The potential energy of a harmonic oscillation when is half way to its and end point is (where `E` it’s the total energy)

Options

A. `(1)/(4)E`
B. `(1)/(2)E`
C. `(2)/(3)E`
D. `(1)/(8)E`

Answer

To solve the problem of finding the potential energy of a harmonic oscillator when it is halfway to its endpoint, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the System**:
- A particle is performing simple harmonic motion (SHM) with amplitude \( A \). The total energy \( E \) of the oscillator is given by the formula:
\[ E = \frac{1}{2} k A^2 \]
where \( k \) is the spring constant.

2. **Determine the Position**:
- When the particle is halfway to its endpoint, it is at position \( x = \frac{A}{2} \).

3. **Calculate Potential Energy**:
- The potential energy \( U \) of a harmonic oscillator at position \( x \) is given by:
\[ U = \frac{1}{2} k x^2 \]
- Substituting \( x = \frac{A}{2} \):
\[ U = \frac{1}{2} k \left(\frac{A}{2}\right)^2 = \frac{1}{2} k \cdot \frac{A^2}{4} = \frac{1}{8} k A^2 \]

4. **Relate Potential Energy to Total Energy**:
- We know the total energy \( E \) is:
\[ E = \frac{1}{2} k A^2 \]
- Therefore, we can express \( k A^2 \) in terms of \( E \):
\[ k A^2 = 2E \]

5. **Substitute Back to Find \( U \)**:
- Now, substitute \( k A^2 \) into the potential energy equation:
\[ U = \frac{1}{8} k A^2 = \frac{1}{8} \cdot 2E = \frac{E}{4} \]

6. **Final Answer**:
- Thus, the potential energy of the harmonic oscillator when it is halfway to its endpoint is:
\[ U = \frac{E}{4} \]