The total mechanical energy of a spring mass system in simple harmonic motion is `E=1/2momega^2 A^2`. Suppose the oscillating particle is replaced by another particle of double the mass while the amplitude A remains the same. The new mechanical energy will

Options

A. become `2E`
B. becoem `E/2`
C. becoem `sqrt2E`
D. remain E

Answer

To solve the problem, we need to analyze how the total mechanical energy of a spring-mass system in simple harmonic motion changes when the mass of the oscillating particle is doubled while keeping the amplitude constant.

### Step-by-Step Solution:

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

2. **Identify the new mass**:
If the original mass is \( m \), the new mass when replaced by another particle of double the mass will be:
\[ m' = 2m \]

3. **Determine the angular frequency (\( \omega \))**:
The angular frequency \( \omega \) is related to the spring constant \( k \) and the mass \( m \) by the formula:
\[ \omega = \sqrt{\frac{k}{m}} \]
When the mass is doubled, the new angular frequency \( \omega' \) becomes:
\[ \omega' = \sqrt{\frac{k}{m'}} = \sqrt{\frac{k}{2m}} = \frac{1}{\sqrt{2}} \sqrt{\frac{k}{m}} = \frac{\omega}{\sqrt{2}} \]

4. **Calculate the new mechanical energy (\( E' \))**:
Substitute the new mass \( m' = 2m \) and the new angular frequency \( \omega' = \frac{\omega}{\sqrt{2}} \) into the energy formula:
\[ E' = \frac{1}{2} m' (\omega')^2 A^2 \]
\[ E' = \frac{1}{2} (2m) \left(\frac{\omega}{\sqrt{2}}\right)^2 A^2 \]
\[ E' = \frac{1}{2} (2m) \left(\frac{\omega^2}{2}\right) A^2 \]
\[ E' = \frac{1}{2} m \omega^2 A^2 \]

5. **Compare the new energy with the original energy**:
We can see that:
\[ E' = E \]
Thus, the new mechanical energy remains the same as the original mechanical energy when the mass is doubled and the amplitude remains constant.

### Final Answer:
The new mechanical energy will be equal to the original mechanical energy \( E \).