The total energy of a particle having a displacement x, executing simple harmonic motion is

Options

A. `prop x`
B. `prop x^(2)`
C. independent of x
D. `prop x^(1//2)`

Explanation

To solve the question regarding the total energy of a particle executing simple harmonic motion (SHM) with displacement \( x \), we can follow these steps:

### Step 1: Understand the Components of Total Energy
The total energy \( E \) of a particle in simple harmonic motion is the sum of its kinetic energy (KE) and potential energy (PE).

### Step 2: Write the Expressions for Kinetic and Potential Energy
The kinetic energy \( KE \) of a particle in SHM is given by:
\[ KE = \frac{1}{2} m \omega^2 (A^2 - x^2) \]
where:
- \( m \) is the mass of the particle,
- \( \omega \) is the angular frequency,
- \( A \) is the amplitude of the motion,
- \( x \) is the displacement from the mean position.

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

### Step 3: Calculate the Total Energy
Now, we can find the total energy \( E \) by adding the kinetic and potential energy:
\[ E = KE + PE \]
Substituting the expressions we have:
\[ E = \left(\frac{1}{2} m \omega^2 (A^2 - x^2)\right) + \left(\frac{1}{2} m \omega^2 x^2\right) \]

### Step 4: Simplify the Expression
Now, simplifying the total energy expression:
\[ E = \frac{1}{2} m \omega^2 (A^2 - x^2) + \frac{1}{2} m \omega^2 x^2 \]
\[ E = \frac{1}{2} m \omega^2 A^2 - \frac{1}{2} m \omega^2 x^2 + \frac{1}{2} m \omega^2 x^2 \]
The \( -\frac{1}{2} m \omega^2 x^2 \) and \( +\frac{1}{2} m \omega^2 x^2 \) cancel each other out:
\[ E = \frac{1}{2} m \omega^2 A^2 \]

### Step 5: Conclusion
From the final expression for total energy \( E = \frac{1}{2} m \omega^2 A^2 \), we can see that the total energy is dependent only on the mass \( m \), angular frequency \( \omega \), and amplitude \( A \). It does not depend on the displacement \( x \).

Thus, the total energy of a particle executing simple harmonic motion is **independent of \( x \)**.

### Final Answer
The correct option is **3) independent of x**.

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