The potential energy of a spring is the minimum when it is

Options

A. compressed
B. extended
C. at its natural length
D. at its natural length but is kept at a height above the ground

Answer

To determine when the potential energy of a spring is at its minimum, we can follow these steps:

### Step 1: Understand the formula for potential energy of a spring
The potential energy (PE) stored in a spring is given by the formula:
\[ PE = \frac{1}{2} k x^2 \]
where:
- \( k \) is the spring constant (a measure of the spring's stiffness),
- \( x \) is the amount of extension or compression from the spring's natural length.

### Step 2: Analyze the conditions for minimum potential energy
From the formula, we can see that the potential energy depends on the square of \( x \). The value of \( x \) can be either positive (extension) or negative (compression).

### Step 3: Determine when \( x \) is at its minimum
The potential energy will be minimized when \( x \) is at its minimum value. Since \( x \) represents the amount of extension or compression, the minimum value of \( x \) is:
\[ x = 0 \]
This means that the spring is neither compressed nor extended.

### Step 4: Relate \( x = 0 \) to the spring's natural length
When \( x = 0 \), the spring is at its natural length. This is the position where the spring is in its equilibrium state, and no external forces are acting on it to compress or extend it.

### Step 5: Conclusion
Thus, the potential energy of a spring is at its minimum when the spring is at its natural length.

### Final Answer
The potential energy of a spring is the minimum when it is at its natural length (Option 3).

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