The value of total mechanical energy of a particle is SHM is

Options

A. Always constant
B. Depend on time
C. `(1)/(2)KA^(2)"cos"^(2)(omegat+phi)`
D. `(1)/(2)mA^(2)"cos"^(2)(omegat+phi)`

Explanation

To find the total mechanical energy of a particle executing simple harmonic motion (SHM), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Total Mechanical Energy**:
- The total mechanical energy (E) in SHM is the sum of kinetic energy (KE) and potential energy (PE) at any point in time.
- Mathematically, this can be expressed as:
\[ E = KE + PE \]

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

3. **Potential Energy in SHM**:
- The potential energy in SHM is given by:
\[ PE = \frac{1}{2} kx^2 \]
- Where \( k \) is the spring constant and \( x \) is the displacement from the mean position.

4. **Total Mechanical Energy in SHM**:
- In SHM, the total mechanical energy remains constant and can be derived from the maximum displacement (amplitude, A) of the motion.
- The total mechanical energy can be expressed as:
\[ E = \frac{1}{2} k A^2 \]
- This equation shows that the total mechanical energy is dependent on the amplitude of the motion.

5. **Conclusion**:
- Therefore, the value of the total mechanical energy of a particle executing simple harmonic motion is constant and can be calculated using the formula:
\[ E = \frac{1}{2} k A^2 \]
- This energy remains constant throughout the motion, as energy continuously transforms between kinetic and potential forms.

### Final Answer:
The total mechanical energy of a particle in simple harmonic motion is constant and given by \( E = \frac{1}{2} k A^2 \).

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