Amplitude of a harmonic oscillator is A, when velocity of particle is half of maximum velocity, then determine position of particle.

Options

A. `x =+-(sqrt(5)A)/(4)`
B. `x =+-(sqrt(3)A)/(5)`
C. `x =+-(sqrt(3)A)/(2)`
D. `x =+-(sqrt(3)A)/(4)`

Explanation

To solve the problem, we need to determine the position of a harmonic oscillator when its velocity is half of the maximum velocity. Let's break down the solution step by step.

### Step 1: Understand the maximum velocity
The maximum velocity \( V_{\text{max}} \) of a harmonic oscillator is given by the formula:
\[ V_{\text{max}} = A \omega \]
where \( A \) is the amplitude and \( \omega \) is the angular frequency.

### Step 2: Determine the given velocity
According to the problem, the velocity \( V \) of the particle is half of the maximum velocity:
\[ V = \frac{1}{2} V_{\text{max}} = \frac{1}{2} A \omega \]

### Step 3: Write the expression for velocity at position \( x \)
The velocity \( V \) of the particle at any position \( x \) in simple harmonic motion is given by:
\[ V = \omega \sqrt{A^2 - x^2} \]

### Step 4: Set the two expressions for velocity equal
Now we can set the two expressions for velocity equal to each other:
\[ \frac{1}{2} A \omega = \omega \sqrt{A^2 - x^2} \]

### Step 5: Cancel \( \omega \) from both sides
Assuming \( \omega \neq 0 \), we can cancel \( \omega \) from both sides:
\[ \frac{1}{2} A = \sqrt{A^2 - x^2} \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides gives:
\[ \left(\frac{1}{2} A\right)^2 = A^2 - x^2 \]
\[ \frac{1}{4} A^2 = A^2 - x^2 \]

### Step 7: Rearrange the equation to solve for \( x^2 \)
Rearranging the equation, we get:
\[ x^2 = A^2 - \frac{1}{4} A^2 \]
\[ x^2 = A^2 \left(1 - \frac{1}{4}\right) \]
\[ x^2 = A^2 \cdot \frac{3}{4} \]

### Step 8: Solve for \( x \)
Taking the square root of both sides gives us:
\[ x = \pm \sqrt{\frac{3}{4}} A = \pm \frac{\sqrt{3}}{2} A \]

### Final Answer
Thus, the position of the particle when its velocity is half of the maximum velocity is:
\[ x = \pm \frac{\sqrt{3}}{2} A \]