Assuming that potential energy of spring is zero when it is stretched by `( x_(0))/( 2)`, its potential energy when it is compressed by `x_(0)` is

Options

A. `( 3)/(8) kx_(0)^(2)`
B. `- ( 3)/( 4) kx_(0)^(2)`
C. `( 3)/( 4) kx_(0)^(2)`
D. `(1)/( 8) kx_(0)^(2)`

Answer

To solve the problem, we need to find the potential energy of a spring when it is compressed by \( x_0 \), given that the potential energy is zero when it is stretched by \( \frac{x_0}{2} \).

### Step-by-Step Solution:

1. **Understanding the Potential Energy of a Spring:**
The potential energy (PE) stored in a spring is given by the formula:
\[ U = \frac{1}{2} k x^2 \]
where \( k \) is the spring constant and \( x \) is the displacement from the mean position.

2. **Condition 1 - Potential Energy at \( \frac{x_0}{2} \):**
We are told that the potential energy is zero when the spring is stretched by \( \frac{x_0}{2} \). Let's denote this position as point A.
\[ U_A = 0 \quad \text{when} \quad x = \frac{x_0}{2} \]

3. **Setting the Mean Position:**
Since the potential energy is defined to be zero at point A, we can assume that point A is the new mean position for our calculations.

4. **Condition 2 - Potential Energy when Compressed by \( x_0 \):**
Now, we need to find the potential energy when the spring is compressed by \( x_0 \). Let's denote this position as point B.
\[ U_B = \frac{1}{2} k x^2 \quad \text{where} \quad x = -x_0 \]
(negative because it is compressed).

5. **Calculating Potential Energy at Point B:**
Substituting \( x = -x_0 \) into the potential energy formula:
\[ U_B = \frac{1}{2} k (-x_0)^2 = \frac{1}{2} k x_0^2 \]

6. **Finding the Change in Potential Energy:**
Since we defined the potential energy at point A (stretched by \( \frac{x_0}{2} \)) to be zero, we need to calculate the potential energy at point A:
\[ U_A = \frac{1}{2} k \left(\frac{x_0}{2}\right)^2 = \frac{1}{2} k \frac{x_0^2}{4} = \frac{1}{8} k x_0^2 \]

7. **Final Calculation of Potential Energy at Point B:**
The potential energy at point B (compressed by \( x_0 \)) is:
\[ U_B = \frac{1}{2} k x_0^2 \]
Thus, the potential energy when compressed by \( x_0 \) is:
\[ U_B - U_A = \frac{1}{2} k x_0^2 - \frac{1}{8} k x_0^2 \]
Finding a common denominator:
\[ U_B - U_A = \frac{4}{8} k x_0^2 - \frac{1}{8} k x_0^2 = \frac{3}{8} k x_0^2 \]

### Conclusion:
The potential energy when the spring is compressed by \( x_0 \) is:
\[ \frac{3}{8} k x_0^2 \]