T)/(12)`

Explanation

To solve the problem of finding the time required for a simple harmonic oscillator to travel from \( x = A \) to \( x = \frac{A}{2} \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Displacement**:
The initial position is \( x = A \) and the final position is \( x = \frac{A}{2} \). The displacement \( y \) can be calculated as:
\[ y = A - \frac{A}{2} = \frac{A}{2} \]

2. **Use the Equation of Motion**:
The equation for displacement in simple harmonic motion is given by:
\[ y = A \cos(\omega t) \]
where \( \omega \) is the angular frequency.

3. **Determine the Angular Frequency**:
The angular frequency \( \omega \) is related to the time period \( T \) by the formula:
\[ \omega = \frac{2\pi}{T} \]

4. **Set Up the Equation**:
Substitute \( y = \frac{A}{2} \) into the equation:
\[ \frac{A}{2} = A \cos\left(\frac{2\pi}{T} t\right) \]

5. **Simplify the Equation**:
Dividing both sides by \( A \) (assuming \( A \neq 0 \)):
\[ \frac{1}{2} = \cos\left(\frac{2\pi}{T} t\right) \]

6. **Find the Angle**:
The cosine of \( \frac{2\pi}{T} t \) equals \( \frac{1}{2} \). The angle whose cosine is \( \frac{1}{2} \) is:
\[ \frac{2\pi}{T} t = \frac{\pi}{3} \]
(This corresponds to the first quadrant solution.)

7. **Solve for Time \( t \)**:
Rearranging the equation gives:
\[ t = \frac{T}{6} \]

### Final Answer:
The time required to travel from \( x = A \) to \( x = \frac{A}{2} \) is:
\[ t = \frac{T}{6} \]