The particle executing simple harmonic motion has a kinetic energy `K_(0) cos^(2) omegat`. The maximum values of the potential 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, we need to analyze the kinetic energy of a particle executing simple harmonic motion (SHM) given by the equation \( K(t) = K_0 \cos^2(\omega t) \). We will find the maximum values of the potential energy and the total energy.

### Step-by-Step Solution:

1. **Understanding Kinetic Energy in SHM**:
The kinetic energy of a particle in SHM is given by the equation \( K(t) = \frac{1}{2}mv^2 \), where \( m \) is the mass and \( v \) is the velocity of the particle. In this case, we are given that \( K(t) = K_0 \cos^2(\omega t) \).

2. **Finding Maximum Kinetic Energy**:
The maximum value of \( K(t) \) occurs when \( \cos^2(\omega t) \) is at its maximum, which is 1. Therefore, the maximum kinetic energy \( K_{\text{max}} \) can be calculated as:
\[ K_{\text{max}} = K_0 \cdot 1 = K_0 \]

3. **Relationship Between Kinetic and Potential Energy**:
In SHM, the total mechanical energy \( E \) is the sum of kinetic energy \( K \) and potential energy \( U \):
\[ E = K + U \]
The total energy in SHM is constant and is equal to the maximum kinetic energy or maximum potential energy.

4. **Finding Maximum Potential Energy**:
Since the maximum kinetic energy is \( K_{\text{max}} = K_0 \), the maximum potential energy \( U_{\text{max}} \) is also equal to \( K_0 \) because in SHM:
\[ U_{\text{max}} = K_{\text{max}} = K_0 \]

5. **Calculating Total Energy**:
The total energy \( E \) in SHM is equal to the maximum kinetic energy (or potential energy):
\[ E = K_{\text{max}} = K_0 \]

6. **Final Results**:
Therefore, the maximum values of potential energy and total energy are both \( K_0 \).

### Conclusion:
The maximum values of the potential energy and the total energy are respectively:
- Maximum Potential Energy \( U_{\text{max}} = K_0 \)
- Total Energy \( E = K_0 \)