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 find the ratio between potential energy (PE) and kinetic energy (KE) of a body executing simple harmonic motion (SHM) at a given displacement, we can follow these steps:

### Step 1: Understand the given information
We are given that the current displacement \( x \) is \( \frac{\sqrt{3}}{2} \) times the amplitude \( A \). Therefore, we can express this as:
\[ x = \frac{\sqrt{3}}{2} A \]

### Step 2: Write the formulas for potential energy and kinetic energy
The potential energy \( PE \) in SHM is given by:
\[ PE = \frac{1}{2} m \omega^2 x^2 \]
The kinetic energy \( KE \) in SHM is given by:
\[ KE = \frac{1}{2} m \omega^2 (A^2 - x^2) \]

### Step 3: Substitute the value of \( x \) into the formulas
Now, substituting \( x = \frac{\sqrt{3}}{2} A \) into the potential energy formula:
\[ PE = \frac{1}{2} m \omega^2 \left(\frac{\sqrt{3}}{2} A\right)^2 \]
\[ PE = \frac{1}{2} m \omega^2 \left(\frac{3}{4} A^2\right) \]
\[ PE = \frac{3}{8} m \omega^2 A^2 \]

Next, substitute \( x \) into the kinetic energy formula:
\[ KE = \frac{1}{2} m \omega^2 \left(A^2 - \left(\frac{\sqrt{3}}{2} A\right)^2\right) \]
\[ KE = \frac{1}{2} m \omega^2 \left(A^2 - \frac{3}{4} A^2\right) \]
\[ KE = \frac{1}{2} m \omega^2 \left(\frac{1}{4} A^2\right) \]
\[ KE = \frac{1}{8} m \omega^2 A^2 \]

### Step 4: Find the ratio of potential energy to kinetic energy
Now, we can find the ratio \( \frac{PE}{KE} \):
\[ \frac{PE}{KE} = \frac{\frac{3}{8} m \omega^2 A^2}{\frac{1}{8} m \omega^2 A^2} \]
The \( m \), \( \omega^2 \), and \( A^2 \) terms cancel out:
\[ \frac{PE}{KE} = \frac{3}{1} \]

### Final Answer
Thus, the ratio of potential energy to kinetic energy is:
\[ \frac{PE}{KE} = 3:1 \]