Two equal masses are attached to the two end of a spring of force constant `k` the masses are pulled out symmetrically to stretch the spring by a length `2x_(0)` over its natural length. Find the work done by the spring on each mass.

Explanation

To find the work done by the spring on each mass when two equal masses are attached to the ends of a spring and the spring is stretched by a length \(2x_0\), we can follow these steps:

### Step 1: Understand the Initial Condition
Initially, the spring is at its natural length, and the masses are at rest. Therefore, the initial potential energy stored in the spring is zero.

**Hint:** Remember that potential energy in a spring is given by \( \frac{1}{2} k x^2 \), where \( x \) is the extension from the natural length.

### Step 2: Determine the Final Condition
When the masses are pulled symmetrically, the spring is stretched by a length of \(2x_0\). Thus, the extension \(x\) of the spring is \(2x_0\).

**Hint:** The total extension of the spring is the distance each mass moves away from the natural length of the spring.

### Step 3: Calculate the Potential Energy Stored in the Spring
The potential energy \(U\) stored in the spring when it is stretched by \(2x_0\) is given by the formula:
\[ U = \frac{1}{2} k x^2 \]
Substituting \(x = 2x_0\):
\[ U = \frac{1}{2} k (2x_0)^2 = \frac{1}{2} k \cdot 4x_0^2 = 2 k x_0^2 \]

**Hint:** Make sure to square the total extension when substituting into the potential energy formula.

### Step 4: Determine the Work Done by the Spring
The work done by the spring on the masses can be calculated as the change in potential energy. Since the initial potential energy is zero, the work done \(W\) by the spring is:
\[ W = U_{\text{initial}} - U_{\text{final}} = 0 - 2 k x_0^2 = -2 k x_0^2 \]

**Hint:** The work done by the spring is negative because the spring does work against the direction of the applied force.

### Step 5: Calculate Work Done on Each Mass
Since there are two equal masses and the work done by the spring is distributed equally, the work done by the spring on each mass is:
\[ W_{\text{each mass}} = \frac{-2 k x_0^2}{2} = -k x_0^2 \]

**Hint:** Remember that the total work done is shared equally between the two masses.

### Final Answer
The work done by the spring on each mass is:
\[ \boxed{-k x_0^2} \]