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 the total mechanical energy of a spring-mass system in simple harmonic motion and how it changes when the mass is altered while keeping the amplitude constant.

### Step-by-Step Solution:

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

2. **Identify the Changes in the System:**
The problem states that the mass of the oscillating particle is doubled (from \( m \) to \( 2m \)), while the amplitude \( A \) remains the same.

3. **Substitute the New Mass into the Energy Formula:**
If we replace \( m \) with \( 2m \), the new mechanical energy \( E' \) can be expressed as:
\[ E' = \frac{1}{2} (2m) \omega'^2 A^2 \]
where \( \omega' \) is the new angular frequency after the mass change.

4. **Consider the Conservation of Energy:**
Since no external work is done on the system, the total mechanical energy must remain constant. Therefore, we can set the new energy equal to the original energy:
\[ E' = E \]

5. **Set Up the Equation:**
Substitute the expressions for \( E' \) and \( E \):
\[ \frac{1}{2} (2m) \omega'^2 A^2 = \frac{1}{2} m \omega^2 A^2 \]

6. **Simplify the Equation:**
Cancel out the common terms \( \frac{1}{2} A^2 \):
\[ 2m \omega'^2 = m \omega^2 \]

7. **Solve for the New Angular Frequency:**
Divide both sides by \( m \) (assuming \( m \neq 0 \)):
\[ 2 \omega'^2 = \omega^2 \]
Now, divide both sides by 2:
\[ \omega'^2 = \frac{\omega^2}{2} \]

8. **Take the Square Root:**
Taking the square root gives:
\[ \omega' = \frac{\omega}{\sqrt{2}} \]

9. **Conclusion about the Mechanical Energy:**
Since the mechanical energy \( E' \) remains equal to \( E \) despite the change in mass, we conclude:
\[ E' = E \]

### Final Answer:
The new mechanical energy will remain the same, \( E' = E \).

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