A particle executes simple harmonic motion with a frequency. (f). The frequency with which its kinetic energy oscillates is.

Options

A. `v/2`
B. v
C. 2v
D. zero

Answer

To solve the problem, we need to understand the relationship between the frequency of simple harmonic motion (SHM) and the frequency of the oscillation of kinetic energy.

### Step-by-Step Solution:

1. **Understand the Motion**: A particle executing simple harmonic motion (SHM) has a certain frequency denoted as \( f \). This frequency is the number of complete cycles the particle completes in one second.

2. **Kinetic Energy in SHM**: The kinetic energy (KE) of a particle in SHM varies as the particle moves. At the mean position (the center of the motion), the particle has maximum velocity, and thus the kinetic energy is at its maximum. Conversely, at the extreme positions (maximum displacement), the velocity is zero, and so the kinetic energy is at its minimum.

3. **Oscillation of Kinetic Energy**: As the particle moves from one extreme position to the other, the kinetic energy goes from minimum to maximum and back to minimum. This means that one complete oscillation of kinetic energy occurs when the particle travels from one extreme to the other and back.

4. **Time Period of Kinetic Energy**: The time taken for the particle to go from one extreme to the other (one complete cycle of kinetic energy) is half of the time period \( T \) of the SHM. Therefore, the time period of the kinetic energy oscillation is \( T/2 \).

5. **Frequency of Kinetic Energy**: The frequency of an oscillation is the reciprocal of the time period. Thus, the frequency of the kinetic energy oscillation can be calculated as:
\[ f_{KE} = \frac{1}{T/2} = \frac{2}{T} \]

6. **Relate to Frequency of SHM**: The frequency of the simple harmonic motion is given by:
\[ f = \frac{1}{T} \]
Therefore, we can express the frequency of kinetic energy in terms of the frequency of SHM:
\[ f_{KE} = 2f \]

7. **Conclusion**: The frequency with which the kinetic energy oscillates is twice the frequency of the simple harmonic motion. Thus, the correct answer is:
\[ \text{Frequency of kinetic energy} = 2 \times \text{Frequency of SHM} \]

### Final Answer:
The frequency with which the kinetic energy oscillates is \( 2f \).

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