Frequency of variation of kinetic energy of a simple harmonic motion of frequency n is

Options

A. 2n
B. n
C. `n/2`
D. 3n

Explanation

To find the frequency of variation of kinetic energy of a simple harmonic motion (SHM) with a given frequency \( n \), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding SHM**: In simple harmonic motion, the motion of the particle oscillates between two extreme positions. The velocity of the particle is maximum at the mean position and zero at the extreme positions.

2. **Kinetic Energy in SHM**: The kinetic energy (KE) of a particle in SHM can be expressed as:
\[ KE = \frac{1}{2} m v^2 \]
where \( m \) is the mass of the particle and \( v \) is its velocity.

3. **Velocity vs. Time**: The velocity of the particle varies sinusoidally with time. If we plot velocity against time, we see that it starts from zero at the extreme position, reaches maximum at the mean position, and returns to zero at the next extreme position.

4. **Kinetic Energy vs. Time**: Since kinetic energy depends on the square of the velocity, the kinetic energy will also vary sinusoidally but will reach its maximum when the velocity is maximum (at the mean position) and will be zero when the velocity is zero (at the extreme positions).

5. **Time Period of SHM**: Let the time period of the SHM be \( T \). The frequency \( n \) is given by:
\[ n = \frac{1}{T} \]

6. **Time Period of Kinetic Energy**: The kinetic energy reaches its maximum and minimum values twice during one complete cycle of SHM. Therefore, the time period of the kinetic energy will be half of the time period of the SHM:
\[ T_{KE} = \frac{T}{2} \]

7. **Frequency of Kinetic Energy**: The frequency of the kinetic energy can be calculated as:
\[ n_{KE} = \frac{1}{T_{KE}} = \frac{1}{\frac{T}{2}} = \frac{2}{T} \]
Substituting \( T = \frac{1}{n} \):
\[ n_{KE} = 2n \]

8. **Conclusion**: Therefore, the frequency of variation of kinetic energy of a simple harmonic motion with frequency \( n \) is:
\[ n_{KE} = 2n \]

### Final Answer:
The frequency of variation of kinetic energy of simple harmonic motion of frequency \( n \) is \( 2n \).