The elastic potential energy of a spring

Options

A. Increases only when it is stretched
B. Decreases only when it is stretched
C. Decreases only when it is compressed
D. Increases whether stretched or compressed

Answer

To find the elastic potential energy of a spring, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Spring Force**:
The force exerted by a spring is given by Hooke's Law, which states that the force \( F \) exerted by a spring is proportional to the displacement \( x \) from its equilibrium position. This can be expressed as:
\[ F = -kx \]
where \( k \) is the spring constant.

**Hint**: Remember that the negative sign indicates that the force exerted by the spring is in the opposite direction to the displacement.

2. **Work Done on the Spring**:
When we stretch or compress the spring, we do work against the spring force. The work done \( W \) in stretching or compressing the spring from an initial position \( x = 0 \) to a position \( x \) is given by:
\[ W = \int_0^x F \, dx = \int_0^x kx \, dx \]

**Hint**: The work done is the integral of the force over the displacement.

3. **Calculating the Work Done**:
Evaluating the integral:
\[ W = \int_0^x kx \, dx = \left[ \frac{1}{2} kx^2 \right]_0^x = \frac{1}{2} kx^2 \]
This work done is stored as elastic potential energy \( U \) in the spring.

**Hint**: The limits of integration help you find the total work done from the initial to the final position.

4. **Elastic Potential Energy**:
The elastic potential energy \( U \) stored in the spring when it is either stretched or compressed by a distance \( x \) is given by:
\[ U = \frac{1}{2} kx^2 \]
This formula shows that the potential energy increases as the displacement \( x \) increases, whether the spring is stretched or compressed.

**Hint**: The formula indicates that potential energy is always positive when \( k \) and \( x \) are positive.

### Conclusion:
The elastic potential energy of a spring is given by the formula:
\[ U = \frac{1}{2} kx^2 \]
This energy increases when the spring is either stretched or compressed.