The paricle executing simple harmonic motion had kinetic energy and the total energy are respectively:

Options

A. `(K_(0))/(2)`and`K_(0)`
B. `K_(0)`and `2K_(0)`
C. `K_(0)`and `K_(0)`
D. 0 and `2K_(0)`

Explanation

To solve the problem regarding the relationship between kinetic energy and total energy of a particle executing simple harmonic motion (SHM), we can follow these steps:

### Step 1: Understand the Definitions
In SHM, the total mechanical energy (E) of the particle is constant and is given by the formula:
\[ E = \frac{1}{2} m \omega^2 A^2 \]
where:
- \( m \) is the mass of the particle,
- \( \omega \) is the angular frequency,
- \( A \) is the amplitude of the motion.

### Step 2: Kinetic Energy in SHM
The kinetic energy (K) of the particle at any position \( x \) is given by:
\[ K = \frac{1}{2} m v^2 \]
where \( v \) is the velocity of the particle. The velocity can be expressed in terms of \( x \) as:
\[ v = \omega \sqrt{A^2 - x^2} \]
Thus, the kinetic energy can be rewritten as:
\[ K = \frac{1}{2} m \omega^2 (A^2 - x^2) \]

### Step 3: Relationship Between Kinetic Energy and Total Energy
From the definitions, we know that the total energy is the sum of kinetic energy (K) and potential energy (U):
\[ E = K + U \]
In SHM, the potential energy (U) is given by:
\[ U = \frac{1}{2} m \omega^2 x^2 \]
Thus, we can express the total energy as:
\[ E = K + \frac{1}{2} m \omega^2 x^2 \]

### Step 4: Analyzing the Maximum Values
At the mean position (where \( x = 0 \)), the kinetic energy is maximum:
\[ K_{\text{max}} = \frac{1}{2} m \omega^2 A^2 = E \]
At the extreme positions (where \( x = A \)), the kinetic energy is zero and potential energy is maximum:
\[ U_{\text{max}} = \frac{1}{2} m \omega^2 A^2 = E \]

### Step 5: Conclusion
Thus, the relationship between kinetic energy and total energy can be summarized as:
- When the particle is at the mean position, \( K = E \) and \( U = 0 \).
- When the particle is at the extreme positions, \( K = 0 \) and \( U = E \).

Given the options in the problem, the correct relationship is:
- Kinetic energy is \( \frac{1}{2} E \) when the potential energy is maximum, and total energy is \( E \).

### Final Answer
The relationship between kinetic energy and total energy of a particle executing SHM can be expressed as:
- Kinetic Energy: \( \frac{1}{2} K_0 \)
- Total Energy: \( K_0 \)