# Elastic Potential Energy

## If a force results in only deformation, with no thermal, sound, or kinetic energy, the work done is stored as elastic potential energy.

#### Key Points

• In order to produce a deformation, work must be done.

• The potential energy stored in a spring is given by  $PE_{el} = \frac{1}{2}k x^2$, where k is the spring constant and x is the displacement.

• Deformation can also be converted into thermal energy or cause an object to begin oscillating.

#### Terms

• A transformation; change of shape.

• The energy possessed by an object because of its motion, equal to one half the mass of the body times the square of its velocity.

• Moving in a repeated back-and-forth motion.

#### Examples

• A mouse trap stores elastic potential energy by twisting a piece of metal; this energy is released when the mouse steps into it.

#### Figures

1. ##### Applied force versus deformation

A graph of applied force versus distance for the deformation of a system that can be described by Hooke’s law is displayed. The work done on the system equals the area under the graph or the area of the triangle, which is half its base multiplied by its height, or W=(1/2)kx^2.

## Elastic Potential Energy

In order to produce a deformation, work must be done. That is, a force must be exerted through a distance, whether you pluck a guitar string or compress a car spring. If the only result is deformation and no work goes into thermal, sound, or kinetic energy, then all the work is initially stored in the deformed object as some form of potential energy. Elastic energy is the potential mechanical energy stored in the configuration of a material or physical system when work is performed to distort its volume or shape. For example, the potential energy PEel stored in a spring is

$PE_{el} = \frac{1}{2} k x^2$

where k is the elastic constant and x is the displacement.

It is possible to calculate the work done in deforming a system in order to find the energy stored. This work is performed by an applied force Fapp. The applied force is exactly opposite to the restoring force (action-reaction), and so $F_{app}=kx$. A graph shows the applied force versus deformation x for a system that can be described by Hooke’s law (Figure 1). Work done on the system is force multiplied by distance, which equals the area under the curve, or $\frac{1}{2}kx^2$ (Method A in the figure). Another way to determine the work is to note that the force increases linearly from 0 to kx, so that the average force is $\frac{1}{2}kx$, the distance moved is x, and thus

$W=F_{app}d=(\frac{1}{2}kx)(x)=\frac{1}{2}kx^2$ (Method B in the figure).

Elastic energy of or within a substance is static energy of configuration. It corresponds to energy stored principally by changing the inter-atomic distances between nuclei. Thermal energy is the randomized distribution of kinetic energy within the material, resulting in statistical fluctuations of the material about the equilibrium configuration. There is some interaction, however. For example, for some solid objects, twisting, bending, and other distortions may generate thermal energy, causing the material's temperature to rise. This energy can also produce macroscopic vibrations sufficiently lacking in randomization to lead to oscillations that are merely the exchange between (elastic) potential energy within the object and the kinetic energy of motion of the object as a whole.

#### Key Term Glossary

average
The arithmetic mean.
##### Appears in these related concepts:
deformation
A transformation; change of shape.
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displacement
A vector quantity that denotes distance with a directional component.
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Displacement
The length and direction of a straight line between two objects.
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distortion
(optics) an aberration that causes magnification to change over the field of view.
##### Appears in this related concept:
elastic
Capable of stretching; particularly, capable of stretching so as to return to an original shape or size when force is released.
##### Appears in these related concepts:
elastic potential energy
The energy stored in a deformable object, such as a spring.
##### Appears in these related concepts:
energy
A quantity that denotes the ability to do work and is measured in a unit dimensioned in mass × distance²/time² (ML²/T²) or the equivalent.
##### Appears in these related concepts:
equilibrium
The state of a body at rest or in uniform motion, the resultant of all forces on which is zero.
##### Appears in these related concepts:
Equilibrium
A state of rest or balance due to the equal action of opposing forces.
##### Appears in these related concepts:
force
A physical quantity that denotes ability to push, pull, twist or accelerate a body which is measured in a unit dimensioned in mass × distance/time² (ML/T²): SI: newton (N); CGS: dyne (dyn)
##### Appears in these related concepts:
Force
A force is any influence that causes an object to undergo a certain change, either concerning its movement, direction or geometrical construction.
##### Appears in these related concepts:
kinetic
Of or relating to motion
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kinetic energy
The energy possessed by an object because of its motion, equal to one half the mass of the body times the square of its velocity.
##### Appears in these related concepts:
Kinetic Energy
The energy associated with a moving particle or object having a certain mass.
##### Appears in these related concepts:
Law
A concise description, usually in the form of a mathematical equation, used to describe a pattern in nature
##### Appears in these related concepts:
motion
A change of position with respect to time.
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oscillation
the act of oscillating or the state of being oscillated
##### Appears in these related concepts:
potential
A curve describing the situation where the difference in the potential energies of an object in two different positions depends only on those positions.
##### Appears in these related concepts:
potential energy
The energy an object has because of its position (in a gravitational or electric field) or its condition (as a stretched or compressed spring, as a chemical reactant, or by having rest mass)
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restoring force
If the system is perturbed away from the equilibrium, the restoring force will tend to bring the system back toward equilibrium. The restoring force is a function only of position of the mass or particle. It is always directed back toward the equilibrium position of the system.An example is the action of a spring. An idealized spring exerts a force that is proportional to the amount of deformation of the spring from its equilibrium length, exerted in a direction to oppose the deformation. Pulling the spring to a greater length causes it to exert a force that brings the spring back toward its equilibrium length. The amount of force can be determined by multiplying the spring constant of the spring by the amount of stretch.
##### Appears in these related concepts:
Restoring force
A variable force that gives rise to an equilibrium in a physical system. If the system is perturbed away from the equilibrium, the restoring force will tend to bring the system back toward equilibrium. The restoring force is a function only of position of the mass or particle. It is always directed back toward the equilibrium position of the system
##### Appears in these related concepts:
solid
A substance in the fundamental state of matter that retains its size and shape without need of a container (as opposed to a liquid or gas).
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static
Fixed in place; having no motion.
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thermal energy
The internal energy of a system in thermodynamic equilibrium due to its temperature.
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work
A measure of energy expended in moving an object; most commonly, force times displacement. No work is done if the object does not move.