A particle executing simple harmonic motion with time period T. the time period with which its kinetic energy oscillates is

Options

A. T
B. 2T
C. 4T
D. `T/2`

Explanation

To determine the time period with which the kinetic energy of a particle executing simple harmonic motion (SHM) oscillates, we can follow these steps:

### Step 1: Understand the Kinetic Energy in SHM
The kinetic energy (KE) of a particle in SHM is given by the formula:
\[ KE = \frac{1}{2} m \omega^2 A^2 \cos^2(\omega t) \]
where:
- \( m \) is the mass of the particle,
- \( \omega \) is the angular frequency,
- \( A \) is the amplitude,
- \( t \) is the time.

### Step 2: Identify the Angular Frequency
The angular frequency \( \omega \) is related to the time period \( T \) by the equation:
\[ \omega = \frac{2\pi}{T} \]

### Step 3: Substitute Angular Frequency into Kinetic Energy Formula
Substituting \( \omega \) into the kinetic energy formula, we get:
\[ KE = \frac{1}{2} m \left(\frac{2\pi}{T}\right)^2 A^2 \cos^2\left(\frac{2\pi}{T} t\right) \]

### Step 4: Analyze the Cosine Function
The term \( \cos^2\left(\frac{2\pi}{T} t\right) \) oscillates between 0 and 1. The function \( \cos^2(x) \) has a period of \( \pi \) (since \( \cos(x) \) has a period of \( 2\pi \)). Therefore, the time period of \( \cos^2\left(\frac{2\pi}{T} t\right) \) is:
\[ T_{KE} = \frac{T}{2} \]

### Step 5: Conclusion
Thus, the time period with which the kinetic energy oscillates is:
\[ \frac{T}{2} \]

### Final Answer
The time period with which the kinetic energy oscillates is \( \frac{T}{2} \).

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