Elastic Potential Energy Calculator (U = ½kx²)
Solve the elastic potential energy formula U = ½kx² for energy, spring constant, or displacement. Enter any two values and get the third with steps.
The elastic potential energy formula
Elastic potential energy is the energy stored in a spring (or any elastic material) when it is stretched or compressed:
- U — elastic potential energy, in joules (J)
- k — spring constant, in newtons per metre (N/m) — always positive
- x — displacement from the spring's natural length, in metres (m)
Enter any two values and leave the third blank; the calculator solves for it and shows the substitution step.
Worked example
A spring with a spring constant of 200 N/m is stretched 0.1 m. How much energy is stored?
- Formula: U = ½kx²
- Substitute: U = ½ × 200 × 0.1²
- Result: U = 1 J
Solving in reverse
The same formula rearranges to find the spring constant or displacement when energy is known:
- Spring constant: k = 2U / x² — enter U and x, leave k blank.
- Displacement: x = √(2U / k) — enter U and k, leave x blank (a square root, so the result is always the non-negative magnitude).
For example, 1 J stored at 0.1 m displacement means k = 2 × 1 / 0.01 = 200 N/m. And 1 J stored in a 200 N/m spring means x = √(2 × 1 / 200) = 0.1 m.
Common mix-ups
- Forgetting the ½. U = kx² (without the one-half) is a common slip that doubles the answer.
- Forgetting to square x. Elastic potential energy is proportional to displacement squared, not displacement itself.
- Entering a negative energy when solving for displacement. Because displacement is found through a square root, a negative U has no real solution.
Where this shows up
Elastic potential energy is derived directly from Hooke's law (F = kx) — the ½kx² formula comes from integrating that linearly increasing spring force over the stretch distance. Like gravitational potential energy, it is a stored form of energy that converts into kinetic energy when released (think of a released slingshot), and the energy stored equals the work done in stretching the spring.
Frequently asked questions
- What is the formula for elastic potential energy?
- Elastic potential energy stored in a stretched or compressed spring is U = ½kx², where U is energy in joules (J), k is the spring constant in newtons per metre (N/m), and x is the displacement from the spring's natural length, in metres (m).
- How do I find the displacement from stored energy?
- Rearrange the formula to x = √(2U / k). Enter the energy and spring constant, leave displacement blank, and the calculator solves it — including the square root — for you.
- Why does elastic potential energy use x² instead of just x?
- Because the spring force grows linearly as you stretch it (F = kx), the work needed to stretch it further also grows — integrating that increasing force over distance gives the ½kx² result, the area of a triangle on a force-vs-displacement graph.
- Can elastic potential energy be negative?
- No. Since x is squared and k is always positive, U = ½kx² can never be negative — energy stored in a spring is always zero or positive, regardless of whether the spring is stretched or compressed.
- What happens if the numbers I enter give no real solution?
- Solving for displacement needs a square root of 2U/k. If you enter a negative energy, that quantity is negative and has no real square root, so the calculator reports that no real solution exists.
- Why must the spring constant be positive?
- The spring constant is a physical stiffness — it cannot be zero (no spring at all) or negative, which has no physical meaning for a real spring.
- What units does this calculator use?
- SI units — joules (J) for energy, newtons per metre (N/m) for the spring constant, and metres (m) for displacement. Convert other units before entering them.