Elastic Forces and Hooke’s Law

Why It Matters

Elastic force appears in force, energy, and oscillation problems. The same spring model is used for equilibrium, elastic potential energy, and spring-mass SHM.

Definition

Hooke’s law states that, within the proportional limit, the extension or compression of a spring is directly proportional to the applied force. The spring force is a restoring force: it acts to return the spring to its natural length.

Key Representations

Magnitude form:

$$F=kx$$

where $k$ is the spring constant and $x$ is the extension or compression from the natural length.

Elastic potential energy is:

$$U_{\text{elastic}}=\frac{1}{2}k{x}^2$$

Force Direction

The spring force acts opposite to the displacement from equilibrium. If the spring is stretched, it pulls back; if compressed, it pushes back.

Graph Interpretation

For a Hooke’s law spring, the force-extension graph is a straight line through the origin. The gradient is the spring constant $k$, and the area under the graph gives elastic potential energy stored.

Common Exam Points

• Hooke’s law applies only within the proportional limit.
• The spring force is restoring, so direction matters.
• The extension $x$ is measured from natural length, not necessarily from an arbitrary origin.
• Elastic potential energy links this topic to Potential Energy and Conservative Forces.

Links

• Related: forces
• Related: work energy and power
• Related: oscillations
• Related: force types and interactions
• Related: potential energy and conservative forces