Two simple harmonic motions `y_(1) = Asinomegat` and `y_(2)` = Acos`omega`t are superimposed on a particle of mass m. The total mechanical energy of the particle is

Options

A. `1/2momega^(2)A^(2)`
B. `momega^(2)A^(2)`
C. `1/4momega^(2)A^(2)`
D. zero

Answer

To find the total mechanical energy of a particle undergoing two superimposed simple harmonic motions (SHM), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Equations of Motion**:
- The two simple harmonic motions are given as:
\[ y_1 = A \sin(\omega t) \]
\[ y_2 = A \cos(\omega t) \]

2. **Understand the Phase Relationship**:
- The second equation can be rewritten using the sine function:
\[ y_2 = A \sin\left(\omega t - \frac{\pi}{2}\right) \]
- This indicates that \(y_2\) is a sine wave that is phase-shifted by \(-\frac{\pi}{2}\) radians (or 90 degrees) relative to \(y_1\).

3. **Determine the Resultant Motion**:
- Since \(y_1\) and \(y_2\) are perpendicular to each other (due to the phase difference of \(\frac{\pi}{2}\)), we can find the resultant amplitude \(A'\) of the combined motion using the Pythagorean theorem:
\[ A' = \sqrt{A^2 + A^2} = \sqrt{2A^2} = A\sqrt{2} \]

4. **Calculate the Total Mechanical Energy**:
- The total mechanical energy \(E\) of a simple harmonic oscillator is given by the formula:
\[ E = \frac{1}{2} m \omega^2 A^2 \]
- For our resultant amplitude \(A' = A\sqrt{2}\), we substitute this into the energy formula:
\[ E = \frac{1}{2} m \omega^2 (A\sqrt{2})^2 \]
- Simplifying this gives:
\[ E = \frac{1}{2} m \omega^2 (2A^2) = m \omega^2 A^2 \]

5. **Final Answer**:
- Thus, the total mechanical energy of the particle is:
\[ E = m \omega^2 A^2 \]