Calculation
E = ½ · k · x²
Energy grows with deflection squared; doubling deflection stores four times as much.
E = ½kx²
Spring Energy Calculator
Energy grows with deflection squared; doubling deflection stores four times as much.
MINTSI
02
Result
Select a target and calculate.
Understand the inputs
- Spring energy E — The elastic energy stored in the spring, released again when the load is removed.
- Spring rate k — The spring rate. Source: the spring's datasheet, or the slope of the force-deflection curve (see the spring-force calculator).
- Deflection x — The deflection relative to the spring's unloaded length.
Example
½ · 1,000 N/m · 0.1² m² = 5 J.
Assumptions and limits
Linear spring without damping or permanent deformation.
Understand Spring Energy Calculator
This calculator finds the missing quantity among spring energy E, spring rate k and deflection x from the other two, for a linear spring. Using bows and catapults, it shows why stored energy grows with the square of deflection — not linearly like spring force itself.
What does this quantity describe?
The elastic energy stored in a linear spring is E = ½·k·x². Unlike spring force, which grows linearly with deflection, stored energy grows quadratically — doubling deflection quadruples stored energy.
Drawing a bow stores potential energy in its bent limbs, which converts to the arrow's kinetic energy on release. Since stored energy grows with the square of the draw length, even a moderate increase in draw length produces a noticeably larger energy gain — one reason bow design and draw length strongly affect arrow speed.
Formula and variables
E = ½ · k · x²
E = ½ · k · x²x = √(2·E/k)k = 2·E/x²
| Symbol / input | Meaning |
|---|---|
| Spring energy E | The elastic energy stored in the spring, released again when the load is removed. |
| Spring rate k | The spring rate. Source: the spring's datasheet, or the slope of the force-deflection curve (see the spring-force calculator). |
| Deflection x | The deflection relative to the spring's unloaded length. |
Choose the inputs correctly
Enter two of the three quantities. Spring energy requires rate and deflection, deflection requires energy and rate (via a square root), rate requires energy and deflection. The formula applies only within the spring's linear-elastic range.
How to use the calculator
Select the target quantity and enter the two known values with units. Since x enters squared, solving for x loses the sign of the original deflection direction.
Worked example
Given k = 1,000 N/m and x = 0.1 m, substitution gives E = 0.5 · 1,000 N/m · 0.1² m² = 5 J.
Understand the result and units
Stored energy of 5 J shows how much work was done compressing the spring to this deflection, and how much can be released again on relaxation. At double the deflection (0.2 m instead of 0.1 m), this energy would quadruple to 20 J, not merely double.
The SI unit is the joule (J = N·m). Spring rate is given in N/m, deflection in metres; if deflection is given in mm, convert to SI units before substituting, since x enters squared.
Why does spring energy grow with the square of deflection?
The work done compressing a spring is the area under the force-deflection curve. Since force F = k·x itself grows linearly with deflection, this area is a triangle with legs x and k·x; its area is ½·x·(k·x) = ½·k·x². Every additional bit of deflection therefore requires not just more distance but also an already-higher force, so the two effects multiply and stored energy grows quadratically rather than linearly.
Typical applications
Spring energy calculations are used to design spring accumulators, catapults and tensioning devices, estimate absorbed impact energy in suspension design, and for safety assessments of preloaded springs.
Assumptions, limits and common mistakes
The formula applies only within the spring's linear-elastic range, where F = k·x holds. Beyond the proportional limit or at solid height, actual stored energy deviates from this simple quadratic relationship, since the force-deflection curve itself is no longer linear there.
Common mistake: Do not confuse the quadratic relationship with a linear one: double deflection means quadruple energy, not double. Also, consistently convert deflection to metres before substituting, since a unit error in x enters the result squared.
Frequently asked questions
Why does energy quadruple at double deflection when force only doubles?
Because energy is the area under the linearly rising force-deflection curve. That area grows with the square of deflection, while force itself grows only linearly.
How are bow draw and arrow speed related?
Energy stored on the draw, E = ½·k·x², converts approximately fully into the arrow's kinetic energy on release; a longer draw length raises stored energy quadratically and so noticeably increases achievable speed.
Does the formula still hold at very large deflections near solid height?
No, beyond the proportional limit or once spring coils touch, the real force-deflection curve deviates from the linear model, so the simple quadratic energy formula no longer holds exactly either.