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 in simple harmonic motion (SHM), we can follow these steps:

### Step 1: Understand the Energy Components
In SHM, the total mechanical energy (E) is the sum of kinetic energy (KE) and potential energy (PE).

### Step 2: Write the Expressions for KE and PE
The kinetic energy (KE) of a particle in SHM is given by:
\[ KE = \frac{1}{2} k (A^2 - x^2) \]
where \( k \) is the spring constant, \( A \) is the amplitude, and \( x \) is the displacement from the mean position.

The potential energy (PE) is given by:
\[ PE = \frac{1}{2} k x^2 \]

### Step 3: Write the Total Mechanical Energy
The total mechanical energy (E) can be expressed as:
\[ E = KE + PE \]
Substituting the expressions for KE and PE:
\[ E = \frac{1}{2} k (A^2 - x^2) + \frac{1}{2} k x^2 \]

### Step 4: Simplify the Expression
Now, we simplify the expression:
\[ E = \frac{1}{2} k (A^2 - x^2) + \frac{1}{2} k x^2 \]
Combining the terms:
\[ E = \frac{1}{2} k A^2 - \frac{1}{2} k x^2 + \frac{1}{2} k x^2 \]
The \( -\frac{1}{2} k x^2 \) and \( +\frac{1}{2} k x^2 \) cancel each other out:
\[ E = \frac{1}{2} k A^2 \]

### Step 5: Conclusion
The total mechanical energy \( E \) in simple harmonic motion is:
\[ E = \frac{1}{2} k A^2 \]
This value is constant and does not depend on the displacement \( x \) or time \( t \).

### Final Answer
The total mechanical energy of a particle in SHM is always constant.

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