Consider the following statements. The total energy of a particles executing simple harmonic motion depends on its
1. amplitude
2. Period
3. displacement of these :

Options

A. 1 and 2 are correct
B. 2 and 3 are correct
C. 1 and 3 are correct
D. 1, 2 and 3 are correct

Answer

To solve the question regarding the total energy of a particle executing simple harmonic motion (SHM) and its dependence on amplitude, period, and displacement, we can follow these steps:

### Step 1: Understand the formula for total energy in SHM
The total energy (E) of a particle executing simple harmonic motion is given by the formula:
\[ E = \frac{1}{2} m \omega^2 A^2 \]
where:
- \(E\) is the total energy,
- \(m\) is the mass of the particle,
- \(\omega\) is the angular frequency,
- \(A\) is the amplitude of the motion.

### Step 2: Relate angular frequency to the period
The angular frequency \(\omega\) is related to the period \(T\) of the motion by the equation:
\[ \omega = \frac{2\pi}{T} \]
This means we can express the total energy in terms of the period.

### Step 3: Substitute \(\omega\) in the energy formula
Substituting \(\omega\) into the energy formula gives:
\[ E = \frac{1}{2} m \left(\frac{2\pi}{T}\right)^2 A^2 \]
This simplifies to:
\[ E = \frac{1}{2} m A^2 \frac{4\pi^2}{T^2} \]
or
\[ E = \frac{2\pi^2 m A^2}{T^2} \]

### Step 4: Analyze the dependence of total energy on amplitude and period
From the final equation, we can see that:
- The total energy \(E\) is **directly proportional** to the square of the amplitude \(A^2\).
- The total energy \(E\) is **inversely proportional** to the square of the period \(T^2\).

### Step 5: Consider the role of displacement
The displacement of the particle at any point in time does not affect the total energy. The total energy remains constant for a given amplitude and mass, regardless of the particle's position in its motion.

### Conclusion
Based on the analysis:
- The total energy depends on the amplitude (1) and the period (2).
- The total energy does not depend on the displacement (3).

Thus, the correct statements are:
1. Amplitude (True)
2. Period (True)
3. Displacement (False)

### Final Answer
The total energy of a particle executing simple harmonic motion depends on its amplitude and period, but not on its displacement. Therefore, the correct options are 1 and 2.

---