The potential energy of a particle executing SHM varies sinusoidally with frequency `f`. The frequency of oscillation of the particle will be

Options

A. `(f)/(2)`
B. `(f)/sqrt(2)`
C. `f`
D. `2f`

Explanation

To solve the problem, we need to determine the frequency of oscillation of a particle executing Simple Harmonic Motion (SHM) based on the information given about its potential energy.

### Step-by-Step Solution:

1. **Understand the Displacement Equation of SHM**:
The displacement \( x \) of a particle in SHM can be expressed as:
\[ x = a \sin(\omega t + \phi) \]
where \( a \) is the amplitude, \( \omega \) is the angular frequency, and \( \phi \) is the phase constant.

2. **Relate Angular Frequency to Frequency**:
The angular frequency \( \omega \) is related to the frequency \( f \) by the equation:
\[ \omega = 2\pi f \]

3. **Potential Energy in SHM**:
The potential energy \( U \) of a particle in SHM is given by:
\[ U = \frac{1}{2} k x^2 \]
where \( k \) is the spring constant.

4. **Substituting Displacement into Potential Energy**:
Substitute the expression for \( x \) into the potential energy equation:
\[ U = \frac{1}{2} k (a \sin(\omega t + \phi))^2 \]
This simplifies to:
\[ U = \frac{1}{2} k a^2 \sin^2(\omega t + \phi) \]

5. **Using the Identity for Sin²**:
We can use the trigonometric identity:
\[ \sin^2(\theta) = \frac{1 - \cos(2\theta)}{2} \]
Applying this identity:
\[ U = \frac{1}{2} k a^2 \cdot \frac{1 - \cos(2(\omega t + \phi))}{2} \]
This can be rewritten as:
\[ U = \frac{1}{4} k a^2 - \frac{1}{4} k a^2 \cos(2(\omega t + \phi)) \]

6. **Identifying the Frequency of Potential Energy**:
The term \( \cos(2(\omega t + \phi)) \) indicates that the potential energy oscillates with a frequency of \( 2\omega \). Therefore, the frequency of oscillation of the potential energy is:
\[ f_{U} = \frac{2\omega}{2\pi} = 2f \]

7. **Finding the Frequency of the Particle**:
Since the potential energy oscillates at frequency \( f_U = 2f \), the frequency of the particle's oscillation \( f' \) is half of that:
\[ f' = \frac{f_U}{2} = \frac{2f}{2} = f \]

### Final Answer:
The frequency of oscillation of the particle is \( f' = \frac{f}{2} \).