Hooke's Law

Definition

Hooke’s Law is a principle of physics that states that the force $F$ needed to extend or compress a spring by some distance $x$ scales linearly with that distance. $F = kx$

How to read: “The force F is equal to the spring constant k times the displacement x.”
Meaning: Applied force scales linearly with displacement from equilibrium; $k$ is the spring constant (stiffness).

Why It Matters

This law is the foundation of mechanical engineering, describing how materials deform under stress and return to their original shape. It allows us to design everything from car suspensions and bridges to the sensitive sensors in our smartphones with predictable safety and performance.

Core Concepts

Linear Relationship: The force is directly proportional to the displacement.
Restoring Force: The force exerted by the spring itself is $F_s = -kx$ $F_{s} = - k x$ , acting in the opposite direction of the displacement to return the system to its equilibrium position.
- How to read: “The restoring force F s is equal to negative k times x.”
- Meaning: The spring pushes back opposite to displacement — negative sign encodes the restoring direction toward equilibrium.
Elastic Limit: The law only holds true as long as the material remains in its “elastic” region. If stretched too far, the material will permanently deform, and the relationship becomes non-linear.
Work Calculation: Because the force is variable, the work $W$ $W$ required to stretch a spring from $x=a$ $x = a$ to $x=b$ $x = b$ is found using an integral: $W = \int_a^b kx \, dx = \frac{1}{2}k(b^2 - a^2)$ $W = \int_{a}^{b} k x d x = \frac{1}{2} k (b^{2} - a^{2})$
- How to read: “The work W is equal to the integral from a to b of k x with respect to x, which is equal to one half k times the quantity b squared minus a squared.”
- Meaning: Work to stretch a spring equals the area under the linear force curve — also $\frac{1}{2}k\Delta x^2$ for extension from $a$ to $b$ .

Definition

Why It Matters

Core Concepts

Connected Concepts

Connected notes