If a body is executing simple harmonic motion and its current displacement is `(sqrt(3))/(2)` times the amplitude from its mean position, then the ratio between potential energy and kinetic energy is

Options

A. `3:2`
B. `2:3`
C. `sqrt(3):1`
D. `3:1`

Explanation

To solve the problem, we need to find the ratio of potential energy (PE) to kinetic energy (KE) for a body executing simple harmonic motion (SHM) when its displacement \( x \) is \( \frac{\sqrt{3}}{2} \) times the amplitude \( A \).

### Step-by-Step Solution:

1. **Identify the given values:**
- Displacement \( x = \frac{\sqrt{3}}{2} A \)
- Amplitude \( A \)

2. **Formulas for potential energy and kinetic energy:**
- The potential energy (PE) in SHM is given by:
\[ PE = \frac{1}{2} k x^2 \]
- The kinetic energy (KE) in SHM is given by:
\[ KE = \frac{1}{2} k (A^2 - x^2) \]

3. **Substitute \( x \) into the potential energy formula:**
- Substitute \( x = \frac{\sqrt{3}}{2} A \) into the potential energy formula:
\[ PE = \frac{1}{2} k \left(\frac{\sqrt{3}}{2} A\right)^2 = \frac{1}{2} k \left(\frac{3}{4} A^2\right) = \frac{3}{8} k A^2 \]

4. **Substitute \( x \) into the kinetic energy formula:**
- First, calculate \( x^2 \):
\[ x^2 = \left(\frac{\sqrt{3}}{2} A\right)^2 = \frac{3}{4} A^2 \]
- Now substitute into the kinetic energy formula:
\[ KE = \frac{1}{2} k \left(A^2 - \frac{3}{4} A^2\right) = \frac{1}{2} k \left(\frac{1}{4} A^2\right) = \frac{1}{8} k A^2 \]

5. **Calculate the ratio of potential energy to kinetic energy:**
- Now we can find the ratio \( \frac{PE}{KE} \):
\[ \frac{PE}{KE} = \frac{\frac{3}{8} k A^2}{\frac{1}{8} k A^2} \]
- The \( k A^2 \) terms cancel out:
\[ \frac{PE}{KE} = \frac{3}{1} = 3 \]

### Final Answer:
The ratio of potential energy to kinetic energy is \( 3:1 \).

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