U = 1/2 k x²

Definition

The equation $$u = \frac{1}{2} k x^2$$ defines the elastic potential energy stored in a compressed or stretched spring. Here, 'u' represents the elastic potential energy, 'k' is the spring constant indicating the stiffness of the spring, and 'x' is the displacement from the spring's equilibrium position. This relationship illustrates how energy is stored in elastic materials when they are deformed.

5 Must Know Facts For Your Next Test

1. The formula shows that elastic potential energy increases quadratically with displacement; doubling the displacement results in four times the stored energy.
2. The spring constant 'k' varies for different materials and determines how stiff or flexible a spring will be when subjected to forces.
3. Elastic potential energy is only stored as long as the deformation does not exceed the elastic limit; beyond this limit, materials may undergo permanent deformation.
4. This equation is crucial for understanding energy conservation in systems involving springs and oscillatory motion, like in pendulums or vibrating systems.
5. Elastic potential energy plays a significant role in various real-world applications, such as in car suspensions, toys like slingshots, and even in engineering designs like bridges.

Review Questions

• How does the displacement 'x' affect the elastic potential energy according to the formula $$u = \frac{1}{2} k x^2$$?
  • In the equation $$u = \frac{1}{2} k x^2$$, displacement 'x' has a quadratic relationship with elastic potential energy 'u'. This means that as you increase 'x', the energy stored increases exponentially. For instance, if you double the displacement from equilibrium, you actually quadruple the stored energy. This illustrates how sensitive elastic potential energy is to changes in displacement.
• Discuss how Hooke's law relates to the concept of elastic potential energy and provide an example of its application.
  • Hooke's law states that the force exerted by a spring is directly proportional to its displacement from equilibrium, expressed as $$F = kx$$. This relationship helps explain how elastic potential energy accumulates when a spring is either compressed or stretched. For example, if you pull back a slingshot (which acts like a spring), you're storing elastic potential energy that can be released as kinetic energy when you let go. The formula for this stored energy is derived directly from Hooke's law.
• Evaluate the implications of exceeding a material's elastic limit on its potential energy storage capabilities and overall behavior.
  • Exceeding a material's elastic limit fundamentally alters its ability to store elastic potential energy as described by $$u = \frac{1}{2} k x^2$$. When this limit is surpassed, materials may not return to their original shape, resulting in plastic deformation or permanent changes. This means any potential energy that was stored during deformation would be lost in forms like heat or sound rather than being fully recoverable. Such behavior can lead to failure in mechanical systems where precise control over energy storage is crucial, affecting everything from simple toys to complex engineering structures.

Related terms

Spring Constant: A measure of a spring's stiffness, represented by 'k' in Hooke's law, indicating how much force is needed to stretch or compress a spring by a unit distance.

Hooke's Law: A principle stating that the force needed to extend or compress a spring is directly proportional to the distance it is stretched or compressed, usually expressed as $$F = kx$$.

Mechanical Energy: The sum of potential and kinetic energy in an object; elastic potential energy is a form of mechanical energy associated with the configuration of an elastic object.