Elastic energy is the energy stored in an elastically deformed body, such as a compressed or stretched spring. It equals Ep = ½ k x², where k is the stiffness and x the deformation, and it is released when the body returns to its shape.
Discover FizziQ
How to measure it in class
Elastic energy is measured indirectly, in two stages: first determine the spring constant k, then verify that the stored energy is indeed released as kinetic energy or as height.
Steps:
- Hang known masses from the spring and measure the extension x for each one: the slope of F = f(x) gives k
- Check that the line stays straight: as soon as it curves, you are leaving the elastic domain and ½kx² no longer applies
- Compress or stretch the spring by a value x measured with a ruler, then release a projectile of known mass m
- Film the launch and track it in FizziQ’s Kinematics module to obtain the initial speed v
- Compare ½kx² and ½mv², or measure the height reached and compare ½kx² with mgh
- Repeat for three values of x, chosen as large as possible since x appears squared, and check that the energy indeed varies as x² and not as x
Scientific activities on this topic
- Pole vault: energy balance - follow the kinetic energy of the run-up stored in the flexing of the pole, then released as height
- Spring constant: k = 4π²m/T²
- Mass-spring oscillator: period
- Damping: three regimes
Learn more
A natural origin
Like any potential energy, elastic energy is defined up to a constant. But, unlike gravitational energy, it has here a natural origin: the natural length of the spring, that is, the undeformed state. This is the universally adopted choice, and it makes elastic Ep always positive, since it varies as x².
Where the factor ½ comes from
Elastic energy is not a postulate: it can be calculated. The restoring force of a spring is F = k x (Hooke’s law, covered in detail in the Hooke’s Law entry). Since this force is not constant - it grows proportionally to the extension - you cannot write W = F × x. You must take the average force over the path, which is kx/2 since F grows linearly from 0 to kx. Hence W = (kx/2) × x = ½kx².
Geometrically, it is the area of the triangle under the line F = kx between 0 and x. This graphical reasoning is accessible from the first years of high school and is worth more than a formula learned by heart: it also explains why the energy varies as x² and not as x.
Only within the elastic domain
The formula ½kx² assumes that Hooke’s law applies, so that the deformation is reversible and proportional to the force. Beyond the elastic limit, the material deforms plastically: it no longer returns to its shape, part of the energy has gone into reorganizing the matter and is lost as heat. An overstretched spring stays elongated, a bent paperclip does not straighten up. The F = f(x) curve then bends, and the stored energy is no longer the area of a triangle but that of the actual surface under the curve.
The hysteresis cycle
Even within the elastic domain, no real material returns 100% of what it has received. By plotting F = f(x) during loading and then unloading, you get two slightly different curves: the area between them is the energy dissipated as heat, called hysteresis. It is very small for a steel spring (a few percent) and large for a rubber - hence the heating of a rolling tire, and the fact that a tennis ball never rises back to its release height.
Orders of magnitude: a mediocre storage medium
A spring stores little energy for its mass. A good steel spring tops out at around 0.3 kJ·kg⁻¹, compared with about 500 kJ·kg⁻¹ for a lithium-ion battery and 45,000 kJ·kg⁻¹ for gasoline. This is why springs are used to store energy briefly and release it quickly (shock absorbers, clockwork mechanisms, bows), never to store a large amount. Tendons, at about 2 kJ·kg⁻¹, do better than steel - which explains the role of the Achilles tendon in the economy of running.
In the curriculum
Elastic energy appears in the first year of the French high school specialty program in mechanical energy balances, and in the final year in the study of the mass-spring oscillator. In the latter case, Ek and elastic Ep are exchanged twice per period, their sum remaining constant in the absence of damping: it is the exact analogue of the swinging pendulum, with ½kx² in place of mgh.
Formula
Elastic energy stored in a spring, origin taken at the natural length:
Ep = ½ × k × x²
The restoring force from which it derives (Hooke’s law):
F = −k × x
Balance for a horizontal mass-spring oscillator without friction:
Em = ½ m v² + ½ k x² = constant
Speed of a projectile launched by a spring compressed by x:
v = x × √(k/m)
where:
- Ep: elastic potential energy (J)
- k: spring constant (N·m⁻¹)
- x: extension or compression relative to the natural length (m)
- F: restoring force (N), the minus sign indicating that it opposes the deformation
- m: mass (kg)
- v: speed (m·s⁻¹)
- Em: mechanical energy (J)
Application examples
- A spring of stiffness k = 200 N·m⁻¹ compressed by 5.0 cm stores ½ × 200 × 0.050² = 0.25 J. Compressed by 10 cm, it stores 1.0 J: doubling the compression quadruples the energy.
- This same spring, released against a 20 g marble, gives it v = 0.05 × √(200/0.020) = 5.0 m·s⁻¹ in the ideal case.
- A vaulting pole absorbs about 2 kJ during the plant phase and returns nearly 90% of it: it is this high efficiency, specific to composite materials, that made it possible to gain nearly one meter on the records compared with bamboo and then steel poles.
- A competition bow at 20 kgf drawn 70 cm stores on the order of 60 J and transfers about 75 to 80% to the arrow, the rest going into the limbs and the string.
- The Achilles tendon returns about 90% of the energy it absorbs at each stride, which reduces the energy cost of running by about one third.
FAQ
Q: Why ½kx² and not kx²? A: Because the restoring force is not constant: it grows from 0 to kx during the deformation. The work is calculated with the average force, kx/2, hence the factor ½. It is the area of the triangle under the line F = kx.
Q: Can elastic energy be negative? A: No. It is written ½kx² with k > 0 and x² ≥ 0: it is always positive or zero, whatever the direction of the deformation. Stretching or compressing by the same amount stores exactly the same energy.
Q: Does a spring return all the energy it was given? A: Almost, but never all of it. A small fraction is dissipated as heat by internal hysteresis - a few percent for steel, much more for rubber. And if you exceed the elastic limit, a significant part goes into permanently deforming the material and is never returned.
Q: Where does the energy of a rubber band that is overstretched and stays deformed go? A: Into heat and reorganization of the matter. You can feel it: stretch a large rubber band quickly against your lip, it warms up noticeably. This is the signature of plastic work, which is irreversible.
Q: What is the difference between elastic energy and gravitational energy? A: They are two forms of potential energy, associated with two different conservative forces. Gravitational energy varies linearly with height (mgh), elastic energy varies as the square of the deformation (½kx²). The two combine in the mechanical energy balance, for example in a pole vault where one is transformed into the other.
Related concepts
Hooke’s Law - Potential Energy - Mechanical Energy - Kinetic Energy - Conservation of Energy - Harmonic Oscillator - Elastic Collision - Resonance Frequency