Elastic potential energy Knowledge-Based Study

Study notes

1. What is Elastic Potential Energy?

   When an external force deforms an object—such as stretching a rubber band, compressing a mattress spring, or bending a bow—work is done on that object. If the material returns to its original shape once the applied force is removed, the deformation is described as elastic.

   The energy stored in an object as a result of reversible deformation is called elastic potential energy ($E_e$). This stored energy represents the capacity of the deformed material to perform mechanical work as it returns to its equilibrium (unstretched or uncompressed) configuration.

   Restoring Force and the Spring Constant

   For an ideal spring operating within its elastic limit, the magnitude of the restoring force $F$ exerted by the spring is directly proportional to its displacement or extension $x$ from equilibrium:

   $$F = kx$$

   • $F$: Magnitude of the restoring or applied force in newtons (N).
   • $k$: Spring constant (or stiffness) in newtons per metre (N/m). A higher value of $k$ indicates a stiffer spring.
   • $x$: Extension or compression from the natural equilibrium length in metres (m).

   Because the force varies linearly with extension from $0$ to its maximum value $kx$, the average force required to produce an extension $x$ is $F_{\text{avg}} = \frac{1}{2}kx$.

   The Elastic Potential Energy Formula

   The work done by an external force in stretching or compressing an ideal spring equals the area under the force-extension graph (a right-angled triangle with base $x$ and height $kx$). Consequently, the elastic potential energy stored is given by:

   $$E_e = \frac{1}{2}kx^2$$

   Alternatively, substituting $F = kx$ allows the energy to be expressed in terms of the maximum force and extension:

   $$E_e = \frac{1}{2}Fx$$

   Key Properties of the Relationship

   • Quadratic dependence on displacement: Because energy depends on $x^2$, doubling the extension or compression quadruples the stored elastic potential energy.
   • Symmetry between tension and compression: Because $x^2$ is always non-negative for real values of $x$, the same amount of energy is stored whether a spring is stretched by a distance $x$ or compressed by the identical distance $x$.

   Interchange with Kinetic and Gravitational Potential Energy

   In an isolated mechanical system with no dissipative forces such as friction or air resistance, total mechanical energy is conserved:

   $$E_{\text{total}} = E_k + E_p + E_e = \text{constant}$$

   Common Scenarios

   • Horizontal spring-mass oscillator: When a compressed spring releases a mass on a frictionless horizontal surface, all stored elastic potential energy is converted into kinetic energy at the equilibrium position ($\frac{1}{2}kx^2 = \frac{1}{2}mv^2$).
   • Vertical spring-mass system: As a suspended mass descends, gravitational potential energy decreases while both kinetic energy and elastic potential energy change according to position.
   • Elastic limit warning: Hooke's law and the formula $E_e = \frac{1}{2}kx^2$ apply only up to the material's limit of proportionality. Beyond the elastic limit, plastic deformation occurs, and some energy is irreversibly dissipated as thermal energy.