Elastic Potential Energy

Definition of Elastic Potential Energy

Elastic potential energy is the energy stored in an object when it is stretched or compressed. This happens when the object undergoes elastic deformation, meaning it changes shape temporarily but can return to its original shape once the force is removed.

Examples include stretching a spring, pulling a rubber band, or compressing a spring in a pen. The energy is stored because work is done to change the shape of the object.

Hooke's Law and Elasticity

Hooke's Law describes the relationship between the force applied to stretch or compress an elastic object and the extension (change in length) produced:

Force applied (F) is directly proportional to extension (x), as long as the elastic limit is not exceeded.

Mathematically, this is:

$$F = kx$$

where:

• F is the force applied in newtons (N)
• k is the spring constant in newtons per metre (N/m)
• x is the extension or compression in metres (m)

The limit of proportionality is the point beyond which the force is no longer proportional to the extension. If the object is stretched beyond this point, it may not return to its original shape.

The elastic limit is the maximum extent to which an object can be stretched or compressed and still return to its original shape. Beyond this, permanent deformation occurs.

Note that the limit of proportionality marks the end of the linear relationship between force and extension, while the elastic limit is the maximum reversible deformation. These points may be close but are not always the same.

Calculating Elastic Potential Energy

The elastic potential energy (E) stored in a stretched or compressed spring (or elastic object) can be calculated using the formula:

$$E = \frac{1}{2} k x^2$$

where:

• E is the elastic potential energy in joules (J)
• k is the spring constant in newtons per metre (N/m)
• x is the extension or compression from the natural length in metres (m)

The extension $x$ is always measured from the spring's natural (unstretched) length.

For example, if a spring with a spring constant of 200 N/m is stretched by 0.05 m, the elastic potential energy stored is:

$$E = \frac{1}{2} \times 200 \times (0.05)^2 = \frac{1}{2} \times 200 \times 0.0025 = 0.25 \text{ J}$$

For instance, stretching a spring with $k = 100$ N/m by 0.02 m stores $E = 0.5 \times 100 \times 0.02^2 = 0.02$ J of elastic potential energy.

Energy Transfer in Elastic Materials

When work is done to stretch or compress an elastic object, energy is transferred to the object and stored as elastic potential energy.

If the object is not stretched beyond its elastic limit, the energy is conserved. This means when the force is removed, the object returns to its original shape and the stored elastic potential energy is released, often converting back into other forms of energy such as kinetic energy.

For example, when you stretch a bow and release it, the stored elastic potential energy in the bow is transferred to the arrow as kinetic energy.

If the object is stretched beyond the elastic limit, some energy is lost as heat or causes permanent deformation, so not all the energy is recovered.