Potential Energy of Spring Formula: U = ½kx², Unit, Examples

The potential energy of spring formula is $U = \frac{1}{2}kx^2$ , where k is the spring constant and x is how far the spring is stretched or compressed from its natural length. Pull a spring of stiffness 250 N/m out by 0.20 m and it holds ½ × 250 × 0.20² = 5.0 J.

The energy is stored in the stretched or squashed coils and comes back out when the spring returns to its natural length. Toy launchers, car suspension and a pogo stick all rely on it. The formula uses Hooke's law, so it holds only up to the spring's elastic limit. It sits beside gravitational energy among the energy formulas of mechanics.

Quick facts

Formula
$U = \frac{1}{2}kx^2$
Using force
$U = \frac{1}{2}Fx = \frac{F^2}{2k}$
SI unit
joule (J)
Dimensional formula
[M¹ L² T⁻²]
Unit of k
N/m, dimensions [M¹ T⁻²]
Sign
Always zero or positive, for stretch or squash
Condition
Within the elastic limit, where Hooke's law holds

Potential energy of spring formula: the symbols

Terms in U = ½kx²
Symbol	What it is	SI unit
U	elastic potential energy stored	J
k	stiffness: newtons required for each metre	N/m
x	how far the length differs from relaxed	m

A stiff spring has a large k. The extension x is measured from the natural length, not from wherever the spring happens to start in the question. Because x is squared, a compression of 3 cm stores the same energy as a stretch of 3 cm.

Derivation from the force against extension graph

By Hooke's law the force needed to hold a spring at extension x is $F = kx$ (the spring itself pulls back with $-kx$ ). The force grows as you stretch, so you cannot just multiply force by distance.

Plot F against x. The points fall on a line from the origin whose gradient is k.
Work done is the area under this line from 0 to x.
The area is a triangle: $W = \frac{1}{2}\times x \times kx = \frac{1}{2}kx^2$ .
This work is stored, so $U = \frac{1}{2}kx^2$ .

With calculus: $W = \int_0^x kx'\,dx' = \frac{1}{2}kx^2$ . Stretching from $x_1$ to $x_2$ takes $\frac{1}{2}k(x_2^2 - x_1^2)$ , which is not the same as $\frac{1}{2}k(x_2 - x_1)^2$ .

Worked examples

Example 1: a door spring

A door closer spring of constant 250 N/m is pulled 20 cm beyond its relaxed length. How much energy does it store?

Convert: x = 20 cm = 0.20 m.

$U = \frac{1}{2}(250)(0.20)^2 = 125 \times 0.040 = 5.0\ \text{J}$

Answer: 5.0 J.

Example 2: a spring launcher (exam style)

A toy launcher has a spring of constant 1.5 kN/m compressed by 4.0 cm. It fires a 30 g ball horizontally on a frictionless track. Find the launch speed.

Convert: k = 1500 N/m, x = 0.040 m, m = 0.030 kg.

$U = \frac{1}{2}(1500)(0.040)^2 = 750 \times 0.0016 = 1.2\ \text{J}$

All of this becomes kinetic energy: $\frac{1}{2}(0.030)v^2 = 1.2$ , so $v = \sqrt{\frac{2(1.2)}{0.030}} = \sqrt{80} = 8.9\ \text{m/s}$ .

Answer: about 8.9 m/s.

Convert kN/m to N/m and centimetres to metres before squaring. A compression of 4 cm entered as 4 makes the energy 10 000 times too large.

Units, graphs and special cases

Units: (N/m) × m² = N m = J. Dimensions: k has [M¹ T⁻²], and multiplying by [L²] gives [M¹ L² T⁻²]. A graph of U against x is a parabola with its lowest point at x = 0, which is why the natural length is the stable equilibrium position.

For a mass on a spring in simple harmonic motion with amplitude A, the total energy is $\frac{1}{2}kA^2$ . It moves between spring energy at the ends and kinetic energy in the middle. For two springs in series the combined constant is smaller than either; in parallel the constants add. A vertical spring holding a hanging mass is stretched by mg/k at equilibrium.

Frequently asked questions

What is the formula for potential energy of a spring?

Use U = ½kx². Put the stiffness k in N/m and the stretch or squash x in metres, and U comes out in joules.

Is spring potential energy ever negative?

No. Because x is squared, the energy is zero at the natural length and positive for any stretch or compression.

What is the unit of spring constant?

It is measured in N/m. Dividing force, [M¹ L¹ T⁻²], by a length leaves the dimensions [M¹ T⁻²].

How does stored energy change if the extension doubles?

It becomes four times as large, since it depends on x². Doubling k at the same extension only doubles the energy.

What is the difference between spring force and spring potential energy?

Spring force, F = −kx, is a vector in newtons that pulls the spring back. Spring potential energy, ½kx², is a scalar in joules that measures the work stored.

Potential energy of spring formula: the symbols

Derivation from the force against extension graph

Worked examples

Units, graphs and special cases

Common mistakes and related pages

Frequently asked questions